Nonlinear Dynamics Perspective of Wolfram's New Kind of Science: Volume 2

A NONlINEAR DYNAMICS PERSPECTIVE OF HOlFRAM'S NEW HlNU OF SCIENCE Volume I1 WORLD SCIENTIFIC SERIES ON NONLINEAR SCIE...

Author: Leon O. Chua

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A NONlINEAR DYNAMICS PERSPECTIVE OF HOlFRAM'S NEW HlNU OF SCIENCE Volume I1

WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Editor: Leon O. Chua University of California, Berkeley Series A.

MONOGRAPHS AND TREATISES

Volume 38: Nonlinear Noninteger Order Circuits & Systems — An Introduction P. Arena, R. Caponetto, L. Fortuna & D. Porto Volume 39: The Chaos Avant-Garde: Memories of the Early Days of Chaos Theory Edited by Ralph Abraham & Yoshisuke Ueda Volume 40: Advanced Topics in Nonlinear Control Systems Edited by T. P. Leung & H. S. Qin Volume 41: Synchronization in Coupled Chaotic Circuits and Systems C. W. Wu Volume 42: Chaotic Synchronization: Applications to Living Systems E. Mosekilde, Y. Maistrenko & D. Postnov Volume 43: Universality and Emergent Computation in Cellular Neural Networks R. Dogaru Volume 44: Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems Z. T. Zhusubaliyev & E. Mosekilde Volume 45: Bifurcation and Chaos in Nonsmooth Mechanical Systems J. Awrejcewicz & C.-H. Lamarque Volume 46: Synchronization of Mechanical Systems H. Nijmeijer & A. Rodriguez-Angeles Volume 47: Chaos, Bifurcations and Fractals Around Us W. Szemplińska-Stupnicka Volume 48: Bio-Inspired Emergent Control of Locomotion Systems M. Frasca, P. Arena & L. Fortuna Volume 49: Nonlinear and Parametric Phenomena V. Damgov Volume 50: Cellular Neural Networks, Multi-Scroll Chaos and Synchronization M. E. Yalcin, J. A. K. Suykens & J. P. L. Vandewalle Volume 51: Symmetry and Complexity K. Mainzer Volume 52: Applied Nonlinear Time Series Analysis M. Small Volume 53: Bifurcation Theory and Applications T. Ma & S. Wang Volume 54: Dynamics of Crowd-Minds A. Adamatzky Volume 55: Control of Homoclinic Chaos by Weak Periodic Perturbations R. Chacón Volume 56: Strange Nonchaotic Attractors U. Feudel, S. Kuznetsov & A. Pikovsky Volume 57: A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science L. O. Chua

WORLD SCIENTIFIC SERIES ON

NONLINEAR SCIENCE\

Series A

VoI. 57

Series Editor: Leon 0. Chua

A NONllNEAR DYNAMICS PERSPECTIVE OF WOlFRAM'S NEW KN I D OF SCIENCE Volume I1

Leon 0 Chua University of California a t Berkeley, USA

vp World Scientific NEWJERSEY

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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ISBN-13 ISBN-10 ISBN-13 ISBN-10 ISBN-13 ISBN-10

978-981-256-977-6 981-256-977-4 978-981-256-976-9 981-256-976-6 978-981-256-642-3 981-256-642-2

Printed in Singapore.

(Vol. I) (Vol. I) (Vol. II) (Vol. II) (Set) (Set)

To Amalia, Ariella, Diana, Dimitri, Jake, Louisa, and Sophia

v

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A PRE-PUBLICATION BOOK REVIEW ♦

Ë A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM’S NEW KIND OF SCIENCE (in 2 Volumes) Leon O Chua (University of California at Berkeley, USA) World Scientific Series on Nonlinear Science, Series A - Vol. 57 vol. 1, 396 pp. Pub. date: June 2006 ISBN 981-256-977-4 Vol. 2, 580 pp pp. Pub. date: June 2007 Set ISBN: 981-256-642-2

“Our over-riding goal is to introduce cellular automata from the perspective of nonlinear dynamics for the lay readers unfamiliar with cellular automata”, writes Prof. Leon Chua. The book is a colorful presentation indeed, which will please everyone with fresh ideas and attractive illustrations. The text is not about cellular automata, it is about a tiny but fundamentally complex class of one-dimensional binary-state three-cell neighborhood cellular automata. The journey starts with Boolean-cube representation of the cell state transition rules. Every rule is shown by a cube such that every state of three-cell neighborhood is uniquely represented by a vertex of the cube. Vertices take values, 0 or 1, of cell-state transition function over corresponding states of the neighborhood. A concept of linear separability of functions and indices of complexity are introduced then. Complexity index of a transition function is a minimal number of parallel planes necessary to separate vertices of the Boolean-cube representing the function into the clusters of the same values. All functions are classified by three possible values of complexity index. It is illustrated that functions with complexity index ∗

Appeared in Journal of Cellular Automata, Volume 2, Number 3. [Reproduced with permission from the journal publisher.] vii

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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science

one exhibit a repetitive behavior while those with index two support mobile self-localizations, gliders, and non-trivially interacting propagating patterns. Few functions having complexity index three, it is claimed, have complex — at the degree of unpredictability — behavior. Introducing a universal neuron is a culmination of the first volume. A universal neuron is a scalar nonlinear differential equation which generates each of 256 rules. The equation has eight parameters, which can be interpreted as synaptic weights hence the name. The parameters of the neuron-equation relate to complexity as follows. One needs just four parameters to represent a cell-state transition function with complexity index one, six parameters to represent a function with index two, and all eight parameters are necessary for functions with index three. 1 An abstract interpretation of universal neuron parameters and cell-state transition functions in terms of genotype is provided, illustrated in the book, could be useful in future studies on evolution of cellular automata. First volume ends with chapter “Predicting the Unpredictable”, where 256 rules are partitioned into 88 global equivalence classes. Two rules are globally equivalent if they have identical nonlinear dynamics for all initial input patterns. Complexity index one is typical for 38 classes, index two for 41 classes, and just nine classes2 have highest index of complexity. Second volume proceeds along, in Chua’s words, “. . . a paradigm shift in research in Cellular Automata, which has hitherto been either empirical h· · ·i or highly abstract. Our approach is both analytical and constructive, made possible by our discovery of an explicit unified formula for h· · ·i characteristic functions, which was derived from an associated nonlinear differential equation, or a non-linear difference equation form.” There we enjoy complete characterization of behavior of studied cellular automata in terms of attractors and invariant orbits. Main findings include exact classification of invertible and non-invertible rules (with period one to three), selecting of Bernoulli rules. Also complete table of rules is provided which can be used to predict automaton global behavior from any initial configuration. Out of 256 rules, 112 non-periodic rules remarkably obey an explicit generalized Bernoulli shift formula, thereby allowing precise prediction of the global (time-asymptotic) dynamics. This fundamental result may have a substantial impact on future research in cellular automata. Indeed, the remaining 18 equivalence classes (including rules 30, 90, 110, etc.) also exhibit a more complex form of Bernoulli shift reminiscent but topologically different from the 112 Bernoulli rules reported so far. It is also incited universal rules exhibit 1/f power spectrum, 3 which is widely accepted as a signature of dynamical complexity in many disciplines, including humanities and arts. Then author invites us to share his findings on fractal geometry of the characteristics function, explicit formulas for generation of characteristics functions from binary bit-strings, geometrical and analytic properties of characteristics functions. We also become acquainted with identification and classification of non-constructible configurations and fixed points. At this point the author introduces “isle of Eden”, a configuration, whose the only predecessor is the configuration itself, and which is a fixed point in global evolution of cellular automaton.4 Rest of the volume deals with time-reversibility and invertibility of cell-state transition rules. These are studied by analyzing, sometimes with the help of generalized Bernoulli maps, attractors of the rules’ global dynamics. Dynamics of each attractor of a time-reversible rule is mirrored, in space and time, by its bilateral twin rule. Over half of the rules, 170 out of 256, are time-reversible in Chua’s framework; other 86 rules are irreversible in the sense that attractors mirror each other only in space not time.

1

At this point one can be sceptical about representation potential of generative complexity at the global dynamics level, Leon Chua however provides certain demonstrative examples to strengthen the idea. 2 Exact structure of the highest-complexity classes, in Wolfram coding, is {27, 83, 39, 53), {29, 71}, {46, 116, 139, 209}, {58, 114, 163, 177}, {78, 92, 121, 197}, {105}, {150}, {172, 228, 202, 216}, {184, 226}. 3 Such particular spectrum in cellular automata may be a result of glider interaction and reproduction. 4 As Leon Chua poetically said, time really stood still on an isle-of-Eden, as in a black hole.

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A test for time-reversibility of attractor is designed, and considered in relation with an idea that having attractor and its mirror we can move between time periods and thus mimic cosmological phenomena in cellular automata. The book appeals to wide auditorium. Apart of hard-core cellular automatists, those studying in non-linear sciences, electronic engineering, mathematics and logics, complexity and emergent phenomena, and possibly even chemistry and biology will certainly discover exciting concepts, analogies and research tools in this refreshing text. Anyone from freshmen to elderly academics will find parts interesting to them. The volumes are somewhat special and exciting because they posses a unique “Chua brand” and show gradual development of ideas and concepts in an educational and entertaining hence mathematically rigorous manner. Andrew Adamatzky (University of the West of England, Bristol, UK ) Editor — Journal of Cellular Automata


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♦

CONTENTS ♦

Ë Preface

vii

Prologue

xi

Volume I Chapter 1. Threshold of Complexity

1

1. Introduction

1

2. Cellular Automata is a Special Case of CNN

3

3. Every Local Rule is a Cube with Eight Colored Vertices

4

4. Every Local Rule is a Code for Attractors of a Dynamical System

6

4.1. Dynamical system for rule 110

78

4.2. There are eight attractors for each local rule

79

5. Every Local Rule has a Unique Complexity Index

83

5.1. Geometrical interpretation of projection σ and discriminant w(σ)

83

5.2. Geometrical interpretation of transition points of discriminant function w(σ)

84

5.3. Geometrical structure of local rules

86

5.4. A local rule with three separation planes

87

5.5. Linearly separable rules

95

5.6. Complexity index

96 xi


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5.7. Every local rule is a member of an equivalence class

102

5.8. Making non-separable from separable rules

110

5.9. Index 2 is the threshold of complexity

111

Chapter 2. Universal Neuron

113

1. Firing and Quenching Patterns

113

2. A Universal Neuron

119

3. Gallery of One-Dimensional Cellular Automata

121

4. Genealogic Classification of Local Rules

190

4.1. Primary and secondary firing patterns

190

4.2. Partitioning 256 local rules into 16 gene families

192

4.3. Each gene family has 16 gene siblings

192

5. The Double-Helix Torus

198

5.1. Algorithm for generating all 16 local rules belonging to each gene family

198

5.2. “8/24” Distribution pattern in gene siblings

198

5.3. Coding local rules on a double helix

206

6. Explaining and Predicting Pattern Features

210

6.1. Gallery of gene family patterns

210

6.2. Predicting the background

210

Chapter 3. Predicting the Unpredictable

229

1. Introduction

229

2. Local versus Global Equivalence

233

3. Predicting the Unpredictable

241

3.1. Paritioning 256 local rules into 89 global equivalence classes

241

3.2. The Vierergruppe V: Key for defining global transformations T† , T, and T∗

248

3.3. Proof of global equivalence

259

4. Predicting the Predictable

264

4.1. The rotation group R

265

4.2. Local equivalence classes 4.3. Finding all rotations which map any N ∈

270 n Sm

to

n any N ∈ Sm

4.4. Truth-table mapping matrices for the rotation group R

295 296

4.5. Combining transformations from the Vierergruppe V and the rotation group R

338

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4.6. Laterally symmetric interaction group for local equivalence classes

338

4.7. Mapping parameter vectors between rule 110 and its locally-equivalent rules

338

5. Concluding Remarks

359

References

361

Index (for Volume I)

363

Volume II Chapter 4.

From Bernoulli Shift to 1/f Spectrum

369

1. Introduction

369

1.1. Computing all 256 rules from one CA difference equation 2. Mapping Local Rules onto Global Characteristic Functions 2.1. CA characteristic functions

371 371 372

2.2. Algorithm for plotting the graph of CA characteristic functions 2.3. A glimpse of some time-τ characteristic functions

372 χτ

3. Transient Regimes and Attractors

N

373 379

3.1. Mapping CA attractors onto time-τ maps

382

3.2. A gallery of time-1 maps and power spectrum

387

3.3. Three general properties of time-1 maps

452

3.4. Invertible time-τ maps

453

4. Period-k Time-1 Maps: k = 1, 2, 3

454

4.1. Period-1 rules

454

4.2. Period-2 rules

459

4.3. Period-3 rules

460

4.4. Invariant orbits

463

5. Bernoulli στ -Shift Rules

463

5.1. Gallery of Bernoulli στ -shift rules

463

5.2. Predicting the dynamic evolution from {β, τ }

475

5.3. Two limiting cases: Period-1 and palindrome rules

487

5.4. Resolving the multivalued paradox

491

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6. Predictions from Power Spectrum

494

6.1. Characteristic features of Bernoulli rules

494

6.2. Turing-universal rules: { 110 , 124 , 137 , 193 } exhibit 1/f power-frequency characteristics

499


499

Chapter 5. Fractals Everywhere

509

1. Characteristic Functions: Global Representation of Local Rules

509

1.1. Deriving explicit formula for calculating χ1N

511

1.2. Graphs of characteristic functions χ1N

528

1.3. Deriving the Bernoulli map from

χ1

170

528

1.4. Deriving inverse Bernoulli map from χ1240

528

1.5. Deriving affine (mod 1) characteristic functions

593

χ1

596

2. Lameray Diagram on on

N

Gives Attractor Time-1 Maps

2.1. Lameray diagram of 170

597


597


597


597


597


603


607

3. Characteristic Functions are Fractals

610

4. Predicting the Fractal Structures

618

4.1. Two-level fractal stratifications 4.1.1. Stratification prediction procedure

618 623

4.1.2. Examples illustrating stratification prediction procedure 4.1.3. {Φ1 , Φ2 , Φ3 , Φ4 } — stratified families

623 625

4.2. Rules having no fractal stratifications

630

4.3. Origin of the fractal structures

630

5. Gardens of Eden

634

6. Isle of Eden

655


656

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Chapter 6.

From Time-Reversible Attractors to the Arrow of Time

657

1. Recap on Time-τ Characteristic Functions and Return Maps

658

2. Rule 62 Has Four Distinct Topological Dynamics

660

2.1. Period-1 attractor Λ1 ( 62 ) and its basin tree I+1 [Λ1 (

62 )]

2.2. Period-3 “isle of Eden” orbits Λ2 ( 62 )

667 675

2.3. Period-3 attractors Λ3 ( 62 ) and their basin trees I+1 [Λ3 (

62 )]

675

2.4. Bernoulli στ -shift attractors Λ4 ( 62 ) and their basin trees I+1 [Λ4 (

62 )]

676

3. Concept of a Time-Reversible Attractor 4. Time-Reversible Rules

700 705

4.1. Relationship between invertible and time-reversible attractors

706

4.2. Time reversible does not imply invertible

706

4.3. Time-reversible implies invertible if it is not period-1

707

4.4. Table of time-reversible rules

707

5. There are 84 Time-Reversible Bernoulli στ -Shift Rules

715

5.1. There are 42 time-reversible Bernoulli στ -shift rules (with |σ| = 1, β = 2σ > 0, and τ = 1) having only one Bernoulli attractor 5.2. Four canonical Bernoulli shift maps

715 732

5.3. There are eight time-reversible time-2 Bernoulli στ -shift rules (with |σ| = 1, β = 2σ > 0, and τ = 2) having only one Bernoulli attractor

732

5.4. There are 32 time-reversible Bernoulli στ -shift rules with two invertible attractors 5.5. Composition of 84 time-reversible Bernoulli rules

732 732

6. What Bit Strings Are Allowed in an Attractor or Invariant Orbit?

765

6.1. Laws governing period-1 bit strings

765


792


792

6.4. Laws governing Bernoulli στ -shift bit strings

792

6.4.1. Shift left or shift right by one bit

792

6.4.2. Unfolding Bernoulli orbits in complex plane

816

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6.4.3. Shift left or shift right by one bit and followed by complementation

816

6.4.4. Shift left or shift right by one bit every two iterations

827

6.4.5. Time-reversible rules with two Bernoulli attractors

827

6.4.6. Time-irreversible rules with two Bernoulli attractors

827

6.4.7. Time-irreversible rules with three Bernoulli attractors

827

6.4.8. Deriving bit string laws for globally-equivalent rules is trivial 7. Mathematical Foundation of Bernoulli στ -Shift Maps

827 867

7.1. Exact formula for time-1 Bernoulli maps for rules 170 , 240 , 85 , or 15

867

7.1.1. Exact formula for Bernoulli right-copycat shift map 170

867

7.1.2. Exact formula for Bernoulli left-copycat shift map 240

871

7.1.3. Exact formula for Bernoulli shift map for 85

873

7.1.4. Exact formula for Bernoulli shift map for 15

874

7.1.5. Exact formula for Bernoulli shift maps with β = 2σ (left shift), or β = 2−σ (right shift), σ = 1 and τ = 1 7.1.6. Exact formula for Bernoulli shift maps for 184

874 875

7.2. Exact formula for time-τ Bernoulli maps for rules 170 , 240 , 85 , or 15

876

7.2.1. Geometrical interpretation of time-τ maps

876

7.2.2. Exact formula for time-2 Bernoulli left-shift map for rule 170

881

7.2.3. Exact formula for time-2 Bernoulli right-shift map for rule 240 7.2.4. Exact formulas for generalized Bernoulli maps

884 888

7.2.5. Analytical proof of period-3 “Isle of Eden” Λ2 ( 62 ) with I = 4 as subshift of time-1 Bernoulli στ -shift map (with σ = 1) φn = 2φn−1 mod ν(I) for rule 170

889


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7.2.6. Analytical proof of time-2 map of Λ2 ( 62 ) with I = 4 is a subshift of time-2 Bernoulli στ -shift map (with σ = −1) of Eq. (85) of rule 240

892

7.2.7. Analytical proof of time-1 map of Λ4 ( 62 ) with I = 4 is a subshift of time-1 Bernoulli στ -shift map (with σ = 2) of Eq. (63) of rule 170 7.3. Λ4 ( 62 ) is a subshift of Λ( 240 ) 8. The Arrow of Time

893 895 899

8.1. Rule 6

899

8.2. Rule 9

899

8.3. Rule 25

899

8.4. Rule 74

910


910

9.1. Attractors of 206 local rules

910

9.2. Time reversality

911

9.3. Paradigm shift via nonlinear dynamics

913

Errata for Volume I

934

Epilogue

935

References

937

Index (for Volumes I and II)

939

xvii


♦

Chapter 4 FROM BERNOULLI SHIFT TO 1/f SPECTRUM ♦

Ë By exploiting the new concepts of CA characteristic functions and their associated attractor time-τ maps, a complete characterization of the long-term time-asymptotic behaviors of all 256 one-dimensional CA rules are achieved via a single “probing” random input signal. In particular, the graphs of the time-1 maps of the 256 CA rules represent, in some sense, the generalized Green’s functions for Cellular Automata. The asymptotic dynamical evolution on any CA attractor, or invariant orbit, of 206 (out of 256) CA rules can be predicted precisely, by inspection. In particular, a total of 112 CA rules are shown to obey a generalized Bernoulli στ -shift rule, which involves the shifting of any binary string on an attractor, or invariant orbit, either to the left, or to the right, by up to 3 pixels, and followed possibly by a complementation of the resulting bit string. The most intriguing result reported in this paper is the discovery that the four Turinguniversal rules 110 , 124 , 137 , and 193 , and only these rules, exhibit a 1/f power spectrum.

1. Introduction

Each of the eight binary bits β0 , β1 , . . . , β7 in the rightmost column of this figure is equal to either 0 or 1. There are 256 distinct combinations of “zeros” and “ones” among the eight binary bits β0 , β1 , . . . , β7 , each one defining a unique Boolean function of three binary variables. One-to-one correspondence of each of these 256 Boolean functions with its associated decimal number 7 N = βk 2k (2)

The basic notations and concepts underlying this tutorial stem from [Chua, 1998; Chua & Roska, 2002] and from Part I [Chua et al., 2002], Part II [Chua et al., 2003], and Part III [Chua et al., 2004]. Throughout this paper we are concerned exclusively with 2-state one-dimensional cellular automata consisting of I + 1 cells, i = 0, 1, 2, . . . , I with periodic boundary conditions, as depicted in Fig. 1(a). Each cell i interacts only with its nearest neighbors i − 1 and i + 1, as depicted in Fig. 1(b). Here ui−1 , ui and ui+1 denote the three inputs needed to compute the single output yi by a three-input nonlinear function yi = N (ui−1 , ui , ui+1 )

k=0

determines a local rule N of the cellular automaton. Each coefficient β0 , β1 , . . . , β7 is uniquely identified, via its coordinates (ui−1 , ui , ui+1 ) from Fig. 1(d) as a vertex of the Boolean cube shown in k is colored in blue if βk = 0 and Fig. 1(e). A vertex in red if βk = 1. For example, the colored vertices in

(1)

Boolean computations by this function are executed according to the truth table depicted in Fig. 1(c). 369

370


Fig. 1. (a) A one-dimensional Cellular Automata (CA) made of (I + 1) identical cells with a periodic boundary condition. Each cell “i” is coupled only to its left neighbor cell (i − 1) and right neighbor cell (i + 1). (b) Each cell “i” is described by a local rule N , where N is a decimal number specified by a binary string {β0 , β1 , . . . , β7 }, βi ∈ {0, 1}. (c) The symbolic truth table specifying each local rule N , N = 0, 1, 2, . . . , 255. (d) By recoding “0” to “−1”, each row of the symbolic truth table in (c) can be recast into a numeric truth table, where γk ∈ {−1, 1}. (e) Each row of the numeric truth table in (d) can be represented as a vertex of a Boolean Cube whose color is red if γk = 1, and blue if γk = −1.

Chapter 4: From Bernoulli Shift to 1/F Spectrum

Fig. 1(e) correspond to the local rule N = 0 • 20 + 1 • 21 + 1 • 22 + 1 • 23 + 0 • 24 + 1 • 25 + 1 • 26 + 0 • 27 = 110. (3) Depending on the context, each variable ui−1 , ui , ui+1 , or yi may assume a symbolic Boolean value “0” or “1”, as depicted in Fig. 1(c), or a numeric value, “−1” or “+1”, as depicted in Fig. 1(d). The symbolic and numeric representations are related to each other as follows1 : Boolean variable − 1 (4) 1 Boolean variable = numeric variable + 1 2 (5) numeric variable = 2

•

In particular, the real variables γ0 , γ1 , . . . , γ7 in Fig. 1(d) are related to the Boolean variables β0 , β1 , . . . , β7 in Fig. 1(c) via the formula γk = 2βk − 1

(6)

Substituting Eq. (6) into Eq. (2), we obtain the following equivalent local rule number 7 1 255 + N = γk 2k (7) 2

each one of the 256 local rules is listed in Table 4 of Part II [Chua et al., 2003]. The eight parameters {z2 , c2 , z1 , c1 , z0 , b1 , b2 , b3 } in this equation can be used to derive the coefficients βk in Fig. 1(c), k = 0, 1, . . . , 7 via the formula: 1 βk = (1 + sgn{z2 + c2 |(z1 + c1 |(z0 + b1 uk,i−1 2 + b2 uk,i + b3 uk,i+1 )|)|}) (9) where the numeric coefficients uk,i−1, uk,i and uk,i+1 are given by row k of the truth table in Fig. 1(d). The state variables uti−1 , uti , and uti+1 in Eq. (8) must be expressed in numeric values −1 and +1. Since this paper (Part IV) will be devoted exclusively to Boolean variables xi ∈ {0, 1}, it is more convenient to express Eq. (8) in terms of xi via Eq. (4); namely, CA Difference Equation 2 : xti ∈ {0, 1} 1 = (1 + sgn{z2 + c2 |(z1 + c1 |(z0 + b1 xti−1 xt+1 i 2 + b2 xti + b3 xti+1 )|)|}) where z0

k=0

(10)

The cellular automaton evolves in discrete time steps t = 0, 1, 2, . . . . The output of the ith cell (in numeric representation) can be calculated analytically from the following nonlinear difference equation [Chua et al., 2004] involving eight parameters: CA Difference Equation 1 : uti ∈ {−1, 1} + b2 uti

+

(8)

b3 uti+1 )|)|}

It is indeed remarkable that one equation suffices to define all 28 = 256 Boolean functions of three variables ui−1 , ui , and ui+1 by simply specifying eight real numbers. Even more remarkable is that the CA Difference equation (8) is robust in the sense that the eight parameter values defining each local rule N form a dense set. One set of parameters {z2 , c2 , z1 , c1 , z0 , b1 , b2 , b3 } for realizing 1

1 1 [z0 − (b1 + b2 + b3 )], z1 z1 , 2 2

1 z2 z2 2

1.1. Computing all 256 rules from one CA difference equation

= sgn{z2 + c2 |(z1 + c1 |(z0 + b1 uti−1 ut+1 i

371

2. Mapping Local Rules onto Global Characteristic Functions Given any local rule N , N = 0, 1, 2, . . . , 255, and any binary initial configuration (or initial state when used in the context of nonlinear dynamics) →

x(0) = [x0 (0), x1 (0), · · · , xI−1 (0), xI (0)]

(11)

for a one-dimensional Cellular Automaton with I+1 cells [see Fig. 1(a)], where xi (0) ∈ {0, 1}, we can → associate uniquely the Boolean string x(0) with the binary expansion (in base 2) of a real number 0 • x0 x1 · · · xI−1 xI on the unit interval [0, 1]; namely, →

x [x0 x1 · · · xI−1 xI ]

→ φ 0 • x0 x1 · · · xI−1 xI

(12)

The Boolean variable is considered as a real number in Eqs. (4) and (5), or in any equation involving real-variable (i.e. nonlogic) operations.

372


where the decimal equivalent of Eq. (12) is given by φ=

I

2−(i+1) xi

(13)

i=0

We will often need to consider also the bilateral image x(0) = [xI (0), xI−1 (0), . . . , x1 (0), x0 (0)] = T†(x(0)) (14)

←

henceforth called the backward Boolean string asso→ ciated with the forward Boolean string x from Eq. (12), where T† is the (I + 1)-dimensional left– right transformation operator defined in Table 13 of [Chua et al., 2004], namely, T† [x0 x1 · · · xI−1 xI ] = [xI xI−1 · · · x1 x0 ]

(15)

←

Each backward Boolean string x in Eq. (14) maps into the real number φ† defined by x → φ† 0 • xI xI−1 · · · x1 x0

←

(16)

where the decimal equivalent of Eq. (16) is given by φ† =

I

2−(I+1)+i xi

(17)

Φ ◦ TN = χN ◦ Φ

(20)

where “◦” denotes the composition operation. Observe that in the limit where I → ∞, the state space Σ coincides with the collection of all bi-infinite strings extending from −∞ to ∞, and I→∞

(21)

In this general case, the CA characteristic function χ N is defined on every point (i.e. real number) φ ∈ [0, 1], thereby including all irrational numbers as well [Niven, 1967].

where ←

x [xI xI−1 · · · x1 x0 ]

2.1. CA characteristic functions For a one-dimensional CA with I + 1 cells, there are ∆ ∆ nΣ = 2I distinct Boolean strings, where I = I + 1. Let Σ denote the state space made of the collection of all nΣ Boolean strings. Each local rule N induces a global map (18)

where each state x ∈ Σ is mapped into exactly one state T N (x) ∈ Σ. Since each state x ∈ Σ corresponds to one, and only one, point φ ∈ [0, 1] via Eq. (13), it follows that the global map (18) induces an equivalent map χ N from the set of all rational numbers R[0, 1] over the unit interval [0, 1] into itself; namely, χ N : R[0, 1] → R[0, 1]

henceforth called the CA characteristic function of N . The one-to-one correspondence between the global map T N and the CA characteristic function χ N is depicted in the diagram shown in Fig. 2, → where Φ denotes the transformation of the state x into the decimal function defined in Eq. (13). This diagram is said to be commutative because

lim R[0, 1] = [0, 1]

i=0

TN : Σ → Σ

Fig. 2. A commutative diagram establishing a one-to-one correspondence between T N and χ N .

(19)

2.2. Algorithm for plotting the graph of CA characteristic functions Since the domain of the CA characteristic function χ N of any local rule N (for finite I) consists of a subset of rational numbers in the unit interval [0, 1], a computer program for constructing the graph of the characteristic function χ N can be easily written as follow: Step 1. Divide the unit interval [0, 1] into a finite number of uniformly-spaced points, called a linear grid, of width ∆φ. For the examples in Sec. 2.3, we choose ∆φ = 0.005. Step 2. For each grid point φj ∈ [0, 1], identify the corresponding binary string sj ∈ Σ. Step 3. Determine the image sj ∈ Σ of sj under N , i.e. find sj = T N (sj ) via the truth table of N .


Step 4. Calculate the decimal equivalent of sj via Eq. (13). Step 5. Plot a vertical line through the abscissa φ N = φj with height equal to sj . Step 6. Repeat steps 1–5 over all (1/∆φ) + 1 grid points. For the examples in Sec. 2.3, there are (1/0.005) + 1 = 201 grid points. For reasons that will be clear later, it is sometimes more revealing to plot the τ th iterated value τ (sj ) T N ◦ T N ◦ · · · T N (sj ) sτj = T N

(22)

τ times

of sj , instead of T N (sj ), at each grid point φj ∈ [0, 1]. For obvious reasons, such a function is called a time-τ CA characteristic function and will henceforth be denoted by χτN . Using this terminology, the algorithm presented above can be used to plot the graph of the “time-1” CA characteristic function χ1N of any local rule N . The same algorithm applies mutatis mutandis, for plotting the graph of the time-τ characteristic function χτN as well. To enhance readability, it is desirable to plot ∆ the M = (1/∆φ)+1 vertical lines of χτN in alternating red and blue colors, henceforth referred to as red and blue φ-coordinates φred and φblue , respectively. The “tip” of each vertical line gives the value of χτN corresponding to each φ coordinate. The system of red and blue lines is defined via the following simple algorithm. For any I, partition all (I +1)-bit binary strings into a red group and a blue group. All members of the red group have a “0” as their rightmost bit. The blue group then consists of all (I +1)-bit binary strings with a “1” as their rightmost bit. Each group has therefore exactly one half of the total number (M = 2I+1 ) of distinct strings, namely, 2I . Since the end (rightmost) bit of each φblue ∈ [0, 1] is equal to a “1”, by construction, it follows that the largest value of φblue is greater than the largest value of φred by exactly 1/2I+1 . This means that the rightmost vertical line must have color blue, and tends to φ = 1 as I → ∞. The rightmost blue line is, for plotting purpose, drawn through φ = 1. We can then divide the interval [0, 1] into (1/∆φ)+1 grid points, where ∆φ is the prescribed resolution. All characteristic functions in Figs. 3–7 are drawn with ∆φ = 0.005, where a tiny red or blue square is drawn around the tip of each vertical line for ease of

373

identification. In other words, the distance between each red line and its adjacent blue line is equal to 0.005. Although higher precision can be easily implemented by a computer, the limited printer resolution will cause adjacent red and blue lines to merge through ink diffusion for ∆φ < 0.005. To construct Figs. 3–7, we chose ∆φ = 0.005, φstart = 0• 00 · · 00 1, and φend = 0• 11 · · 11 1. · · 65 0 s 65 1 s Our choice leads to exactly 100 red vertical lines (located at 0.005, 0.015, 0.025, . . . , 0.995, ∆red = 0.01) with binary base-2 expansion φred = 0 • β1 β2 β3 · · · β65 0, βi ∈ {0, 1}, which interleave with 101 blue vertical lines (located at 0.000, 0.01, 0.02, . . . , 1.00, ∆blue = 0.01) with binary base-2 1, β ∈ {0, 1}. expansion φblue = 0 • β1 β2 β3 · · · β65 i The 201 red and blue lines shown in the characteristic functions in Figs. 3–7 represent only their approximate positions on [0, 1] because the resolution of their exact positions is determined by the value of I, which is chosen to be 65 in Figs. 3–7. This means that our state space Σ is “coarse grain” and contains only 266 distinct 66-bit binary strings, each one representing a unique rational number on [0, 1], of which only 201 are actually drawn in these figures to avoid clutter. Since an arbitrary rational number on [0, 1] requires an arbitrarily large (though finite) value of I for an exact base-2 expansion (i.e. I → ∞ in Eq. (13)), a fine grain characteristic function χτN which includes all possible rational numbers φ ∈ [0, 1] in its domain would be impractical to plot on paper, or even store on any computer memory. However, the characteristic functions (calculated with I = 65) shown in Figs. 3 to 7 are adequate for most purposes. Increasing the value of I is equivalent to “sandwiching” more vertical lines in between the existing lines drawn in these figures.

2.3. A glimpse of some time-τ characteristic functions χτN Let us take a glance at some representative examples of CA characteristic functions. For brevity, we will henceforth refer to “time-1” CA characteristic functions simply as “characteristic functions”. Example 1.

χ1128

The graph of the characteristic function χ1128 of 128 is shown in Fig. 3(a). This is among the

374

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science (a)

(b)

Fig. 3. Time-1 CA characteristic functions χ1128 and χ1200 for local rules 128 and 200 , respectively. Although only 201 points (enclosed by tiny squares) are shown, the abscissa (φ coordinate) of each point is calculated with a 66-bit string resolution.


375

(a)

(b)


376


(b)



377

(a)

(b)

Fig. 6. Time-1 CA characteristic functions χ151 and time-2 CA characteristic function χ 251 for local rule 51 . Although only 201 points (enclosed by tiny squares) are shown, the abscissa (φ coordinate) of each point is calculated with a 66-bit string resolution.

378


(b)

Fig. 7. Time-1 CA characteristic functions χ162 and time-3 CA characteristic function χ362 for local rule 62 . Although only 201 points (enclosed by tiny squares) are shown, the abscissa (φ coordinate) of each point is calculated with a 66-bit string resolution.


simplest characteristic functions. Observe that no vertical lines intersect the unit-slope main diagonal except at φ 128 = 0• 00 and φ 128 = 1• 00 (where the bar over a sequence of binary bits denotes repetition of these bits ad infinitum). These two period-1 fixed points give rise to a homogeneous “0” dynamic pattern D 128 [0• 00] (homogeneous blue color) in the former, and a homogeneous “1” dynamic pattern D 128 [1• 00] (homogeneous red color) in the latter. The qualitative dynamics of these two orbits, however, are dramatically different. The orbit from D 128 [0• 00] is an attractor in the sense of nonlinear dynamics [Alligood et al., 1996] because it has a nonempty basin of attraction BΛ , which in this case consists of all points in the closed-open unit interval [0, 1). The orbit from φ 128 = 1• 00 is an example of both an invariant orbit, and a Garden of Eden, to be defined in Sec. 3. Example 2.

χ1200

The graph of the characteristic function χ1200 of 200 is shown in Fig. 3(b). In this case, observe that there are many vertical lines which terminate exactly on the main diagonal. There are therefore many period-1 fixed points which imply the presence of many period-1 attractors. This is characteristic of local rules belonging to Wolfram’s class 1 rules [Wolfram, 2002]. We will return to this class of attractors in Sec. 3. Example 3.

χ1170 χ1

The graph of the characteristic function 170 of 170 is shown in Fig. 4(a). Note that there are no period-1 fixed points except at φ 170 = 0• 00 and φ 170 = 1• 00. Observe also the vertices of all vertical lines fall on one of two parallel lines with slope = 2. This is an example, par excellence, of the classic Bernoulli shift [Nagashima & Baba, 1999], a subject to be discussed at length in Sec. 5. Example 4.

379

blue vertical lines terminate on the upper parallel straight lines. Since the blue and red vertical lines interleave but do not intersect each other, χ1240 is a well-defined single-valued function. In fact, a careful examination of χ1170 and χ1240 in Fig. 4 will reveal that these two piecewise-linear functions are inverse of each other. Subsets of both characteristic functions in Fig. 4 are typical of Wolfram’s class 2 rules. Example 5.

χ130

The graph of the characteristic function χ130 of 30 is shown in Fig. 5(a). This complicated characteristic is typical of all local rules belonging to Wolfram’s class 3 CA rules. Example 6.

χ1110

The graph of the characteristic function χ1110 of 110 is shown in Fig. 5(b). This rather exotic characteristic exhibits many features typical of Wolfram’s class 4 rules. Example 7.

χ151 and χ251

The graphs of the “time-1” characteristic function χ151 and “time-2 ” characteristic function χ251 of 51 are shown in Figs. 6(a) and 6(b), respectively. Observe that while there is only one period-1 fixed point in χ151 , every vertical line terminates on the main diagonal of χ251 . This implies that 51 has a dense set of period-2 invariant orbits. Such local rules will be studied in Sec. 4.4. Example 8.

χ162 and χ362

The graphs of the “time-1” characteristic function χ162 and “time-3 ” characteristic function χ362 of 62 are shown in Figs. 7(a) and 7(b), respectively. Observe that while there are no period-1 fixed points in χ162 , there are many vertical lines which landed on the main diagonal of χ362 . This implies that 62 has many period-3 attractors. Such local rules will be studied in Sec. 4.3.

χ1240

The graph of the characteristic function χ1240 of 240 is shown in Fig. 4(b). There are no period-1 fixed points except at φ 240 = 0• 00 and φ 240 = 1• 00.2 The “double-valued” appearance is only illusory because all red vertical lines terminate on the lower straight lines of slope = 1/2, and all

3. Transient Regimes and Attractors For a CA with finite I, the state space contains exactly nΣ 2I distinct states, where I = I + 1. It follows that given any initial state

x(0) = x0 (0) x1 (0) · · · xI−1 (0) xI (0) (22)

2 The leftmost vertical line should actually be drawn at φ = ε ≈ 0. Printer resolution precludes our showing the correct value χ1240 (0.00) = 0.00.

380


the dynamic pattern D N [x(0)] evolving from the initial state x(0) under any local rule N must eventually repeat itself with a minimum period TΛ , where the Attractor period

TΛ ≤ 2I+1

(23)

depends only on the local rule N , and is independent of the initial state x(0), assuming x(0) belongs to the basin of attraction of a period-TΛ attractor Λ to be defined below. Definition 1.

Transient Regime and Transient Duration: Given any local rule N , and any initial configuration x(0), let Tδ be the smallest nonnegative integer such that

the dynamic pattern D N [x(0)] is called the transient regime originating from the initial state x(0) and the time (Tδ −1) is called the transient duration. Definition 2.

then the TΛ denoted by

Period-TΛ Attractor: If Tδ > 1, consecutive rows of D N [x(0)]

Λ N (x(0)) x(Tδ ) ∪ x(Tδ + 1) · · · ∪ x(Tδ + (TΛ − 1))

(25)

is called a period-TΛ attractor of the local rule N originating from the initial configuration x(0). The set BΛ of all initial states x(0) which tend to the attractor Λ is called the basin of attraction of Λ.

Since x(t), t = 0, 1, 2, . . . , Tδ − 1, will never recur again for all t ≥ Tδ , the first Tδ consecutive rows of

To illustrate the above definitions, consider first the dynamic pattern D 62 [xa (0)] shown in Fig. 8(a). For the initial configuration xa (row 0 in Fig. 8(a)), we find Tδ = 51 and TΛ = 3. Hence, the transient

(a)

(b)

x (Tδ + TΛ ) = x (Tδ )

(24)

Fig. 8. Illustrations of the transient regime and transient duration of rule 62 originating from two different initial configurations xa and xb .


regime originating from xa of the dynamic pattern D 62 [xa ] consists of the first 51 rows in Fig. 8(a). The period-3 orbit is clearly seen by the alternating color backgrounds. For the dynamic pattern D 62 [xb ] shown in Fig. 8(b), observe that the initial configuration xb (row 0 in Fig. 8(b)) gives rise to a longer transient duration Tδ = 83. However, since xa and xb in Fig. 8 were chosen to belong to the basin of attraction of Λ, the period TΛ of the periodic orbit in Figs. 8(a) and 8(b) must be the same, namely, TΛ = 3, as can be easily verified by inspection of the dynamic pattern in Fig. 8. For some local rules, the period TΛ can be much larger than the transient duration, as depicted in the two dynamic patterns D 99 [xa ] and D 99 [xb ] in Figs. 9(a) and 9(b) for local rule N = 99 . Observe that Tδ = 14 and TΛ = 71 for D 99 [xa ]. Similarly, Tδ = 43 and TΛ = 71 for D 99 [xb ].

(a)

381

In fact, there are local rules such as 110 and 30 , and their global equivalence classes, where TΛ can tend to infinity as T → ∞ (for I = ∞). In such cases, it is no longer useful to talk about a transient regime and we will simply refer to the entire dynamic pattern D N [x(0)] as an orbit originating from x(0). The “basin of attraction” BΛ of an attractor Λ must contain, by definition, at least one point not belonging to Λ. It is possible, however, for some periodic orbits to have no basin of attraction. Definition 3. Invariant Orbits:

An orbit Γ whose basin of attraction is the empty set is called an invariant orbit. It follows from Definition 3 that an invariant orbit must have a zero transient duration, i.e. Tδ = 1.

(b)

Fig. 9. Illustrations of the transient regime and transient duration of rule 99 originating from two different initial configurations xa and xb .

382


Local Equivalent Class S41 is Invariant: The orbits of all six rules { 15 , 51 , 85 , 170 , 204 , 240 } belonging to the local equivalence class S41 [Chua et al., 2003] are invariant. Proposition 1.

Proof.

We will see in Table 2 and Sec. 5.4 that, for finite I, every point in Fig. 4(a) is a point on a periodic orbit of 170 whose dynamics consist of shifting each initial string by one pixel to the left. Consequently, there is no transient regime in this case and hence all orbits of 170 are invariant orbits. Since the shifting operation is preserved under the rotational transformations of the local equivalence class S41 listed in Table 25(o) of [Chua et al., 2003], it follows that all orbits of 15 , 51 , 85 , 204 , and 240 are invariant as well. It has been verified by exhaustive computer simulation that only the six rules belonging to S41 are endowed with only invariant orbits. In general, invariant orbits have noninvariant neighboring orbits. We have seen earlier a special case of an invariant orbit consisting of only a single point; namely, φ 128 = 1• 00 in Fig. 3(a). Observe that in addition to having no basin of attraction, φ 128 = 1• 00 has no preimage (predecessor). Such special initial configuration is called a garden of Eden [Moore, 1962]. Observe that no garden of Eden can be a periodic orbit with a period TΛ > 1, otherwise any point on the orbit is a predecessor of its next iterate. A period-1 garden of Eden is therefore a truly unique specie worthy of its own name, henceforth dubbed an isle of Eden. Indeed, we can generalize this unique phenomenon, which does not exist in continuous dynamical systems (such as ODE), to define a “period-k ” isle of Eden from the kth iterated characteristic function χkN of N . A gallery of period-k isles of Eden of all one-dimensional cellular automata will be presented in Part V of this tutorial series.

3.1. Mapping CA attractors onto time-τ maps Since invariant orbits are not attractors, they are not robust in the sense that precisely specified initial states must be used to observe them. Since one of the most fundamental problems in nonlinear dynamics is to analyze and predict their longterm behaviors as t → ∞, we will develop some novel and effective techniques for analyzing and

predicting global qualitative behaviors of robust CA attractors. In general, each CA local rule N can exhibit many distinct attractors Λi , i = 1, 2, . . . , Ω, as demonstrated in Figs. 3–7. Each attractor represents a distinct operating mode and must be analyzed as a separate dynamical system. In order to exploit the lateral symmetry exhibited by many bilateral pairs N and N † T † [N ] of local rules, where T † denotes the left–right transformation operation defined in [Chua et al., 2004], it is more revealing to represent and examine each attractor from two spatial directions, namely, a forward (left → right) direction and a backward (right → left) direction. Since each CA attractor Λ is periodic (for finite I) with some period TΛ , it is usually represented by displaying TΛ consecutive binary bit strings s1 , s2 , . . . , sTΛ , as illustrated in Figs. 8 and 9. In order to exploit the analytical tools from nonlinear dynamics [Alligood et al., 1996; Shilnikov et al., 1998], it is essential that we transcribe these rather unwieldy pictorial data into an equivalent nonlinear time series. Such a one-to-one transcription is precisely defined by Eqs. (13) and (17) via the commutative diagram shown in Fig. 2. Hence, → each forward Boolean string x is mapped bijectively onto a real number φ ∈ [0, 1] via Eqs. (12) and ← (13). Similarly, each backward Boolean string x(0) is mapped bijectively onto a real number φ† ∈ [0, 1] via Eqs. (16) and (17). Each “period-TΛ ” attractor Λ defined by a pattern made of TΛ consecutive Boolean strings is therefore mapped onto a forward time series ϕ = [φ0 , φ1 , φ2 , . . . , φTΛ ],

φi ∈ [0, 1],

(26)

henceforth called a forward orbit, and a backward time series ϕ† = [φ†0 , φ†1 , φ†2 , . . . , φ†TΛ ],

φ†i ∈ [0, 1],

(27)

henceforth called a backward orbit, where the length of each time series (resp. period of each orbit) is equal to TΛ . It follows from the definition of the period of an attractor that TΛ = 1 for all “period-1” attractors in Fig. 3(b), TΛ = 2 for all “period-2 ” attractors in Fig. 6(b), and TΛ = 3 for all “period-3 ” attractors in Fig. 7(b). Attractors associated with local rules belonging to Wolfram’s Classes 3 and 4 can have an extremely large period TΛ , a number greater than the number of elementary particles in the universe even for a relatively small I = 100.


From a computational perspective, such time series is effectively infinite in length. The qualitative dynamics associated with an attractor can often be uncovered and understood by plotting the following two attractor-induced time-τ maps [Alligood et al., 1996] associated with the forward time series ϕ and the backward time series ϕ† , respectively: Forward time-τ map

(28)

ρτ [N ] : φn−τ → φn

Backward time-τ map ρ†τ [N ] : φ†n−τ → φ†n

(29)

For each local rule N , the forward time-τ map ρτ and the backward time-τ map ρ†τ are defined explicitly via the time-τ characteristic function χτN as follow: ρτ (φn−τ ) = χτN (φn−τ )

(30)

ρ†τ (φ†n−τ ) = χτN (φ†n−τ )

(31)

Explicit coordinates (φn−τ , φn ) and (φ†n−τ , φ†n ) for plotting each point of the forward time-τ map ρτ [N ] and the backward time-τ map ρ†τ [N ], are listed in Table 1 for τ = 1, 2, and 3. When τ = 1, the resulting time-1 maps [Alligood et al., 1996; Hirsch & Smale, 1974] ρ1 [N ] and ρ†1 [N ] are sometimes called first-return maps in the literature because they behave like Poincare return maps [Poincare, 1897]. Figure 10 shows the Poincare first-return map interpretation of four forward time-1 maps ρ1 [200], ρ1 [51], ρ1 [62], and ρ1 [170] of CA rules 200 , 51 , 62 , and 170 , repectively. In each case, the Poincare cross-section is the same unit square [0, 1] × [0, 1] we have encountered earlier in our definition of the CA characteristic function χ1N in Eq. (19). Only one out of many attractors is shown for each time-1 map in Fig. 10. In Fig. 10(a), only one period-1 attractor of rule 1 200 is shown (labeled as point ). The domain of the time-1 map ρ1 [200] in this trivial case con1 and all iterates sists of only the single point {},

383

1 One can intermap trivially onto the fixed point . 1 as the point where a planet intersects pret point an imaginary Poincare cross-section once every revolution. Figure 10(b) shows a period-2 attractor (out of many others) of local rule 51 . The orbit of the circulating planet intersects the Poincare crosssection at two points. The domain of the time-1 1 2 1 2 and } where ρ1 () → map ρ1 [51] is {, 2 1 → . ρ1 () Figure 10(c) shows a period-3 attractor of local rule 62 . The circulating orbit is seen to intersect the Poincare cross-section at three points. The 1 domain of the time-1 map ρ1 [62] consists of {, 2 3 1 2 2 3 , } where ρ1 () → , ρ1 () → , and 3 1 → . ρ1 () Figure 10(d) shows a Bernoulli σ1 -shift attractor (to be discussed in depth in Sec. 5) of local rule 170 where the domain of the time-1 map ρ1 [170] consists of all points on the two parallel lines with slope equal to 2 for the case I = ∞. For finite I > 60, the attractor consists of almost all points on these two lines separated by tiny gaps ∆ < 10−18 and is therefore not discernible. The domain in the case I = ∞ consists of the entire unit interval 1 2 3 . . . , ) 6 , , are [0, 1]. Only a few iterates (, shown to avoid clutter. One can associate the complicated orbit in Fig. 10(d) with the trajectory of a comet, which in this case would visit almost all points on these two parallel lines, as originally envisioned by Poincare. In so far as the qualitative dynamics is concerned, it suffices to examine the evolution of the time-1 map induced by the orbit in Fig. 10. To illustrate this important insight discovered by Poincare, let us examine the forward time-1 map of a period-3 1 2 3 in , attractor of 62 consisting of points , Fig. 11(a), as well as the associated backward time-1 1 2 and . 3 map in Fig. 11(b) consisting of points , In order to illustrate what we mean by the fundamental principle which asserts that CA rules belonging to the same global equivalence class εκm [Chua et al., 2004] must have identical qualitative dynamics, we also show the forward time-1 map for 118 in Fig. 11(c), and the backward time-1 map for 118 in Fig. 11(d), where 118 and 62 belong to same equivalence class ε222 derived in [Chua et al., 2004]. Since 118 and 62 are related by a left–right transformation operator T† , i.e. T † [62] = 118 , it follows from the theory of global equivalence class developed in [Chua et al., 2004], that the two rules 62 and 118 have identical qualitative behaviors.

384

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 1.

Explicit coordinates (φn−τ , φn ) for defining time-τ maps for τ = 1, 2, 3.


385

Fig. 10. Poincare return map interpretation of four forward time-1 maps. (a) Period-1 map ρ1 [200]. (b) Period-2 map ρ1 [51]. (c) Period-3 map ρ1 [62]. (d) Bernoulli σ1 -shift map ρ1 [170].

In particular, they have, qualitatively, the same transient regimes, the same attractors, and the same invariant orbits, modulo a bijection. Moreover, their dynamics must also be mapped onto each other, as depicted by the diagram shown in Fig. 11. This well-known geometrical construction is called a Lameray diagram [Shilnikov et al., 1998], named after the French mathematician Lameray who first discovered its pedagogical value in the eighteenth century. It is also called a cobweb diagram [Alligood et al., 1996] because it resembles the web spun by a spider. Important Observation Every Point on the forward time-1 map ρ1 : φn−1 → φn , or the backward time-1 map ρ†1 : φ†n−1 → φ†n , of any CA rule N is a point on the characteristic function χ1N .

In other words, the CA characteristic function χ1N is a complete and global representation of N . It is complete because it contains all information needed to derive the dynamic evolutions from any initial state by simply drawing a Lameray (cobweb) diagram directly on χ1N ! It is global because each point on χ1N codes for an entire configuration of length I + 1, and not just for one pixel if the local rule were used instead. Clearly, the points defining the time-1 maps ρ1 [N ] and ρ†1 [N ] are subsets of the points defining the characteristic function χ1N . It follows from the above observation that every point on a time-τ map of N is a point on the time-τ characteristic function χτN . Remarks

1. If we imagine the three points on the time1 map ρ1 [62] in Fig. 11(a) as points on a unimodal function (e.g. logistic map) [Alligood

386


Fig. 11. Cobweb diagram showing the evolution of 62 and 118 from any state of a period-3 attractor in forward time (a and c), and backward time (b and d). Points kiand kj denotes corresponding instants of time.

et al., 1996], then we can associate this particular period-3 attractor of 62 as a period-3 point of a continuous map f : [0, 1] → [0, 1] which we know is chaotic because “period-3 implies chaos” [Alligood et al., 1996]. 2. It follows from Remark 1 above that every forward and backward time-1 map exhibited in Table 2 of Sec. 3.2 can be interpreted as a period-TΛ attractor of a continuous map f : [0, 1] → [0, 1] over the unit interval [0, 1].

3. It follows from Remark 2 above that for every CA rule N , N = 0, 1, 2, . . . , 255, and finite I, we can construct a continuous one-dimensional map f N : [0, 1] → [0, 1] which has a period-TΛ point coinciding with a period-TΛ attractor, or invariant orbit, of rule N . 4. It follows from Remark 3 above that since all attractors, or invariant orbits, of a CA rule N are disjoint sets of points over [0, 1], we can always construct a polynomial P N (x), x ∈ [0, 1],


which passes through all of these points. In fact, we can invoke the canonical representation from [Chua & Kang, 1977] to derive an explicit equation P N (x) involving the absolute value function as the only nonlinearity, such that P N (x) passes through the union of all points associated with all attractors Λi , i = 1, 2, . . . , Ω of N . The continuous real-valued function P N (x ) : [0, 1] → [0, 1], henceforth called a “Rule N induced function,” contains each attractor, Λi , i = 1, 2, . . . , Ω, of N as a period-TΛi point. Since P N (x) can be constructed to include also attractors not observed from N , it clearly has much richer nonlinear dynamics. Hence, for finite I, all attractors and invariant orbits of each of the 256 CA rules can be imbedded into a single continuous real-valued function over the unit interval [0, 1].

3.2. A gallery of time-1 maps and power spectrum The qualitative dynamics and long-term asymptotic behaviors of each attractor of a local rule N can often be predicted from one or more of its time-τ maps ρτ [N ], τ = 1, 2, . . . . In fact, a total of 224 out of 256 local rules have attractors that resemble those shown in Fig. 10, or their “compositions”. For an in-depth study of some of these rules in Sec. 5, and in Part V, as well as for future reference, a gallery of the forward time-1 map ρ1 [N ] and the backward time-1 map ρ†1 [N ] of up to three distinct attractors are exhibited in Table 2. For local rules with several qualitatively different attractors, their time-1 maps are printed in different colors. Unlike in Figs. 10 and 11, the points are not labeled to avoid clutter. For each rule N in Table 2, the forward time-1 map ρ1 [N ] is printed in the left column and the backward time-1 map ρ†1 [N ] is printed in the right column. All points with the same color (red, blue, or green) pertain to an attractor of the same color. The power spectrum of the forward time series ϕ of Eq. (26) associated with the red “forward ” time-1 map is calculated using the Mathcad software and printed in the middle column.3 We will see in Sec. 6 that the power spectrum reveals additional valuable and insightful information which cannot be extracted from time-τ maps. 3

387

Table 2 contains 256 three-component frames, henceforth referred to in this paper as CA attractor vignettes, corresponding to the 256 local rules. Each vignette N provides a signature of the type of attractors inhabiting a CA local rule N . Except for the six local rules 15 , 51 , 85 , 170 , 204 , and 240 (to be discussed in Sec. 4.4), whose dynamic patterns are invariant orbits, all other vignettes contain information on “robust” CA attractors. The simplest vignette shows the time-1 map (in red) of only one attractor (e.g. vignette 2 ). In this case, the power spectrum pertains to the forward time-1 map ρ1 [2] depicted in the left column. We will show in Sec. 6 that some spectrum harbors additional albeit nonrobust dynamic modes. Vignette 11 of Table 2 shows two time-1 maps (colored in red and blue, respectively) corresponding to two distinct types of attractors, called Bernoulli attractors, to be analyzed in Sec. 5. In this case, the power spectrum pertains to the red forward time-1 map ρ1 [11] depicted in the left column. Vignette 25 shows three time-1 maps (colored in red, blue, and green, respectively) corresponding to three distinct types of attractors to be analyzed in Sec. 5. In this case, the power spectrum, as always, pertains to the red forward time-1 map ρ1 [25] depicted in the left column. An exception to our 3-color code applies to time-1 maps of rules with a continuum of period-1 and period-2 attractors. Since such attractors are qualitatively similar, only a dull blue color is used to indicate various clusters of period-1 and period-2 points. In addition, the location of two typical period-1 points are identified as solid dots (painted in light red and light blue color) in both forward and backward time-1 maps of such period-1 attractors (e.g. ρ1 [4] and ρ†1 [4] for N = 4). Note that the background color of the power spectrum of all period-1 time-1 maps are painted yellow with only a bold red line emerging at f = 1 signifying the absence of any other frequency components. Similarly, two typical pairs of solid points are painted red and blue at the precise locations where they are located in both forward and backward time-1 maps of period-2 attractors for those rules harboring a continuum of period-2 attractors (e.g. N = 5, 51, etc). All other period-2 points form clusters and are painted in dull blue color. The power spectrum of all period-2 time-1 maps consists of a bold red line located at f = 1/2.

The power spectrum of the corresponding backward time-1 map is qualitatively identical and is therefore redundant.

388


Table 2. Gallery of forward time-1 maps ρ1 [N ] and backward time-1 maps ρ†1 [N ] for attractor Λ1 (red), Λ2 (blue), and Λ3 (green).

Chapter 4: From Bernoulli Shift to 1/F Spectrum Table 2.

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The twin rules (N, N † ), N = 0, 1, 2, . . . , 255, are globally equivalent and listed in Table 1 of [Chua et al., 2004]. It is therefore not surprising that their forward and backward time-1 maps are related.

All power spectra in Table 2 are calculated with I = 450. In some more intricate cases, such as 110 , 54 , etc., a larger value of I ≥ 900 is used. An (I + 1)-bit random bit string generated by a Borland Delphi random function software is used as our initial configuration (i.e. initial state). This bit string is not repeatable in view of its random nature. Since a sufficiently long random bit-string should in principle contain all possible combinations distributed over different portions of the string, we can expect that most of the robust modes of each local rule N will emerge in the subsequent iterations. Indeed, all vignettes in Table 2 are repeatable with different random bit strings. In the case where there are multiple attractors with widely-separated basins of attractions we must repeat our simulations with different carefully chosen initial states. We usually choose initial configurations containing various periodic subconfigurations of different periods. It is important that such choices do not provoke the nonlinear dynamics from escaping into another basin of attraction. To enhance our chances of uncovering most of the robust modes, we usually found it useful to randomize the periodicity and relative positions of the various subconfigurations. To obtain a reliable power spectrum at very low-frequency ranges, we have significantly extended our simulation time for some rules, such as 110 , 137 , etc., in order to obtain a sufficiently long time series of length up to n = 216 = 65536. Such lengthy simulations also call for a corresponding increase in I because f = 1/(I + 1) represents the lowest observable frequency component. For rules in Table 2 which exhibit a 1/f -spectrum, namely, the four universal computing rules 110 , 124 , 137 , and 193 [Chua et al., 2004] discussed in Sec. 6.2, the determination of their low-frequency spectra in Table 2 requires an immense amount of simulation times.

Time-1 map Property 1: Dual mapping Correspondence (1) The forward time-1 map ρ1 [N ] of N is identical to the backward time-1 map ρ†1 [N † ] of N † : ρ1 [N ] = ρ†1 [N † ]

(2) The forward time-1 map ρ1 [N † ] of N † is identical to the backward time-1 map ρ†1 [N ] of N : ρ1 [N † ] = ρ†1 [N ]

Time-1 map Property 2: Bilateral mapping invariance The forward time-1 map ρ1 [N ] and the backward time-1 map ρ†1 [N ] of any bilateral CA rule N are identical. There are 64 bilateral CA rules. They are listed in Table 8 of [Chua et al., 2004]. For all of these rules, their vignettes in Table 2 have identical left and right frames (e.g. 0 , 1 , 4 , 5 , 18 , 19 , etc.).

Following are some fundamental relationships exhibited by time-1 maps between various local rules. Let N † T † [N ] denote the local rule obtained by applying the left–right transformation operator T† to N [Chua et al., 2004]. We will henceforth call N † the lateral twin of N , and vice versa. Time-1 map property 3 is true for all rules except { 57 ,

99

(33)

The proof follows from Eqs. (13), (17), (28) and (29). As an example, compare vignette 110 and its lateral twin vignette 124 in Table 2. Observe the left frame of vignette 110 and the right frame of vignette 124 are identical. Similarly, the left frame of vignette 124 and the right frame of vignette 110 are also identical. It is instructive for the reader to verify the Dual mapping Correspondence by comparing the twin vignettes of all rules in Table 2, thereby obtaining a “constructive”, albeit less rigorous, proof. If N is bilateral in the sense that N † † T [N ] = N , i.e. N is a fixed point of the left– right transformation T† , then we have the following Corollary:

3.3. Three general properties of time-1 maps

4

(32)

Time-1 map Property 3: π-rotation mapping symmetry 4 1. The forward time-1 map ρ1 [N ] of N and the forward time-1 map ρ1 [N ] of N ,

184

,

226

}.

∆

=

T [N ]


453

are related by a 180◦ rotation about the center.5 2. The backward time-1 map ρ†1 [N ] of N and the backward time-1 map ρ†1 [N ] of N T [N ] are related by a 180◦ rotation about the center.

Similarly, the backward time-τ map ρ†τ [N ] : [0, 1] → [0, 1] defined in Eq. (30) (for τ =1) is said to be invertible over [0, 1] iff

Remarks

Remarks

1. The time-1 map property 3 can be verified by inspection of Table 2. 2. The above three properties are stated for time-1 maps for simplicity. The same properties hold also for time-τ maps for all τ . 3. We have verified, by computer simulations, that the above three properties are consistent with all time-1 maps listed in Table 2. It is truly remarkable that such consistency is achieved by using only one random configuration “probing” string for each attractor. 4. Our computer simulation results have provided a resounding validation of Wiener’s brilliant insight of using random signals as probes for nonlinear system characterizations [Wiener, 1958].

1. It is important to keep in mind that each time-τ map is associated with one, and only one, attractor. We will see in Example 3 below that time-τ maps corresponding to different attractors of the same rule N may exhibit different invertibility property. 2. For finite I, the domain of the functions ρτ [N ] and ρ†τ [N ] in Definition 3 must be restricted to a subset of all rational numbers on [0, 1].

3.4. Invertible time-τ maps Since the period TΛ of any attractor of N is the smallest integer where the orbit repeats itself, no two points in the domain of the functions ρτ [N ] and ρ†τ [N ] can map to the same point, it follows that both maps ρτ [N ] and ρ†τ [N ] are bijective, and hence have a well-defined single-valued inverse map [ρτ [N ]]−1 and [ρ†τ [N ]]−1 , respectively. More than half (146 out of 256) of all one-dimensional CA rules exhibit the following important mathematical property which makes the nonlinear dynamics of these rules tractable. Definition 3. Invertible Time-τ map (I = ∞): The

forward time-τ map ρτ [N ] : [0, 1] → [0, 1] defined in Eq. (29) (for τ = 1) is said to be invertible over [0, 1] iff ρτ [N ] = [ρ†τ [N ]]−1 5

(34)

ρ†τ [N ] = [ρτ [N ]]−1

(35)

Geometrical Interpretation of Invertible time-1 maps For τ = 1, the two conditions (34) and (35) are equivalent to the condition that the set of points, henceforth called the graphs of ρ1 [N ], and ρ†1 [N ], in the left and right frames of vignette N , are mirror images (i.e. reflection) of each other relative to the main diagonal. Example 1.

Consider vignette 3 of Table 2. Its left and right frames have only one color (red). Hence 3 has only one robust attractor.6 Since the graph of ρ1 [3] on the left and the graph of ρ†1 [3] on the right of vignette 3 are reflections of each other about the main diagonal, the time-1 maps ρ1 [3] and ρ†1 [3] are invertible. Example 2. Consider vignette 11 . The two colors in the left and right frames imply that 11 has at least two robust attractors. But since both graphs of the same color are mirror images about the diagonal, both pairs of time-1 maps of 11 are invertible. Example 3. Consider vignette 110 . The red color graphs on the left and the right sides of vignette 110 are clearly not mirror images of each other. Hence, the forward time-1 map ρ1 [110] and the backward time-1 map ρ†1 [110] are not invertible.

The symbol T denotes the global complementation operator defined in [Chua et al., 2004]. Indeed, the two forward time-1 maps {ρ1 [N ], ρ1 [N ]} form a two-element Abelian group whose group multiplication operation consists of a 180◦ rotation about the center. Similar property applies to the two backward time-1 maps {ρ†1 [N ], ρ†1 [N ]}. Both are examples of the abstract element group C2 . 6 For each vignette in Table 2, we have shown only time-1 maps of robust attractor prototypes. Many rules have attractors that can only be observed with specially chosen initial configurations.

454


Example 4.

Finally, consider vignette 62 . There are at least two attractors. The graphs of the red color time-1 maps ρ1 [62] and ρ†1 [62] consisting of only three red dots7 are not mirror symmetric about the main diagonal. It follows that the red color forward and backward time-1 maps of 62 are not invertible. In contrast, the blue color time-1 maps ρ1 [62] and ρ†1 [62] consisting of a large ensemble of points exhibit reflection (mirror) symmetry about the diagonal and hence the two blue color time-1 maps of 62 are invertible. Remarks

1. A forward time-1 map ρ1 [N ] is invertible if, and only if, its associated backward time-1 map ρ†1 [N ] is invertible. 2. Since the “composition” between two invertible functions is also an invertible function, it follows that if a forward time-1 map ρ1 [N ], or a backward time-1 map ρ†1 [N ], is invertible, then so are their associated time-τ maps ρτ [N ] and ρ†τ [N ], for any integer τ .

4. Period-k Time-1 Maps: k = 1, 2, 3 In this section we organize local rules into three separate groups based on the global qualitative behaviors of their time-1 maps, which were derived from random initial configurations. Each time-1 map is the outcome of a single random initial state. Unlike the 256 dynamic patterns presented in [Wolfram, 2002] and [Chua et al., 2003], which have no predictive ability because the “probing” input signal consists of only a single red center pixel, the time-1 maps in Table 2 can be used, with complete confidence, to predict the long-term behaviors due to any initial configurations. Time-1 maps are, qualitatively, reminiscent of the classic Green’s function from theoretical physics, the impulse response from linear circuit and system theory [Chua et al., 1987] and the Brownian motion response a la Wiener [Wiener, 1958], where in all cases, a single testing signal is enough to predict the response to any initial configurations.

4.1. Period-1 rules Our research on time-1 maps of period-1 attractors has found that there are a total of 93 (out of 256) one-dimensional CA rules from Table 2 with robust period-1 modes in the sense that almost all random initial states will converge to a period-1 configuration; namely, a fixed point. These 93 rules can be logically partitioned into four distinct families whose members are listed in Tables 3 and 4, respectively. These rules are organized in accordance with the theory of global equivalence class εκm developed in [Chua et al., 2004].8 Since all members of a given equivalence class εκm have identical global dynamical behaviors, it suffices to examine and analyze in depth only one member of each class. Since Table 3 contains 45 rules which exhibit invertible time-1 maps, we will henceforth refer to these rules as invertible rules for simplicity. These rules are invertible because their forward time-1 maps ρ1 [N ] and backward time-1 maps ρ†1 [N ] are identical with respect to both color and position, along the main diagonal, and hence they satisfy Definition 3 in a trivial way. Observe that since there are only 20 global equivalence classes in Table 3, only 20 out of the 45 invertible rules have qualitatively distinct global dynamical response to arbitrary initial states, including transient, attractor, and invariant orbit regimes, and their respective basins of attraction (for attractors). Table 4 contains 24 noninvertible period-1 rules from Table 2. Since they can be partitioned into six global equivalence classes, only six representative noninvertible period-1 rules warrant an indepth analysis. Observe that the rules in Table 4 are noninvertible because each fixed point of ρ1 [N ] in the left frame of vignette N does not map into the same point in ρ†1 [N ] in the right frame of vignette N . For example, the red fixed point in the left frame of vignette 12 and its corresponding fixedpoint in the right frame of vignette 12 are two different points, and hence are not mirror images of each other, relative to the main diagonal. A careful examination of the vignettes in Table 2 corresponding to the 45 time-1 maps from Table 3 reveals that there are 12 rules from Table 3 which must tend to a homogeneous “0” (colored

Note that the domain of the two red color time-1 maps ρ1 [62] and ρ†1 [62] consists of only three rational numbers, obtained by projecting the three red points onto [0, 1]. 8 Throughout this paper, each rule N is coded in red, blue or green color, in accordance with the complexity index [Chua et al., 2002] κ = 1, 2, or 3, respectively. 7


45 Invertible period-1 rules, among them only 29 are bilateral.

455

456


24 Noninvertible period-1 rules.

blue in Fig. 12) attractor and another 12 rules which must tend to a homogeneous “1” (colored red in Fig. 13), for almost all initial states. These 24 homogeneous rules are collected in Tables 5

and 6, respectively. Since all time-1 maps in Table 5 consists of a fixed point at φn = 0• 00, we can predict that all dynamic patterns from the 12 rules in Table 5 must tend to a homogeneous “0”

Fig. 12. All rules belonging to Table 5 tend to a homogeneous blue (“0”) state, regardless of the initial state, chosen randomly. Each pattern has 67 rows and 11 columns.


457

Fig. 13. All rules belonging to Table 6 tend to a homogeneous red (“1”) state, regardless of the initial state, chosen randomly. Each pattern has 67 rows and 11 columns.

Table 5. 12 Homogeneous “0” (blue) Rules. All are invertible but only four are bilateral.

Table 6. 12 Homogeneous “1” (red) Rules. All are invertible but only four are bilateral.

458


(blue) pattern after Tδ iterations. Simulating these 12 rules from a random initial state leads to the 12 dynamic patterns shown in Fig. 12, which confirm our prediction of a homogeneous blue steady state. A similar analysis of the 12 rules in Table 6 shows a common fixed point at φ = 1• 00, which implies a homogeneous “1” (red) steady state response, as confirmed by the simulation results shown in Fig. 13. A comparison of Tables 3 and 5 shows that all nonbilateral (i.e. N = N † ) period-1 rules from Table 3 are members of Tables 5 and 6, which can exhibit only trivial homogeneous “0” and “1”, respectively, patterns. Hence, all invertible nonhomogeneous period-1 rules are bilateral. However, there are 16 invertible but nonbilateral period-1 rules; they all yield trivial homogeneous “0” or “1” patterns and are listed in Tables 5 and 6. As an illustration, the dynamic patterns D N [x(0)] of three invertible (and bilateral ) period1 rules selected from Table 3 are displayed in the left column of Fig. 14; namely, N = 4 , 77 , and 232 . Observe that since 223 in Table 3 belongs to the same global equivalence class ε15 as that of 4 , it has the same qualitative behaviors as 4 [Chua et al., 2004], and need

not be examined. Three additional period-1 patterns chosen from three noninvertible and nonbilateral rules listed in Table 4 ( 44 , 78 , and 172 ) are displayed in the right column of Fig. 14. By the same principle of global equivalence, we can predict that the three rules 100 , 203 , 217 behavior as ∈ ε216 must have the same qualitative 44 . Similarly, the three rules 92 , 141 , 197 ∈ ε35 must have the same qualitative behaviors as 78 , and the three rules 228 , 202 , 216 ∈ 3 ε8 must have the same qualitative behaviors as 172 . Except for the 12 homogeneous “0” rules in Table 5 and the 12 homogeneous “1” rules in Table 6, all other period-1 rules in Tables 3 and 4 consist of clusters of period-1 points distributed over different locations on the main diagonal of the respective vignettes in Table 2. To demonstrate that the three time-1 map properties from Sec. 3.3 hold for all period-1 attractors, two typical period-1 points are highlighted as red and blue dots in each period-1 vignette in Table 2. Observe that the red and blue dots occupy identical positions in the left and the right frames of each vignette for all bilateral period-1 rules (e.g. 4 , 36 , 72 , etc.), as predicted by the bilateral

Fig. 14. Gallery of six period-1 dynamic patterns. The patterns on the left are invertible and bilateral. Those on the right are noninvertible and nonbilateral. Each pattern has 68 rows and 26 columns. The initial configurations (row 0) are chosen randomly.


mapping invariance. Clusters associated with nonbilateral period-1 rules in the left frame are different from those in the right frame of corresponding vignettes, (e.g. 12 , 13 , 44 , etc.). However, the π-rotation mapping symmetry implies that the left frame of vignette N must be identical to the right frame of vignette T [N ], modulo 180◦ rotation about the center. Indeed, the left frame of vignette 12 and the right frame vignette T [12] = 207 are related by a 180◦ rotation, as predicted. While there are many nonhomogeneous period-1 attractors, each represented by a point belonging to some dull blue cluster in Table 2, there is only one attractor (shown in red ) in all, except eight (namely, 40 , 96 , 168 , 224 , 235 , 249 , 234 , and 248 ), homogeneous period-1 rules in Table 2. These eight exceptions, shown in blue, are endowed with a second attractor of a more complicated type (called a Bernoulli σ1 -shift) to be discussed in Sec. 5. Since there are a total of 69 period-1 CA rules (45 in Table 3 and 24 in Table 4), and since 24 among them (12 in Table 5 and 12 in Table 6) have only one period-1 attractors, namely, 12 homogeneous “0” attractors and 12 homogeneous “1” attractors, there are altogether 45 period-1 CA rules having many distinct period-1 points clustered in disconnected groups along the main diagonal in the left and right frames of their associated vignettes in Table 2 (printed in dull blue color). For finite I, these period-1 point are rational numbers on (0, 1). Since rational numbers are denumerable [Niven, 1967], these period-1 points are sparsely distributed and almost every point on (0, 1) are not period-1.

4.2. Period-2 rules An examination of Table 2 shows that there are 17 invertible CA rules possessing period-2 attractors. They are listed in Table 7, organized into 10 global equivalent classes. All 17 rules in Table 7 are bilateral, i.e. N = N † . In addition, there are eight noninvertible CA rules from three global equivalent classes possessing period-2 attractors, as exhibited in Table 8. Observe that all of these rules are nonbilateral. Each period-2 attractor is manifested by two isolated points, symmetrically positioned with respect to the main diagonal in both forward and backward time-1 maps in Table 2. As in the period-1 case, in general there are many distinct period-2 attractors for each period-2 CA rule, and they tend

459

to be organized in various disconnected clusters; they are depicted in dull blue color in each period-2 vignette in Table 2. In addition, two prototype period-2 points are singled out and printed in red and blue colors, respectively, at their precise locations (within the resolution of the printer). The 17 rules in Table 7 are invertible because corresponding points in their forward and backward time-1 maps in the corresponding left and right vignette frames in Table 2 are symmetric, with respect to both color and position, about the main diagonal. Observe also that the left and right vignette frames in Table 2 of all 17 rules in Table 7 are identical, as predicted by the “bilateral mapping invariance” (time-1 map property 2). The eight rules in Table 8 are noninvertible because their forward and backward time-1 maps are not symmetric with respect to the main diagonal. For example, the left and right frames of vignette 28 in Table 2 are not mirror symmetric with respect to the main diagonal. Observe, however, that the right frame of vignette 28 is identical to the left frame of vignette 70 ( 70 = T † [ 28 ]) in Table 2, as predicted by the “dual mapping Correspondence” (time-1 map property 1). Similarly, the left frame of vignette 28 is identical to the right frame of vignette 70 . Observe next that the left frame of vignette 28 and the left frame of vignette 199 = T [ 28 ] are related by a 180◦ rotation about the center, as predicted by the “π-rotation mapping symmetry” (time-1 map property 3). Similarly, the right frame of vignette 28 and the right frame of vignette 199 are related by a 180◦ rotation. As an illustration, the dynamic patterns D N [x(0)] of three invertible (and bilateral ) period-2 rules selected from Table 7 are displayed in the left column of Fig. 15, namely, N = 33 , 51 , and 108 . Observe that since 123 in Table 7 belongs to the same global equivalence class ε210 as that of 33 , it has the same qualitative behaviors as 33 . Three additional period-2 patterns chosen from three noninvertible and nonbilateral rules listed in Table 8 ( 28 , 29 , 198 ) are displayed in the right column of Fig. 15. By the same principle of global equivalence, we can predict that the three rules 70 , 199 , 157 ∈ ε28 must have the same qualitative behavior as 28 . Similarly, rule 71 ∈ ε32 must have the same qualitative behaviors as 29 , and rule 156 ∈ ε240 must have the same qualitative behaviors as 198 .

460


Table 8.

17 Invertible period-2 rules (all are bilateral rules).

8 Noninvertible period-2 rules (all are nonbilateral rules).

Finally, we note from Table 2 that all eight noninvertible and nonbilateral period-2 rules in Table 8 possess an additional form of symmetry; namely, the forward time-1 map ρ1 [N ] is related to the backward time-1 map ρ†1 [N ] by a 180◦ rotation about the center. In other words, the left and right vignette frames of each rule in Table 8 are related by a 180◦ rotation about the origin. We will henceforth

call this rather rare property a self π-rotation symmetry.

4.3. Period-3 rules A comprehensive examination of Table 2 shows that there are only four CA rules that possess robust period-3 attractors, namely, 62 , 118 , 131 , and


461

Fig. 15. Gallery of six period-2 dynamic patterns. The patterns on the left are invertible and bilateral. Those on the right are noninvertible and nonbilateral. Each pattern has 68 rows and 26 columns.

145 . An examination of the vignettes of these four rules in Table 2 show that they are nonbilateral and noninvertible. Since all four rules belong to the same global equivalence class ε222 , as depicted in Table 9, it follows that it suffices to conduct an in-depth analysis of only one of these four rules. Since we have already been exposed to 62 in Fig. 11, let us continue to use this rule for illustrations. Figure 16 shows the dynamic pattern D N [x(0)] of 62 , 118 , 131 , and 145 for different choices of initial states which give rise to qualitatively similar evolution patterns. Observe that each pattern converges to a period-3 attractor after some transient time Tδ whose value depends on the initial states. The presence of a robust period-3 mode in 62 can be predicted from the power spectrum of the forward time series ϕ (defined in Eq. (26)) where a sharp peak centered at f = 1/3 is clearly discernible. Table 9.

In addition to the robust period-3 time-1 maps depicted (in red) in vignettes 62 , 118 , 131 , and 145 , there is a second robust attractor, depicted in blue in Table 2 which can only be understood by examining its associated time-2 map ρ2 [62] to be discussed in Sec. 5. Just like period-1 and period-2 rules, there are many other robust period-3 attractors in 62 , 118 , 131 , and 145 . They are distributed over the unit square as dull blue clusters. Unlike the dynamics patterns associated with period-1 and period-2 rules where different subpatterns evolve independently from one another, and do not interact with one another, we see from 62 that several subpatterns can interact and compete in the sense that one subpattern usually emerges as the winner, after annihilating other competing subpatterns. Such interactions occur in the dynamics of

4 Noninvertible period-3 rules (all are nonbilateral rules).

462


Fig. 16. Evolutions of four globally equivalent rules 62 , 118 , 131 , and 145 . All patterns consist of 82 rows and 66 columns.


62 because it can support subpatterns which propagate to the left, and to the right, thereby colliding with each other in due time.

4.4. Invariant orbits For finite I, there can be only a finite number TΛ of points on the forward and backward time-1 maps of any rule N . This implies that the domain of time-1 maps are generally extremely sparse even for rules with a very large period TΛ . Geometrically this means that a vertical line drawn through an arbitrary point φn−1 ∈ (0, 1) will almost never intersect the time-1 map of most rules. In other words, one expects to see many gaps in most time-1 maps in Table 2. A careful examination of all 256 rules in Table 2 reveals that there are six, and only six rules, which have a dense time-1 map; namely, { 15 , 51 , 85 , 170 , 204 , 240 }. Observe the graph of the time-1 map of these six rules project to almost all points on the unit interval [0, 1]. In the limit, these points actually tend to a continuum so that the time-1 map ρ1 [N ] coincides with the characteristic function χ1N . In this case, every point φn−1 ∈ [0, 1] is a point on a periodic orbit of N , and there are no transient regimes in these six rules. We have therefore the following Proposition. All dynamic patterns D N [x(0)] of

the six rules 15 , 51 , 85 , 170 , 204 , and 240 , belonging to the local equivalence class S41 are invariant orbits.

5. Bernoulli στ -Shift Rules In addition to the 98 rules we have listed so far (45 invertible period-1 rules in Table 3, 24 noninvertible period-1 rules in Table 4, 17 invertible period-2 rules in Table 7, eight noninvertible period-2 rules in Table 8, and four noninvertible period-3 rules in Table 9) where we can predict their global longterm dynamical behaviors, there are 112 additional rules whose attractors can be precisely predicted by invoking the remarkably simple symbolic dynamics exhibited by the well-known Bernoulli shift map [Tang et al., 1983; Nogashima & Baba, 1999]. In particular, we will show in this section that the

463

evolution of each initial configuration of these 112 rules can be predicted by shifting it either to the left, or to the right, by 1, 2 or 3 pixels, and possibly followed by a complementation (i.e. change of color).

5.1. Gallery of Bernoulli στ -shift rules Among the 256 CA rules listed in Table 2, there are 112 rules, henceforth called “Bernoulli σ τ -Shift Rules”, which have simple Bernoulli-shift dynamics. They are extracted from Table 2 and reorganized into three separate Tables. Table 10 contains 84 invertible Bernoulli στ -shift rules, organized as members of 24 global equivalence classes εκm , and listed in column 1. Observe that the second attractor (blue in Table 2) of the four period-3 rules 62 , 118 , 131 , 145 belonging to ε222 are members of Table 10 and hence are also Bernoulli rules. The four members of each equivalence class are listed in columns 2–5. The color chosen for each rule in these columns follows the same code for the complexity index [Chua et al., 2002]. The index κ for each class is listed in column 6. Since some rules in Table 10 have 2 Bernoulli9 attractors, this information is indicated by an “ ” sign in columns 7 and 8, depending on whether the forward time-1 map ρ1 [N ](φn−1 → φn ), or the forward time-2 map ρ2 [N ](φn−2 → φn ) is required to represent the attractor in order to uncover its Bernoulli-shift attractor. The number of Bernoulli attractors for each rule N is listed in column 9. The last column 10 provides an index of table number (13-1 to 13-9) where the characterizing features of each equivalence class belonging to Table 10 can be found. Table 11 contains 20 noninvertible Bernoulli στ -shift rules with two Bernoulli attractors, organized in the same format as Table 10. In this table, all rules have two Bernoulli attractors. The relevant Bernoulli στ -shift maps in this table consist of either the forward time-2 map ρ2 [N ](φn−2 → φn ), or the forward time-3 map ρ3 [N ](φn−3 → φn ). Table 12 contains eight noninvertible Bernoulli στ -shift rules with three Bernoulli attractors displayed in a similar format except for two new columns replacing the former “attractor-number”

9 To avoid clutter, we will henceforth refer to a Bernoulli στ -shift map, rule, or attractor simply as a Bernoulli map, Bernoulli rule or Bernoulli attractor, respectively. We also abuse our language and use the same name “Bernoulli attractor k ”, k = 1, 2, 3, to mean the kth family of attractors which share the same qualitative global dynamics, such as “shift left by 2 pixels”.

464


Table 10.

84 Invertible Bernoulli στ -shift rules with one or two Bernoulli attractors.


20 Noninvertible Bernoulli στ -shift rules with two Bernoulli attractors.

Table 12.

8 Noninvertible Bernoulli στ -shift rules with three Bernoulli attractors.

column due to space limitation. Observe that there are four distinct Bernoulli στ -shift maps represented in Table 12, namely, the forward time-1, time-2, time-3 and time-5 maps ρτ (φn−τ → φn ), τ = 1, 2, 3, 5. In order to state the Bernoulli shifting algorithm in an unambiguous way, we have collected all those rules from Table 10 which evolve in accordance with the same shifting mode into the same group, and have identified each by a shift-mode ID code B N [α, β, τ ] in column 1 of Table 13. All Bernoulli rules from Table 13 obeying the same ID code are listed in the rightmost column 5 of Table 13. Since all rules having the same ID code B N [α, β, τ ] exhibit a qualitatively similar power spectrum, the spectrum of only the first member N of each group is chosen as a prototype. For example, N = 2 and 16 in Table 13-1, 11 in Table 13-3, 14 in Table 13-4, . . . and 62 in Table 13-9. The power spectrum and the forward time-1 map ρ1 [N ](φn−1 → φn ) from vignette N of Table 2 are reproduced in columns 2 and 3, respectively, of Table 13. In addition, the power spectrum from column 2 is partitioned into a low (red), mid (green), and high (blue) frequency range in column 4, where various characteristic features of rule

465

N are identified and annotated. The bold color line segments shown in some of the annotated spectrum represent the average spectrum calculated over a narrow range via a least-square method. For some rules, such as 58 , 3 , 17 , 35 , . . . , etc., the forward time-2 map ρ2 [N ](φn−2 → φn ) is plotted in column 3 instead of ρ1 [N ](φn−1 → φn ) because it is this map which reveals its Bernoulli character. Indeed, the forward time-1 maps of these rules do not reveal any interesting features! The hidden Bernoulli character of these maps emerges, however, as soon as one glances at the τ = 2 characteristic function χ2N . A careful analysis of the forward time-τ maps in column 3 of Table 13 shows that all points (red dots) from the time-τ map (left frame) of vignette N of Table 2 fall exactly on parallel light blue lines with a slope 1 1 1 (36) β ∈ ± , ± , ± , ±2, ±4, ±8 8 4 2 For reasons that will soon be obvious, we will henceforth refer to the parallel light blue lines of each Bernoulli rule N listed in Table 13 as the Bernoulli στ -shift map of N , even though only a subset of

Table 13-1.

B N [α, β, τ ] characterization for 84 invertible Bernoulli rules with one or two Bernoulli attractors.

466

Table 13-2.

(Continued )

467

Table 13-3.

(Continued )

468

Table 13-4.

(Continued )

469

Table 13-5.

(Continued )

470

Table 13-6.

(Continued )

471

Table 13-7.

(Continued )

472

Table 13-8.

(Continued )

473

Table 13-9.

(Continued )

474


these lines belongs to the graph of the corresponding time-τ maps. It is truly remarkable that the forward time-τ map of all Bernoulli rules in Table 13 is always a subset of a Bernoulli στ -shift map with τ ∈ {1, 2}. This amazing result can be confirmed by exhaustive computer simulations of all 256 CA rules. A cursory examination of Table 13 shows that some rules N , such as 11 , 14 , 56 , etc., occur twice, one for each attractor in N . To distinguish between these two attractors, we introduce an integer α ∈ {1, 2}, where α = 1 if N has only one attractor, and α = 2 if it has 2. Let us summarize the above coding scheme as follow: The ID code B N [α, β, τ ] of each Bernoulli shift mode in Table 13 is uniquely identified by three parameters {α, β, τ }, where α ∈ {1, 2} denotes the number of attractors in N , β (defined in Eq. (36), denotes the slope of the Bernoulli στ -shift map (parallel light blue lines), and τ denotes the relevant forward time-τ map shown in column 3. Following exactly the same organization format, we reorganize the 20 noninvertible rules from Table 11 into Table 14. Since all rules from Table 11 have two attractors, α ∈ {1, 2} in Table 14. Observe, however, that unlike Table 13 where τ ∈ {1, 2}, we now have τ ∈ {2, 3}. In other words, the red points in column 3 are not merely copied from Table 2, but must now be calculated. Finally, the eight noninvertible rules in Table 12 are reorganized into Table 15. Since all rules belonging to Table 12 have three attractors, we have α ∈ {1, 2, 3}. As a departure from the previous format, a new column 5 is added in Table 15 displaying a typical dynamic pattern for each attractor in order to demonstrate that the attractors have qualitatively distinct characters, each one having a different basin of attraction. Observe also that τ ∈ {2, 3, 5} in Table 15.

5.2. Predicting the dynamic evolution from {β, τ } The dynamic evolution of any one of the 112 Bernoulli rules from Tables 10–12 can be predicted uniquely from only two parameters in view of the following theorem:

475

στ -Shift Theorem Let B N [α, β, τ ] be the ID code of Bernoulli rule N . Let Theorem 1.

st {xt0

xt1

xt2

···

xtI }

(37)

denote any (I + 1)-bit initial state (configuration) and let st+τ {xt+τ 0

xt+τ 1

xt+τ 2

···

xt+τ I }

(38)

denote the evolved state of N at time t + τ , τ = 1, 2, 3, . . .. Then st+τ can be derived from the following: στ -Shifting rule Case 1. β = 2σ > 0,

τ = n,

n = 1, 2, 3, . . .

(a) σ = 1, 2, 3, . . . st+n is obtained by shifting st to the left by “σ” pixels. (b) σ = −1, −2, −3, . . . st+n is obtained by shifting st to the right by “|σ|” pixels. Case 2. β = −2σ < 0,

τ = n,

n = 1, 2, 3, . . .

Same as Case 1 but followed by complementing the color of all pixels.

Proof.

Due to space limitation, the formal proof of this theorem will be given in Part V.

Applying the στ -shifting rule from the above σ τ -shifting rule theorem, we obtained the explicit σ τ -shifting dynamics in Table 16 for each of the 112 Bernoulli rules from Tables 10–12. Each row in Table 16 spells out the precise instruction for predicting the first, second, third, or fifth iteration of any Bernoulli rule N (with ID code B N [α, β, τ ]) (from any initial state (configuration) st on any attractor) corresponding to τ = 1, 2, 3, or 5, respectively. If N has more than one attractor, one row is devoted to each attractor and the appropriate row to pick depends on which basin of attraction does the initial state belong to. To predict the next

Table 14-1.

B N [α, β, τ ] characterization for 20 noninvertible Bernoulli rules with two Bernoulli attractors.

476

Table 14-2.

(Continued )

477

Table 14-3.

(Continued )

478

Table 14-4.

(Continued )

479

Table 14-5.

(Continued )

480

Table 15-1.

B N [α, β, τ ] characterization for eight noninvertible Bernoulli rules with three Bernoulli attractors.

481

Table 15-2.

(Continued )

482

Table 16.

Explicit στ -shift dynamics for Bernoulli Rules.

483

Table 16.

(Continued )

484

Table 16.

(Continued )

485

Table 16.

(Continued )

486


τ -iteration from any binary bit-string initial state (configuration) st on any attractor of any Bernoulli rule N , one simply transcribes the dynamics specified from the entries marked by a “cross” ( ) in row N . Let us use this direct “read-out” procedure to predict the attractor evolution dynamics of the four Bernoulli rules N = 74 , 99 , 85 and 11 from the dynamic patterns D N [x(0)] exhibited in Fig. 17. Let st denote any row on the attractor regime (i.e. pick t > Tδ ) of these four patterns. The dynamical evolution of 74 from st can be predicted by looking at the first subrow (corresponding to attractor α = 1) of row N = 74 of Table 16 and read out the following precise evolution rules for 74 : Shift string st to the left by 1 pixel to obtain the first iteration st+1.

(39)

Repeating the same procedure we obtain the same pattern shown in the upper left corner of Fig. 17. Similarly, if we go to the second subrow (corresponding to attractor α = 2) of row N = 99 in Table 16, we would read out the following evolution rule for 99 : Shift string st to the right by 1 pixel to obtain the first iteration st+1.

(40)

Repeating the same procedure we obtain the periodic pattern (after the transient regime) shown in the upper right corner of Fig. 17, where the highlighted area denotes the attractor regime. For N = 85 , we read out from row N = 85 in Table 16: Shift string st to the left by 1 pixel and then complementing (changing color) to obtain the first iteration st+1.

(41)

Repeating the same procedure we obtain the dynamic pattern in the lower left corner of Fig. 17. Finally, For N = 11 , we read out from subrow 2 (corresponding to attractor α = 2) of row N = 11 in Table 16: Shift string st to the right by 1 pixel and then complementing (changing color) to obtain the first iteration st+1. 10

487

Repeating the same procedure we obtain the pattern in the lower right corner of Fig. 17. The four Bernoulli rules chosen in Fig. 17 are all described by a forward time-1 map ρ1 [N ](φn−1 → φn ); i.e. τ = 1. Consider next the four rules 74 (α = 2),10 3 , 6 , and 9 shown in Fig. 18 where τ = 2. For 74 , we go to the second sub-row of row N = 74 in Table 16 to read out: Shift string st to the left by 2 pixels to obtain the second iteration st+2.

(43)

Repeating the same procedure we obtain the same pattern shown in the upper left corner of Fig. 18, where only even rows are printed out for easier verification of the above evolution procedure. In order to generate all rows, we would need to iterate also from row st+1 . The other three patterns in Fig. 18 are obtained by exactly the same read out procedure. Finally, Fig. 19 shows the dynamic patterns of four Bernoulli rules 74 (α = 3), 9 , 25 (α = 2) and 25 (α = 3) with τ = 5. To predict the periodic pattern of 74 (α = 3), we go to the third subrow of row N = 74 and read out: Shift string st to the right by 3 pixels to obtain the third iteration st+3 .

(44)

Repeating the same procedure we obtain the pattern shown in the upper left corner of Fig. 19, where only every third row is printed out for easier verification. Clearly, in order to obtain the entire attractor, it is necessary to repeat the above procedure from rows st+1 and st+2 as well. To appreciate the predictive power of the στ -shifting rule, it would be instructive to apply Table 16 to all 112 Bernoulli rules exhibited in Table 5 of Part II [Chua et al., 2003].

5.3. Two limiting cases: Period-1 and palindrome rules

Notice that σ = 0 is not included in the above formulation of the στ -shifting rule. It is possible (42) to enlarge the class of Bernoulli rules significantly by including σ = 0 as a limiting case without

Although we have chosen the same rule 74 from Fig. 17, our initial state here belongs to the basin of attraction of attractor α = 2.

488


Fig. 17.

Dynamic patterns of Bernoulli rules 74 , 99 , 85 , and 11 . The dimension of each pattern is 66 rows × 84 columns.


489

Fig. 18. Dynamic patterns of Bernoulli rules 74 (α = 2), 3 , 6 , and 9 . The dimension of each pattern is 66 rows × 84 columns. Only alternate rows n = 0, 2, 4, . . . are shown.

490


Fig. 19. Dynamic patterns of the four Bernoulli rules 74 (α = 3), 9 , 25 (α = 2) and 25 (α = 3). Successive rows represent three iterations in all cases except 25 (α = 3) where only every fifth iteration is printed. The dimension of each pattern is 66 × 84 for 74 (α = 3) and 25 (α = 2), 67 × 84 for 9 , and 65 × 84 for 25 (α = 3).


changing the original statements of the στ -shifting rule. Limiting Case 1: β = +1 = 20

In this case, since σ = 0 and β > 0, both statements (a) and (b) in Case 1 of the στ -shifting rule imply that no pixel is shifted either left or right. In other words, any point on an attractor of N which follows the στ -shifting rule with β = +1 is a fixedpoint, and hence a period-1 attractor. Conversely, any period-1 point on a period-1 attractor satisfies the στ -shifting property with σ = 0. We can thereby conclude: Corollary 1. The dynamics of all period-1 rules

from Tables 3–6 satisfies the στ -shift rule with σ = 0, for all period-1 initial states. Limiting Case 2: β = −1 = −20

In this case, since σ = 0 and β < 0, both statements (a) and (b) in Case 2 of the στ -shifting rule are satisfied, implying that although any initial state st on an attractor does not shift its position, it changes sign in each iteration. This implies that st is a period-2 point. The converse, however, is not true. In fact, only certain rather special period-2 patterns exhibit the above “sign-alternating” property. In particular, the following result can be proved to be the only period-2 rules satisfying limiting case 2: Corollary 2. Period-2 Palindromes: All period-2

rules which obey the στ -shift property are necessarily palindromes in the sense that any initial configuration string st corresponding to an attractor must be symmetrical with respect to the center of st .11 Among the 25 period-2 rules listed in Tables 7 and 8, only the seven bilateral rules { 19 , 23 , 50 , 51 , 55 , 178 , 179 } are palindromes and only these seven rules obey the στ -shift property with β = −20 .

491

is a subset of a multivalued function consisting of 2|σ| parallel straight lines with slope equal to β = 2−σ , σ = 1, 2, 3. This observation seems to contradict the fact that all time-τ maps must be single-valued functions on the unit interval. This paradox can be resolved by observing that every point on the graph must project to a different point on the horizontal axis. In other words, if we draw very thin projection lines through all points on the graph, as in Figs. 3–7, these lines will merely interleave each other. This will always be the case even as I → ∞; namely, between every two projection lines, there will be more lines, sandwiched in between, ad infinitum. It is mind boggling to imagine a function with such an intricate structure [Niven, 1967]. How can one guarantee the iteration of such Bernoulli rules for large values of I can be reliably carried out on a computer? How can one be sure that the inevitable computing errors due to truncation will not affect the outcome for large values of I? The answer to the above questions comes from our στ -shifting rule. For |β| < 1, st+1 is always obtained by shifting st to the right by |σ| = 1, 2 or 3 pixels. Hence every bit of st will eventually arrive at the right boundary located at φn−τ = φend ≈ 1.0 (depending on I). If the end bit is equal to a “1”, such as φt = 0• 001010 · · · 0101

(45)

then the στ -shifting rule and the periodic boundary condition in Fig. 1 imply (assuming |σ| = 1) the following unambiguous outcomes: φt+1 = 0• 1001010 · · · 010

(46)

φt+2 = 0• 01001010 · · · 01

(47)

t+3

φ

= 0• 101001010 · · · 0

(48)

.. . etc.

5.4. Resolving the multivalued paradox A cursory glance of Tables 13–15 reveals that the graph of the forward time-τ map ρτ [N ] of all Bernoulli rules with β < 1 (equivalently, σ < 0)12

It is now clear that the rightmost bit at time t determines the leftmost bit at time t + 1, assuming |σ| = 1. Hence, there is never any loss of accuracy because the computation is discrete, not continuous.13

11 The palindromes st in Corollary 2 are assumed to be represented as binary bit strings. Corollary 2 is also true for a decimal st as I → ∞. 12 For each Bernoulli rule B N [α, β, τ ] with β = 2σ > 0, its bilateral twin B † [α, β † , τ ] has β † = 2−σ < 1. N 13 Assuming the computer has a sufficiently long word length.

492


It is instructive to illustrate the above right shifting rule on rule 240 whose characteristic function χ1240 is shown in Fig. 4(b). Now recall the first few digits of the decimal expansion of a binary bit string →

x = [x0 x1 · · · xI−1 xI ]

(49)

is given by Eq. (13); namely, 1 1 1 1 1 φ = x0 + x1 + x2 + · · · + I xI−1 + I+1 xI 2 4 8 2 2 = 0.5x0 + 0.25x1 + 0.125x2 + · · · 1 1 (50) + I xI−1 + I+1 xI 2 2 It follows from Eqs. (49) and (50) that if the leftmost bit is a “0”, i.e. x0 = 0, then φ < 0.5 and the lower straight line (with slope β = 1/2) in Fig. 4(b) for 240 will be selected. On the other hand, if the leftmost bit is a “1”, i.e. x0 = 1, then Eq. (50) implies that φ > 0.5 and hence the upper branch in Fig. 4(b) will be selected.

In other words, if the end (rightmost) bit is = 1 at time t, then the Bernoulli σ1 -shifting rule of 240 will shift the end bit “1” to the right, thereby reappearing as the first bit in the next iteration in view of the periodic boundary condition indicated in Fig. 1(a). Since the first bit in the next =1, we have φt+1 > 0.5 iteration now reads xt+1 0 and the dynamics must follow the upper branch of χ1240 . Conversely, if the end bit is xtI = 0, then the Bernoulli right shifting rule for 240 will shift the = 0 in bit “0” to make the first bit equal to xt+1 0 the next iteration, and the dynamics must follow the lower branch of χ1240 . The right shifting rule of 240 dynamics described above is illustrated with the help of the Lameray (cobweb) diagram shown in Fig. 20. 1 and 6 are extremely Observe that the two points close to each other. The decimal coordinate xtI

φ0 = 0.673768048097057 . . . 1 is calculated from the following 66of point bit string (the same I = 65 is used throughout Figs. 3–7)

101011000111110000010000000100111010101010100111010110010101110101 via Eq. (50). Observe the end (rightmost) bit of the above bit string is a “1”. To obtain the next iteration via the σ1 -shifting rule for N = 240 in Table 16, we simply shift the above bit string by one pixel (since τ = 1) to the right (since σ< 0),

which, in view of the periodic boundary condition depicted in Fig. 1(a), is equivalent to inserting a “1” at the leftmost position of the above right-shifted string to obtain the following 66-bit string

110101100011111000001000000010011101010101010011101011001010111010 in the next iteration, whose decimal value (calculated from Eq. (50)) is equal to φ1 = 0.836884024048529. . . It is truly amazing that while the two decimal numbers φ0 and φ1 above reveal no discernible relationship between them, their binary codes betray the hidden secret of rule 240 as simply a trivial Bernoulli right shift of one pixel! Note the cobweb diagram starting from the 6 evolves into an entirely different nearby point orbit, a manifestation of extreme sensitivity. Indeed it is well known that the Bernoulli shift (for I → ∞) 14

is as chaotic as a coin toss, [Nagashima & Baba, 1999] and its chaotic attractor has a Lyapunov exponent [Devaney, 1992] λ=β=2>1

(51)

To understand how the outcome of the Bernoulli rule 240 emulates an ideal “coin toss” Gedanken experiment, let us look at the evolution of the “inverse” Bernoulli rule 170 which must have identical dynamics,14 as I → ∞. The characteristic function χ1240 : [0, 1] → [0, 1] of the Bernoulli rule 170 in Fig. 4(a) can be

For an infinitely long bit string (I → ∞), it is more illuminating to show that there is a one-to-one correspondence between the iterates of the Bernoulli rule 170 and the outcome of an ideal coin-toss.


493

i undergoing 1 , 2 , 3 . . . , 9 10 Fig. 20 Cobweb diagram of Bernoulli rule 240 showing a succession of ten iteration points , 1 and 6 appear as a single point since they differ by only 0.003537. the σ1 -shifting evolution dynamics. Points

described analytically by χ1170 = 2 φ 170 mod 1

(52)

for all φ 170 ∈ [0, 1]. This means that every point of the unit interval [0, 1] corresponds to a semi-infinite binary bit string: [x0 x1 x2 · · · xi · · · xI−1 xI ] → 0• x0 x1 x2 · · · xI−1 xI

(53)

where I → ∞. Now since, except for a set of measure zero (corresponding to the set of all rational numbers), every point in (0, 1) is an irrational15 number [Niven, 1967] whose binary expansion can be identified with a particular coin toss experiment, the ensemble of all possible ideal coin-toss experiments must correspond to the set of all points on [0, 1]. Hence, to exhibit any member of this coin-toss ensemble using the Bernoulli rule 170 , we simply choose an arbitrary point from the unit interval [0, 1]; namely, Eq. (53), and apply the Bernoulli left shifting rule 15

σ1 [ 170 ] to read out the first digit xt+n from each 0 iteration t + n, n = 1, 2, . . . , ∞. The outcome of this binary output string is clearly a member of the ideal coin-toss ensemble. It is in the above sense of a Gedanken experiment that we claim the Bernoulli rule 170 , and its inverse rule 240 , is as chaotic as an ideal coin toss, as I → ∞. It also follows from the above discussion that the Bernoulli rules 170 and 240 are both ergodic [Billingsley, 1978] in the sense that the iterates φ1 , φ2 , . . . , φt from almost every initial state φ0 ∈ [0, 1] would visit every point, infinitely often, on the two parallel lines (with slope β = 2 for 170 and slope β = 1/2 for 240 ) of the characteristic functions χ1170 and χ1240 in Fig. 4 as t → ∞. This is the reason why the time-1 map ρ1 [170] and the characteristic function χ1170 (resp. ρ1 [240] and χ1240 ) are identical functions. In other words, there are no gaps in the graph of ρ1 [170] . An examination of Table 2 shows that there are two other Bernoulli rules whose graphs also coincide

The hallmark of an irrational number is that its binary expansion in Eq. (53) contains every possible finite sequences of bit “0” and bit “1” infinitely often [Billingsley, 1978].

494


with their characteristic functions; namely, rule 15 and 85 . An examination of Table 2 shows that only four out of 112 Bernoulli rules are ergodic over the unit interval [0,1]. All other Bernoulli rules have gaps,16 as is evident from their graphs (indicated by red or blue dots) in Tables 13–15. The four ergodic Bernoulli rules 15 , 85 , 170 , and 240 are invariant in the sense that their orbits are invariant orbits (recall Sec. 4.4). Observe that although their characteristic functions χ1N have only two fixed points, there are infinitely many fixed points χτN as τ → ∞. This follows from Eq. (54) (for 170 ) that the graph of the τ th-iterated characteristic function χτ170 = 2τ φ 170 mod 2

(54)

consists of 2τ parallel lines with slope β = 2τ .17 Since each intersection of these lines with the main diagonal is a fixed point of χτ170 , it follows that as τ → ∞, the number of fixed points tend to infinity. Note, however, that the fixed points of all Bernoulli rules with β > 0 are unstable because the slope of their characteristic function at these points have a slope β > 1. Hence, unless one chooses the exact coordinates of these fixed points, they are not observable. Such fixed points are therefore said to be not robust.

6. Predictions from Power Spectrum The graph of the time-1 map of each attractor in Table 2 is derived using a randomly-generated initial configuration. The fact that the same graphs are generated as long as the random configurations are in the corresponding basins of attraction confirms the validity of using randomly-generated bit strings as “probing” inputs [Wiener, 1958]. In addition to providing the graphs of both forward and backward time-1 maps for each vignette in Table 2 we have also recorded the power spectrum for the forward time-1 map ρ1 [N ] and displayed it in the center frame of each vignette. The spectra of the period-1, period-2 and period-3 rules in Table 2 do not provide any new information. They merely confirm the periodicity of the attractors. In this final section, we will examine the spectra of the 112 Bernoulli rules in Tables 10–12. 16

6.1. Characteristic features of Bernoulli rules The power spectrum of one prototype member of each group B N [α, β, τ ] of Bernoulli rules having identical Bernoulli σ1 -shift maps is displayed in Column 2 of Tables 13–15, respectively. A careful analysis of these spectra reveals certain generic features of all Bernoulli rules belonging to each of the 34 distinct groups B N [α, β, τ ] listed in Table 13– 15. Such generic features include the presence of various robust periodic modes, as well as the rate of increase or decrease of the power spectrum at various frequency ranges. This information is highlighted and annotated in column 4 of each group in Tables 13–15. The presence of a sharp spike at some frequency fp indicates that the Bernoulli rule has a robust natural oscillating mode at this frequency. As a demonstration of its prediction ability, Fig. 21 shows the power spectrum of Bernoulli rules 14 and 81 . An inspection of these two spectra reveals a spike at f = 1/4 in both cases. These spikes imply the presence of a stable and hence robust period4 point. The location of the four period-4 points of 14 are identified on the characteristic function χ114 , in Fig. 22(a). Observe that the points on the two attractors associated with B 14 [1, 2, 1] (red) and B 14 [2, −1/2, 1] (blue) of the Bernoulli rule 14 in Fig. 21, are found at the tip of a sub-group of vertical lines in Fig. 22(a), as expected. Observe that there are many points in Fig. 22(a) which do not lie on the attractors. Here, we have superimposed the two red parallel lines from Fig. 21 where attractor 1 resides, and the two blue parallel lines where attractor 2 resides onto χ114 in Fig. 22. It is interesting to note 1 , 2 , 4 3 that although the four period-4 points , in Fig. 22 lie at the intersection of these two sets of parallel lines, they are not a part of attractors 1 and 2 because points belonging to two different attractors cannot intersect, by definition of an attractor. Note that it is generally not possible to identify period-τ points of N directly from the characteristic function χ1N unless τ = 1. A much more effective way to search for period-τ points of any rule N is to plot the graph of the τ th-iterated characteristic function χτN and look for points on χτN which lie on the main diagonal χτN = φ N . As

We conjecture that the set Λi [N ] corresponding to each attractor α of all noninvariant Bernoulli rules are Cantor sets [Alligood et al., 1996]. 17 See B 25 [2, 8, 3] in Table 15-1 for an example when τ = 3.

495 Fig. 21.

Spectra of Bernoulli rules 14 and 81 and their σ1 -shift map showing locations of four period-4 fixed points of ρ414 and ρ481 , respectively.

496


(a)

(b) Fig. 22.

Characteristic function

χ114

(top) and time-1 map ρ114 (bottom) of Bernoulli rule 14 .


(a)

(b) Fig. 23.

Fourth-iterated characteristic function χ414 (top) and time-4 map ρ414 (bottom) of Bernoulli rule 14 .

497

498 Fig. 24. The power spectrum of four global equivalent rules 110 , 124 , 137 , and 193 capable of universal computation all exhibit a 1/f power-frequency characteristics.


an illustration, we plot χ414 in Fig. 23 (top) and note that indeed there are four points, labeled 1 2 , 3 4 , , and , which lie precisely on the main diagonal, and the coordinates of these four points are precisely those identified earlier in Fig. 22, as expected. It is even more illuminating to examine the time-4 map ρ4 [14] of 14 , shown in the bottom of Fig. 23. This is a map showing ρn−4 → ρn , i.e. only every fourth iterates of χ114 are printed. Observe that, by definition, all points of χ414 must lie on the 16 parallel lines of slope β = 24 = 16 for attractor 1, and β = 2−4 = 1/16 for attractor 2. Observe that although we were able to identify the precise location of the four period-4 points, we would not have undertaken the time-consuming procedure had we not known that a period-4 point must exist for both Bernoulli rules 14 and 81 . The power-spectrum therefore provides valuable clues on what to look for.

6.2. Turing-universal rules: { 110 , 124 , 137 , 193 } exhibit 1/f power-frequency characteristics A careful examination of the power spectrum of all 256 rules in Table 2 reveals that the four globallyequivalent Turing universal rules 110 , 124 , 137 , and 193 , and only these four rules, exhibit a 1/f power-frequency characteristics with a slope equal to approximately −1.5, as exhibited in Fig. 24 [Schroeder, 1991]. This interesting observation suggests that there might exist a fundamental relationship between universal computation and the ubiquitous 1/f phenomena.

7. Concluding Remarks We have completely characterized the long-term (time-asymptotic) behaviors of 206 one-dimensional CA rules with three inputs. Each CA rule can have several attractors and invariant orbits. A single randomly chosen initial state (configuration) is used as a probe to determine uniquely the precise characteristics of the attractor whose basin of attraction contains the “probing” random configuration. A CA rule N is either bilateral (when T † [N ] = N ), or nonbilateral. It can be either invertible (when its forward and backward time-1 maps are symmetrical with respect to the main diagonal) or noninvertible.

499

There are 45 invertible and 24 noninvertible period-1 rules. Each period-1 rule generally has a continuum of period-1 attractors (as I → ∞), clustered along the main diagonal. Among the period-1 rules, there are 12 rules which always tend to the homogeneous “0” attractor, and another 12 rules which always tend to the homogeneous “1” attractor, regardless of the initial state (configuration), except for the isles of Eden states possessed by 40 , 96 , 235 , and 249 . There are 17 invertible period-2 rules all of which are bilateral. There are also eight noninvertible period-2 rules, all of which are nonbilateral. Each period-2 rule generally has a continuum of period-2 attractors (as I → ∞), clustered symmetrically with respect the main diagonal. There are four nonbilateral period-3 rules which can be either invertible or noninvertible. There are 112 Bernoulli rules whose asymptotic behavior is completely characterized by a στ -shifting rule. The period-4 attractor exhibited by 14 and 81 represents a very interesting bifurcation point separating two distinct attractors that warrant further in-depth analysis. The remaining 50 rules consist of 18 noninvertible but bilateral rules (listed in Table 17) and 32 noninvertible and nonbilateral rules (listed in Table 18). The qualitative long-term dynamics of these rules will be studied in Part V. An in-depth analysis on the characterizations of the long-term behaviors of these rules represents challenging future research problems. By invoking the global equivalence principle developed in [Chua et al., 2004], the above list of 50 currently intractable rules reduces to the study of only 10 noninvertible but bilateral rules and only eight noninvertible and nonbilateral rules. A compendium of the characteristic properties and relevant data of all 256 CA rules are collected in Table 19. Except for the 18 rules listed in Table 17, the attractors of all bilateral rules have been completely characterized and annotated in appropriate columns of Table 19. In addition, all invertible attractors of CA rules are completely characterized. Such attractors are closely related to the time-reversal concept from physics, and will be discussed in-depth in Part V. It is important to emphasize that Table 19 can be used to predict, by inspection, the global asymptotic dynamics (as t → ∞) from any initial state (configuration) belonging to any robust attractor

500


Table 18.

18 Noninvertible bilateral rules.

32 Noninvertible nonbilateral rules.

Table 19.

Summary of characteristic properties and identification data for 256 CA rules.

501

Table 19.

(Continued )

502

Table 19.

(Continued )

503

Table 19.

(Continued )

504

Table 19.

(Continued )

505

Table 19.

(Continued )

506


listed in this table. Moreover, for the six “orbitinvariant ” rules 15 , 51 , 85 , 170 , 204 and 240 , Table 19 actually predicts their complete dynamical evolutions over all times, i.e. both transient and steady state regimes, and for all initial states. This follows by default because all orbits of these six rules are invariant, and therefore do not have a transient regime. The main result of this paper is no doubt the gallery of graphs of both forward and backward time-1 maps of all 256 CA rules. Since these graphs

507

do not depend on the initial state (configuration), they completely characterized the long-term asymptotic behaviors of all rules, including the 50 complex rules listed in Tables 17 and 18. They are, in some sense, the generalized Green’s functions for cellular automata. Perhaps the most intriguing unsolved problem is to discover the relationship between the four Turing-universal rules 110 , 124 , 137 , and 193 , and the ubiquitous 1/f power spectrum exhibited by these four rules, and only these rules.


♦

Chapter 5 FRACTALS EVERYWHERE ♦

Ë This fifth installment is devoted to an in-depth study of CA Characteristic Functions, a unified global representation for all 256 one-dimensional Cellular Automata local rules. Except for eight rather special local rules whose global dynamics are described by an affine (mod 1 ) function of only one binary cell state variable, all characteristic functions exhibit a fractal geometry where self-similar two-dimensional substructures manifest themselves, ad infinitum, as the number of cells (I + 1) → ∞. In addition to a complete gallery of time-1 characteristic functions for all 256 local rules, an accompanying table of explicit formulas is given for generating these characteristic functions directly from binary bit-strings, as in a digital-to-analog converter. To illustrate the potential applications of these fundamental formulas, we prove rigorously that the “right-copycat ” local rule 170 is equivalent globally to the classic “left-shift ” Bernoulli map. Similarly, we prove the “left-copycat ” local rule 240 is equivalent globally to the “right-shift ” inverse Bernoulli map. Various geometrical and analytical properties have been identified from each characteristic function and explained rigorously. In particular, two-level stratified subpatterns found in most characteristic functions are shown to emerge if, and only if, b1 = 0, where b1 is the “synaptic coefficient” associated with the cell differential equation developed in Part I. Gardens of Eden are derived from the decimal range of the characteristic function of each local rule and tabulated. Each of these binary strings has no predecessors (pre-image) and has therefore no past, but only the present and the future. Even more fascinating, many local rules are endowed with binary configurations which not only have no predecessors, but are also fixed points of the characteristic functions. To dramatize that such points have no past, and no future, they are henceforth christened “Isles of Eden”. They too have been identified and tabulated.

1. Characteristic Functions: Global Representation of Local Rules

itself which uniquely maps each input binary string {x0 , x1 , x2 , . . . , xI } represented in decimal form

The fundamental concept of the time-1 characteristic function χ1N : [0, 1] → [0, 1]

φ=

I

2−(i+1) xi

(2)

i=0

(1)

into an output binary string

of a local rule N is defined in Part IV [Chua et al., 2005] as a function from the unit interval [0, 1] into

1

χ N (φ) = 509

I i=0

2−(i+1) yi

(3)

510


Cell (I-2) Cell Cell (I-1) I

input u it−1 u it u it+1 =

(-1,1,-1) 2 4

6 0 2 =1

(1,1,-1)

N (uit−1 , uit , uit+1 )

(-1,1,1)

(1,1,1) 7 0 (-1,-1,-1)

(1,-1,-1)

uit−1 uit uit+1 uit +1 0

0

0

0

1

0

0

1

2

0

1

0

3

0

1

1

4

1

0

0

5

1

0

1

6

1

1

0

7

1

1

1

u it

(1,-1,1) 4

u it−1

output uit +1

Local Rule N

22 =

24 = 16

Cell Cell Cell 2 0 1

Symbolic Truth Table

uit +1

26 = 64

Cell (i+1)

Cell Cell (i-1) i

5

β0 β1 β2 β3 β4 β5 β6 β7

Numeric Truth Table 3

27 = 128 1 (-1,-1,1) 25 = 32

23 =

uit−1 uit uit+1 uit +1

8

21 = 2

u it+1

0

-1

-1

-1

1

-1

-1

1

2

-1

1

-1

3

-1

1

1

4

1

-1

-1

5

1

-1

1

6

1

1

-1

7

1

1

1

γ0 γ1 γ2 γ3 γ4 γ5 γ6 γ7


Chapter 5: Fractals Everywhere

Σ Φ

[0, 1]

TN

χN 1

Table 4 of [Chua et al., 2003]. Since χ1N is defined in terms of binary variables xi ∈ {0, 1}, let us apply the conversion relationship Eq. (4) from [Chua et al., 2005] namely,

Σ Φ

1 (ui + 1) 2 and define the step function xi =

[0, 1]

Fig. 2. A commutative diagram establishing a one-to-one correspondence between T N and χ1N .

in decimal representation, as I → ∞, where {y0 , y1 , y2 , . . . , yI } is the output binary string determined from the local rule N of the one-dimensional cellular automata under a periodic boundary condition, as shown in Fig. 1. Every input binary string is assumed to be finite but whose length can be chosen to be arbitrarily large. For practical calculation, the domain of the characteristic function χ1N consists of only a large but finite number of equally-spaced real numbers inside [0, 1]. In the limit I → ∞, the domain of χ1N coincides with the entire unit interval [0, 1]. In this case, every point φ ∈ [0, 1] corresponds uniquely to an infinite binary string. Conversely, each infinite binary string corresponds to a unique point on [0, 1]. This one-to-one correspondence between each binary string in the set Σ of all infinite binary strings and each real number in [0, 1] is depicted in Fig. 2, where φ is defined via Eq. (2). Observe that since the time-1 characteristic function χ1N maps each infinite binary string into another infinite binary string after one iteration of the local rule N , it is a global representation.1

1.1. Deriving explicit formula for calculating χ1N Recall from Eq. (8) of [Chua et al., 2004] that the output of each local rule N can be calculated from the formula (4) where the eight parameters {z2 , c2 , z1 , c1 , z0 , b1 , b2 , b3 } determining each local rule N is given in 1

511

(5)

(6) to rewrite Eq. (4) into

(7) where 1 z0 [z0 − (b1 + b2 + b3 )] 2 z1

1 z1 2

z2

1 z2 2

(8)

in Eq. (3), Substituting Eq. (7) for yi = xt+1 i and deleting the superscripts, we obtain the following explicit formula for calculating the characteristic function χ1N for any local rule N , N = 0, 1, 2, . . . , 255:

(9) where {z2 , c2 , z1 , c1 , z0 , b1 , b2 , b3 } are defined in Eq. (8) and Table 4 of [Chua et al., 2003]. Applying Eq. (9) to all 256 local rules, we obtain the explicit formulas listed in Table 1 for calculating the corresponding time-1 characteristic functions χ1N , N = 0, 1, 2, . . . , 255, where the binary string begins from φ = 0 corresponding to {0, ↑ x0

0, ↑ x1

0, ↑ x2

···

0}, ↑ xI

We can define a time-k characteristic function χkN : [0, 1] → [0, 1] in exactly the same way where the output string is

calculated after every “k” iterations under rule N .

Table 1. Explicit formulas for calculating characteristic functions χ1N in terms of binary strings {x0 , x1 , x2 , . . . , xI }. Each row is partitioned into four equal parts, where each part is color coded either in blue, if the characteristic function has no stratification (see Tables 2 and 5), or in pink, otherwise.

512

Table 1.

(Continued )

513

Table 1.

(Continued )

514

Table 1.

(Continued )

515

Table 1.

(Continued )

516

Table 1.

(Continued )

517

Table 1.

(Continued )

518

Table 1.

(Continued )

519

Table 1.

(Continued )

520

Table 1.

(Continued )

521

Table 1.

(Continued )

522

Table 1.

(Continued )

523

Table 1.

(Continued )

524

Table 1.

(Continued )

525

Table 1.

(Continued )

526

Table 1.

(Continued )

527

528


to φ = 1 corresponding to {1, ↑ x0

1, ↑ x1

1, ↑ x2

···

1}. ↑ xI

1.3. Deriving the Bernoulli map from χ1170 As an application of the explicit formulas listed in Table 1, let us apply the characteristic function χ1170 of 170 to an (I + 1)-bit binary string {x0 , x1 , x2 , . . . , xI } with decimal representation

1.2. Graphs of characteristic functions χ1N For future reference, we have plotted the time1 characteristic functions χ1N for all 256 local rules with I = 65, and displayed them in Table 2 showing only 201 points for each rule to avoid clutter. In other words, each graph of χ1N in Table 2 shows only 201 values of χ1N , each one calculated from a 66-bit binary string. To enhance clarity, every pair of adjacent points in each graph are plotted as a small “red” square “ ” and a small “blue” square “ ” on top of alternating red and blue color bars emanating from each value of φ ∈ [0, 1] corresponding to the 201 uniformly distributed points with spacing ∆φ = 0.005. A careful examination of Table 2 reveals that adjacent pairs of points of χ1N are either located in “close proximity” of each other, or they exhibit an “abrupt jump” from each other. We will henceforth refer to those subintervals where adjacent red and blue squares are close to each other as “smooth”, and those exhibiting “abrupt jumps” as “discontinuous”. A careful analysis of these subintervals reveal that they extend over a minimum range of ∆φ = 0.25 for all 256 rules. For example, only the second subinterval φ ∈ [0.25, 0.50) of rule 2 is discontinuous. On the other hand, the first and second subintervals [0, 0.25) and [0.25, 0.50) of rule 3 are discontinuous. For rule 110 , we find only the fourth subinterval [0.75, 1.00] is discontinuous. For rule 30 , we find all four subintervals [0, 0.25), [0.25, 0.50), [0.50, 0.75) and [0.75, 1.00) are discontinuous. Since these properties are quite useful for understanding the global dynamics of local rules, we have divided the area to the right of the equality sign of each characteristic function χ1N in Table 1 into the above four corresponding equal parts, and painted each part with a light blue background color if the corresponding subinterval has smooth adjacent red and blue squares, or in a light pink background color if adjacent red and blue squares exhibit discontinuous jumps from each other.

φ=

I

2−(i+1) xi

(10)

i=0

to obtain 1

χ 170 (φ) =

I

−(i+1)

2

xi+1 =

i=0

I+1

2−j xj + x0 − x0

j=1

I+1 −(j+1) =2 2 xj − x0 j=0

2φ, φ < 0.5 = 2φ − 1, φ ≥ 0.5

(11)

as I → ∞. It follows from Eq. (11) that the characteristic function χ1170 of the rule 170 converges to the well-known Bernoulli map [Billingsley, 1978] χ1170 (φ) = 2φ mod 1

(12)

as I → ∞. Since the output of each pixel “i” of rule 170 in Table 1 is given simply by yi = xi+1 , the local rule 170 consists of simply copying the “state” of the pixel “i + 1” of the right-neighboring pixel. We will henceforth call rule 170 the right-copycat rule. The graph of the right-copycat rule 170 is shown in Fig. 3.

1.4. Deriving inverse Bernoulli map from χ1240 Let us apply the characteristic function χ1240 of 240 from Table 1 to the (I + 1)-bit binary string φ defined in Eq. (10) to obtain 1

χ 240

I

I−1 1 −(j+1) (φ) = 2 xi−1 = 2 xj 2 i=0 j=−1 I−1 1 −(j+1) 1 = 2 xj + xI 2 2 −(i+1)

j=0

1 1 = φ + xI 2 2 as I → ∞.

(13)

Table 2.

χ

1 0

χ

1

1

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

529

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

0

1 2

1

0.9

0

χ

Gallery of characteristic functions.

χ

1

3

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

2

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

1

φ

3

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 4

χ

1

5

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

530

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

4

1 6

1

0.9

0

χ

(Continued )

χ

1

7

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

6

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

5

φ

7

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 8

χ

1

9

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

531

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

8

1 10

1

0.9

0

χ

(Continued )

χ

1

11

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

10

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

9

φ

11

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 12

χ

1

13

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

532

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

12

1 14

1

0.9

0

χ

(Continued )

χ

1

15

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

14

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

13

φ

15

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 16

χ

1

17

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

533

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

16

1 18

1

0.9

0

χ

(Continued )

χ

1

19

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

18

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

17

φ

19

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 20

χ

1

21

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

534

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

20

1 22

1

0.9

0

χ

(Continued )

χ

1

23

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

22

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

21

φ

23

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 24

χ

1

25

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

535

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

24

1 26

1

0.9

0

χ

(Continued )

χ

1

27

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

26

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

25

φ

27

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 28

χ

1

29

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

536

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

28

1 30

1

0.9

0

χ

(Continued )

χ

1

31

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

30

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

29

φ

31

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 32

χ

1

33

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

537

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

32

1 34

1

0.9

0

χ

(Continued )

χ

1

35

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

34

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

33

φ

35

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 36

χ

1

37

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

538

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

36

1 38

1

0.9

0

χ

(Continued )

χ

1

39

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

38

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

37

φ

39

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 40

χ

1

41

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

539

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

40

1 42

1

0.9

0

χ

(Continued )

χ

1

43

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

42

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

41

φ

43

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 44

χ

1

45

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

540

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

44

1 46

1

0.9

0

χ

(Continued )

χ

1

47

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

46

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

45

φ

47

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 48

χ

1

49

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

541

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

48

1 50

1

0.9

0

χ

(Continued )

χ

1

51

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

50

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

49

φ

51

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 52

χ

1

53

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

542

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

52

1 54

1

0.9

0

χ

(Continued )

χ

1

55

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

54

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

53

φ

55

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 56

χ

1

57

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

543

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

56

1 58

1

0.9

0

χ

(Continued )

χ

1

59

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

58

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

57

φ

59

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 60

χ

1

61

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

544

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

60

1 62

1

0.9

0

χ

(Continued )

χ

1

63

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

62

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

61

φ

63

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 64

χ

1

65

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

545

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

64

1 66

1

0.9

0

χ

(Continued )

χ

1

67

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

66

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

65

φ

67

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 68

χ

1

69

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

546

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

68

1 70

1

0.9

0

χ

(Continued )

χ

1

71

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

70

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

69

φ

71

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 72

χ

1

73

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

547

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

72

1 74

1

0.9

0

χ

(Continued )

χ

1

75

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

74

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

73

φ

75

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 76

χ

1

77

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

548

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

76

1 78

1

0.9

0

χ

(Continued )

χ

1

79

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

78

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

77

φ

79

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 80

χ

1

81

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

549

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

80

1 82

1

0.9

0

χ

(Continued )

χ

1

83

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

82

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

81

φ

83

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 84

χ

1

85

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

550

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

84

1 86

1

0.9

0

χ

(Continued )

χ

1

87

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

86

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

85

φ

87

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 88

χ

1

89

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

551

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

88

1 90

1

0.9

0

χ

(Continued )

χ

1

91

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

90

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

89

φ

91

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 92

χ

1

93

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

552

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

92

1 94

1

0.9

0

χ

(Continued )

χ

1

95

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

94

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

93

φ

95

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 96

χ

1

97

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

553

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

96

1 98

1

0.9

0

χ

(Continued )

χ

1

99

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

98

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

97

φ

99

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 100

χ

1

101

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

554

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

100

1 102

1

0.9

0

χ

(Continued )

χ

1

103

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

102

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

101

φ

103

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 104

χ

1

105

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

555

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

104

1 106

1

0.9

0

χ

(Continued )

χ

1

107

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

106

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

105

φ

107

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 108

χ

1

109

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

556

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

108

1 110

1

0.9

0

χ

(Continued )

χ

1

111

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

110

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

109

φ

111

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 112

χ

1

113

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

557

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

112

1 114

1

0.9

0

χ

(Continued )

χ

1

115

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

114

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

113

φ

115

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 116

χ

1

117

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

558

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

116

1 118

1

0.9

0

χ

(Continued )

χ

1

119

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

118

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

117

φ

119

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 120

χ

1

121

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

559

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

120

1 122

1

0.9

0

χ

(Continued )

χ

1

123

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

122

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

121

φ

123

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 124

χ

1

125

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

560

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

124

1 126

1

0.9

0

χ

(Continued )

χ

1

127

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

126

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

125

φ

127

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 128

χ

1

129

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

561

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

128

1 130

1

0.9

0

χ

(Continued )

χ

1

131

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

130

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

129

φ

131

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 132

χ

1

133

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

562

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

132

1 134

1

0.9

0

χ

(Continued )

χ

1

135

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

134

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

133

φ

135

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 136

χ

1

137

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

563

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

136

1 138

1

0.9

0

χ

(Continued )

χ

1

139

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

138

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

137

φ

139

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 140

χ

1

141

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

564

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

140

1 142

1

0.9

0

χ

(Continued )

χ

1

143

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

142

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

141

φ

143

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 144

χ

1

145

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

565

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

144

1 146

1

0.9

0

χ

(Continued )

χ

1

147

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

146

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

145

φ

147

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 148

χ

1

149

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

566

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

148

1 150

1

0.9

0

χ

(Continued )

χ

1

151

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

150

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

149

φ

151

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 152

χ

1

153

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

567

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

152

1 154

1

0.9

0

χ

(Continued )

χ

1

155

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

154

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

153

φ

155

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 156

χ

1

157

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

568

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

156

1 158

1

0.9

0

χ

(Continued )

χ

1

159

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

158

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

157

φ

159

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 160

χ

1

161

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

569

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

160

1 162

1

0.9

0

χ

(Continued )

χ

1

163

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

162

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

161

φ

163

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 164

χ

1

165

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

570

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

164

1 166

1

0.9

0

χ

(Continued )

χ

1

167

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

166

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

165

φ

167

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 168

χ

1

169

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

571

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

168

1 170

1

0.9

0

χ

(Continued )

χ

1

171

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

170

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

169

φ

171

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 172

χ

1

173

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

572

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

172

1 174

1

0.9

0

χ

(Continued )

χ

1

175

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

174

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

173

φ

175

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 176

χ

1

177

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

573

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

176

1 178

1

0.9

0

χ

(Continued )

χ

1

179

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

178

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

177

φ

179

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 180

χ

1

181

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

574

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

180

1 182

1

0.9

0

χ

(Continued )

χ

1

183

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

182

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

181

φ

183

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 184

χ

1

185

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

575

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

184

1 186

1

0.9

0

χ

(Continued )

χ

1

187

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

186

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

185

φ

187

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 188

χ

1

189

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

576

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

188

1 190

1

0.9

0

χ

(Continued )

χ

1

191

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

190

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

189

φ

191

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 192

χ

1

193

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

577

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

192

1 194

1

0.9

0

χ

(Continued )

χ

1

195

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

194

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

193

φ

195

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 196

χ

1

197

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

578

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

196

1 198

1

0.9

0

χ

(Continued )

χ

1

199

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

198

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

197

φ

199

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 200

χ

1

201

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

579

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

200

1 202

1

0.9

0

χ

(Continued )

χ

1

203

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

202

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

201

φ

203

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 204

χ

1

205

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

580

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

204

1 206

1

0.9

0

χ

(Continued )

χ

1

207

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

206

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

205

φ

207

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 208

χ

1

209

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

581

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

208

1 210

1

0.9

0

χ

(Continued )

χ

1

211

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

210

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

209

φ

211

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 212

χ

1

213

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

582

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

212

1 214

1

0.9

0

χ

(Continued )

χ

1

215

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

214

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

213

φ

215

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 216

χ

1

217

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

583

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

216

1 218

1

0.9

0

χ

(Continued )

χ

1

219

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

218

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

217

φ

219

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 220

χ

1

221

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

584

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

220

1 222

1

0.9

0

χ

(Continued )

χ

1

223

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

222

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

221

φ

223

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 224

χ

1

225

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

585

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

224

1 226

1

0.9

0

χ

(Continued )

χ

1

227

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

226

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

225

φ

227

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 228

χ

1

229

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

586

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

228

1 230

1

0.9

0

χ

(Continued )

χ

1

231

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

230

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

229

φ

231

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 232

χ

1

233

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

587

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

232

1 234

1

0.9

0

χ

(Continued )

χ

1

235

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

234

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

233

φ

235

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 236

χ

1

237

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

588

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

236

1 238

1

0.9

0

χ

(Continued )

χ

1

239

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

238

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

237

φ

239

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 240

χ

1

241

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

589

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

240

1 242

1

0.9

0

χ

(Continued )

χ

1

243

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

242

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

241

φ

243

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 244

χ

1

245

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

590

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

244

1 246

1

0.9

0

χ

(Continued )

χ

1

247

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

246

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

245

φ

247

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 248

χ

1

249

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

591

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

248

1 250

1

0.9

0

χ

(Continued )

χ

1

251

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

250

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

249

φ

251

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0

Table 2.

χ

1 252

χ

1

253

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

592

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

252

1 254

1

0.9

0

χ

(Continued )

χ

1

255

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

254

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

253

φ

255

1 0.9

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.9

0

0

0


function for all finite I, for xI uniquely specifies whether the upper branch (if xI = 1), or the lower branch (if xI = 0) should be selected.

1

χ 170

1.5. Deriving affine (mod 1) characteristic functions

1

0.5

0

593

0.5

1

φ 170

Fig. 3. The characteristic function of the “right-copycat” rule 170 converges to the Bernoulli map.

Since xI ∈ {0, 1} is the rightmost bit of the binary string {x0 , x1 , x2 , . . . , xI }, it follows that the graph of the characteristic function χ1240 (φ) consists of two parallel branches, parameterized by the last binary bit xI , as shown in Fig. 4. Since the output of each pixel “i” of rule 240 in Table 1 is given simply by yi = xi−1 , the local rule 240 consists of simply copying the “state” of the pixel “i − 1” of the left-neighboring pixel. We will henceforth call rule 240 the left-copycat rule. An examination of the graph of the characteristic functions χ1170 in Fig. 3 and χ1240 in Fig. 4 shows that they are symmetrical with respect to the main diagonal. In other words, the two graphs are inverse of each other. Observe that although the graph of χ1240 in Fig. 4 appears to be a double-valued function, it is actually a well-defined single-valued

We have shown in Secs. 1.3 and 1.4 that the characteristic functions χ1170 and χ1240 are affine (mod 1) functions for finite I. An examination of Table 2 reveals that there are only eight affine (mod 1) characteristic functions, namely, χ10 , χ115 , χ151 , χ185 , χ1170 , χ1204 , χ1240 and χ1255 . The explicit formula for each of these characteristic functions can be easily derived from Table 1 as in Secs. 1.3 and 1.4. Table 3 lists the explicit formulas defining these eight affine (mod 1 ) local rules and their corresponding global characteristic function. The graph of each characteristic function χ1N is shown in Table 4. Remark 1.1. Since there exist infinitely many dis-

tinct sets of parameters {z2 , c2 , z1 , c1 , z0 , b1 , b2 , b3 } [Chua et al., 2003] from which the explicit formula Eq. (9) represents the same characteristic function χ1N , N = 0, 1, 2, . . . , 255, the equations listed in Table 1 represent only one of many equivalent explicit formulas. In fact, we need to list explicit formulas for only 128 local rules since the simple transformation in the following proposition allows us to generate corresponding explicit formulas for the remaining 128 rules. Proposition 1.1. Given an explicit formula 1

χN =

I

1

2−(i+1) {f (xi−1 , xi , xi+1 )}

(9a)

i=0

for local rule N , the following corresponding equa∆ tion gives an explicit formula for local rule N = 255 − N :

χ 1240

χ1255–N =

0.5

I

2−(i+1) {−f (xi−1 , xi , xi+1 )}

(9b)

i=0

Proof.

Equation (9b) follows directly from Eq. (9a) and the identity

0

0.5

1

φ

240

Fig. 4. The characteristic function of the “left-copycat” rule 240 converges to the inverse Bernoulli map.

{−f } = 1 − {f }, as can be verified by direct substitution.

(9c)

594


Table 3. Explicit formulas defining the eight affine (mod 1) local rules and their associated characteristic functions. The bar above a binary bit means taking its complement, i.e. 0 = 1 and 1 = 0.

Affine (mod 1) Rule Number N

0

Explicit formula for

Local Rule

χ 1N

Formula

xin +1 = 0

χ 10 (φ ) = 0 χ

(φ )

 − =  − 

1 φ +1 2 1 1 φ + 2 2

15

xin +1

51

xin +1 = xin

χ 15 1 (φ ) = − φ + 1

85

xin +1

χ 18 5 (φ ) = 

170

xin +1 = xin+1

χ 11 7 0 (φ ) = 2 φ

204

xin+1 = xin

χ 12 0 4 (φ ) = φ

=

=

xin−1

xin+1

240

xin +1

255

xin +1 = 1

=

xin−1

χ

1 15

1 240

(φ )

,

if ,

if

xI = 0 xI = 1

 − 2 φ +1

,

if

x1 = 0

 − 2 φ + 2

,

if

x1 = 1

m od 1

1 2φ =  1 1  φ +  2 2

χ 12 5 5 (φ ) = 1

,

if

xI = 0

,

if

xI = 1

Chapter 5: Fractals Everywhere Table 4.

χ0

Characteristic functions χ1N of Affine (mod 1) Rules.

1

1

0.5

χ 255 00.5 1

1

0

0.5

1

φ0

0

χ 85

0.5

1

0.5

1

φ

204

0.5

1

φ

240

1

1

0.5

0

χ 51

255

1

1

1

χ 15

φ

0.5

0.5

1

φ 15

0.5

0

1

1

0.5

χ 204 0.5

φ 85

1

1

0

0.5

1

φ 51

0

1

1

χ 170 0.5

χ 240 0.5 1

1

0

0.5

1

φ 170

0

595

596


Example 1.1. Given the following explicit formula I

1

χ 110 =

−(i+1)

2

i=0

−1 + +1xi−1 + 2xi

1 − 3xi+1 − 2

(9d)

for N = 110 , we apply Eq. (9b) to obtain the following explicit formula I 1 −(i+1) 2 +1 − +1xi−1 + 2xi χ 145 = i=0

1 − 3xi+1 − 2

(9e)

for N = 255 − 110 = 145 . Note that the above formula is different from the one listed in Table 1, since the formulas in Table 1 are constructed independently of Proposition 1.1. Proposition 1.2. The graphs of the characteris-

tic functions for χ1N and χ1255–N are symmetric with respect to the horizontal axis χ1N = 0.5. In particular, χ1N (φ) = 1 − χ1255–N (φ)

(9f)

Proof.

Equation (9f) follows directly from Proposition 1.1 and the identity (9c).

Remark 1.2. It would be instructive for the reader

to verify the graphs of χ1N in Table 2 for N = 128, 129, . . . , 255 can be obtained by flipping the graphs of χ1N for N = 127, 126, 125, . . . , 0, respectively, about the φ N -axis, and then translating them upward by ∆χ1N = 1. It follows from [Chua et al., 2004] that only 89 characteristic functions, out of 256, in Table 1, give qualitatively distinct global dynamics. All other characteristic functions can be generated trivially by applying the three global transformations T† , T and T∗ from Klein’s Vierergruppe.

Remark 1.3.

Example 1.2. Applying T† , T and T∗ to Eq. (9d),

we obtain: 1

†

1

χ 124 = T [χ 110 ] =

I i=0

−(i+1)

2

1 + 2xi + 1xi+1 − 2

−1 + −3xi−1 (9g)

1

1

χ 137 = T[χ 110 ] =

I

−(i+1)

2

i=0

+1 − −1xi−1

1 − 2xi + 3xi+1 − 2 χ1193 = T∗ [χ1110 ] =

I

2−(i+1)

i=0

(9h) +1 − +3xi−1

1 − 2xi − 1xi+1 − 2

(9i)

Remark 1.4. By choosing the characteristic function

of only one local rule from each of the 15 local equivalence classes listed in the left column of Table 20 of [Chua et al., 2004], we can apply the appropriate rotation transformation listed in Table 1 of [Chua et al., 2004], and use Proposition 1.1 to derive an explicit characteristic function formula for each of the remaining 241 local rules.

2. Lameray Diagram on χ1N Gives Attractor Time-1 Maps We have demonstrated in [Chua et al., 2005] that the dynamics on each attractor of a local rule N is uniquely characterized by the forward time-1 return map ρ1 [N ] : φn−1 → φn

(14)

starting from any point on the attractor. For the 69 period-1 rules listed in Tables 3 and 4 of [Chua et al., 2005], such time-1 maps consist of only one point on the main diagonal of the characteristic function χ1N , and is therefore trivial. For the 25 period-2 rules listed in Tables 7 and 8, such time-1 maps consist of two points on the main diagonal of χ1N , and is also trivial. Similarly, the time-1 map of the four period-3 rules listed in Table 9 consist of three points, as illustrated in Fig. 11 of [Chua et al., 2005]. The most interesting and nontrivial time-1 maps considered so far are the 112 generalized Bernoulli στ -shift rules listed in Tables 10–12 of [Chua et al., 2005]. The time-1 maps of these rules, as well as those corresponding to the remaining 50 rules listed in Tables 17 and 18 of [Chua et al., 2005] consist in general of an uncountable [Devaney, 1992] number of points φ ∈ [0, 1], assuming I → ∞. An instructive way to analyze the dynamics of such time-1 maps is to plot the Lameray (cobweb) diagram starting from any generic initial point, and observe how its “cobweb” loci evolves from this


initial point on the characteristic function χ1N . Let us study some of these rules.

2.1. Lameray diagram of 170 The dynamic pattern of the first 65 iterations of rule 170 (with complexity index κ = 1) starting from φ0 = 0.0253584 is shown in Fig. 5(a). The first 12 iterations of the Lameray diagram constructed from the characteristic function χ1170 are shown in Fig. 5(b) for ease of visualization. The continued iterations of the Lameray diagram for 170 up to n = 63 are shown in Fig. 5(c). The color code on top of Fig. 5(a) denotes the iteration number n = 0, 1, 2, . . . , 63. A comparison of the loci (corner points not on the diagonal) in Fig. 5(c) with the forward time1 map of 170 in Table 2 (p. 1106) of [Chua et al., 2005] clearly shows that they all fall on the graph of time-1 map. Indeed, as n → ∞, this loci is seen (not shown) to converge on the complete graph of the forward time-1 map of 170 . Indeed, in this example, all points on the Lameray diagram, including any initial point, are seen to fall on the associated attractor. Note also that all points on the loci of the Lameray diagram of any rule N must be a subset of the associated characteristic function χ1N , by construction.

2.2. Lameray diagram of 240 The dynamic pattern of the first 65 iterations of rule 240 (with complexity index κ = 1) starting from φ0 = 0.0253584, is shown in Fig. 6(a). The first 12 iterations of the Lameray diagram constructed from the characteristic function χ1240 are shown in Fig. 6(b) for ease of visualization. The continued iterations of the Lameray diagram for 240 up to n = 63 are shown in Fig. 6(c). A comparison of the loci in Fig. 6(c) with the forward time-1 map of 240 in Table 2 (p. 1124) of [Chua et al., 2005] clearly shows that they all fall on the graph of the time-1 map. Indeed, as n → ∞, this loci is seen (not shown) to converge on the complete graph of the forward time-1 map of 240 . Indeed, in this example, all points on the Lameray diagram, including any initial point, are seen to fall on the associated attractor.

2.3. Lameray diagram of 2 The dynamic pattern of the first 65 iterations of rule 2 (with complexity index κ = 1) starting from

597

φ0 = 0.64368 is shown in Fig. 7(a). The first six iterations of the Lameray diagram constructed from the characteristic function χ12 are shown in Fig. 7(b) and the continued iterations up to n = 63 are shown in Fig. 7(c). A comparison of the loci in Fig. 7(c) with the forward time-1 map of 2 in Table 2 (p. 1064) of [Chua et al., 2005] shows that except for the initial point (which did not fall on the attractor but belongs to the basin of attraction of the associated attractor), all other points of the loci fall on the time-1 map of 2 . To demonstrate this result holds for any other initial point belonging to the same basin of attraction, Fig. 8 shows the Lameray diagram of 2 (starting from a different initial point φ0 = 0.343184) over n = 25, 30, 35, 50, 75, 100, 150, 200 and 250 iterations, respectively. Again except for the initial point, the loci of the Lameray diagram is seen to converge on the forward time-1 map of 2 in Table 2 of [Chua et al., 2005].

2.4. Lameray diagram of 3 The dynamic pattern of the first 65 iterations of rule 3 (with complexity index κ = 1) starting from φ0 = 0.64368 is shown in Fig. 9(a). The first ten iterations of the Lameray diagram constructed from the characteristic function χ13 are shown in Fig. 9(b) and the continued iterations up to n = 63 are shown in Fig. 9(c). A comparison of the loci in Fig. 9(c) with the forward time-1 map of 3 in Table 2 (p. 1064) of [Chua et al., 2005] shows that except for the initial point (which did not fall on the attractor but belongs to its basin of attraction), all other points of the loci fall on the time-1 map of 3 . To demonstrate this result holds for any other initial point belonging to the same basin of attraction, Fig. 10 shows the Lameray diagram of 3 (starting from a different initial point φ0 = 0.556412) over n = 5, 10, 15, 20, 35, 50, 100, 150 and 250 iterations, respectively. Again, except for the initial point, the loci of the Lameray diagram is seen to converge on the forward time-1 map of 3 in Table 2 of [Chua et al., 2005].

2.5. Lameray diagram of 46 The dynamic pattern of the first 65 iterations of rule 46 (with complexity index κ = 3) starting from φ0 = 0.895314 is shown in Fig. 11(a). The first

N=

170

φ 0 = 0.0253584

,

0

0

0 1 2 3 4 5 6 7 8 9

n 10

63 9

1

φn

5

6 8

0.5

4

20

7

3 2

0

30

0

598

n = 12

0.5

φ n-1 1

0.5

φ n-1 1

1

φn

40

50

0.5

60 0 Fig. 5.

10

20

30

40

50

60

i

0 0

n = 63

(a) Dynamic pattern of 170 from φ0 = 0.0253584. (b) Lameray diagram for first 12 iterations. (c) Lameray diagram over 63 iterations.

N=

240

φ 0 = 0.0253584

,

0

0

0 1 2 3 4 5 6 7 8 9

n 10

63 1 7

φn

2

6

5 1

0.5

8 3

20 4

9

0

30

0

599

n = 12

0.5

φ n-1 1

0.5

φ n-1 1

1

φn

40

50

0.5

60 0 Fig. 6.

10

20

30

40

50

60

i

0 0

n = 63

(a) Dynamic pattern of 240 from φ0 = 0.0253584. (b) Lameray diagram for first 12 iterations. (c) Lameray diagram over 63 iterations.

N=

2

,

φ 0 = 0.64368

0

63 1

0

φn

n 10

0.5

20

0

30

0

600

n=6

0.5

φ n-1 1

0.5

φ n-1 1

1

φn

40

50

0.5

60 0 Fig. 7.

10

20

30

40

50

60

i

0 0

n = 63

(a) Dynamic pattern of 2 from φ0 = 0.64368. (b) Lameray diagram for first six iterations. (c) Lameray diagram over 63 iterations.


N=

φ 0 = 0.343184

2 ,

1

0

250

1

1

φn

φn

φn

0.5

0.5

0.5

0 0

n = 25

0.5

φ n-1 1

1

0 0

n = 30

0.5

φ n-1 1

1

0 0

φn

φn

0.5

0.5

0.5

0

n = 50

0.5

φ n-1 1

1

0 0

n = 75

0.5

φ n-1 1

1

0

φn

0.5

0.5

0.5

n = 150

0.5 Fig. 8.

φ n-1 1

φ n-1 1

n = 100

0.5

φ n-1 1

0.5

φ n-1 1

1

φn

0

0.5

0

φn

0

n = 35

1

φn

0

601

0 0

n = 200

0.5

φ n-1 1

0 0

n = 250

Nine snapshots of the Lameray diagram of 2 starting from φ0 = 0.343184.

ten iterations of the Lameray diagram constructed from the characteristic function χ146 are shown in Fig. 11(b) and the continued iterations up to n = 63 are shown in Fig. 11(c). A comparison of the loci in Fig. 11(c) with the forward time-1 map of 46 in Table 2 (p. 1075) of [Chua et al., 2005] shows that except for the initial point (which did not fall on the attractor but

belongs to its basin of attraction), all other points of the loci fall on the time-1 map of 46 . To demonstrate this result holds for any other initial point belonging to the same basin of attraction, Fig. 12 shows the Lameray diagram of 46 (starting from a different initial point φ0 = 0.888034) over n = 7, 15, 25, 50, 75, 100, 150, 200 and 250 iterations, respectively. Again, except for

N=

3

,

φ 0 = 0.64368

0

63 1

0

φn

n 10

0.5

20

0

30

0

602

n = 10

0.5

φ n-1 1

0.5

φ n-1 1

1

φn

40

50

0.5

60 0 Fig. 9.

10

20

30

40

50

60

i

0 0

n = 63

(a) Dynamic pattern of 3 from φ0 = 0.64368. (b) Lameray diagram for first ten iterations. (c) Lameray diagram over 63 iterations.


N=

φ 0 = 0.556412

3 ,

1

0

250

1

1

φn

φn

φn

0.5

0.5

0.5

0 0

n=5

0.5

0

φ n-1 1

0

1

n = 10

0.5

φ n-1 1

1

0 0

φn

φn

0.5

0.5

0.5

0

n = 20

0.5

0

φ n-1 1

0

1

n = 35

0.5

φ n-1 1

1

0

φn

0.5

0.5

0.5

n = 100

0.5 Fig. 10.

φ n-1 1

φ n-1 1

n = 50

0.5

φ n-1 1

0.5

φ n-1 1

1

φn

0

0.5

0

φn

0

n = 15

1

φn

0

603

0 0

n = 150

0.5

φ n-1 1

0 0

n = 250


the initial point, the loci of the Lameray diagram is seen to converge on the forward time-1 map of 46 in Table 2 of [Chua et al., 2005].

2.6. Lameray diagram of 110 The dynamic pattern of the first 65 iterations of rule 110 (with complexity index κ = 2) starting

from φ0 = 0.40653 is shown in Fig. 13(a). The first five iterations of the Lameray diagram constructed from the characteristic function χ1110 are shown in Fig. 13(b) and the continued iterations up to n = 63 are shown in Fig. 13(c). A comparison of the loci in Fig. 13(c) with the forward time-1 map of 110 in Table 2 (p. 1091) of [Chua et al., 2005] shows that except for some initial

N=

46

φ 0 = 0.895314

,

0

63 1

0

φn

n 10

0.5

20

0

30

0

604

n = 10

0.5

φ n-1 1

0.5

φ n-1 1

1

φn

40

50

0.5

60 0 Fig. 11.

10

20

30

40

50

60

i

0 0

n = 63



N = 46 ,

φ 0 = 0.888034

1

0

250

1

1

φn

φn

φn

0.5

0.5

0.5

0 0

n=7

0.5

φ n-1 1

1

0 0

n = 15

0.5

φ n-1 1

1

0 0

φn

φn

0.5

0.5

0.5

0

n = 50

0.5

φ n-1 1

1

0 0

n = 75

0.5

φ n-1 1

1

0

φn

0.5

0.5

0.5

n = 150

0.5 Fig. 12.

φ n-1 1

φ n-1 1

n = 100

0.5

φ n-1 1

0.5

φ n-1 1

1

φn

0

0.5

0

φn

0

n = 25

1

φn

0

605

0 0

n = 200

0.5

φ n-1 1

0 0

n = 250


set of points (which may not fall on the attractor associated with the loci but belongs to its basin of attraction), all other points of the loci fall on the time-1 map of 110 associated with the corresponding attractor. To demonstrate this result holds for any other initial point belonging to the same basin of attraction, Fig. 14 shows the Lameray diagram of 110

(starting from a different initial point φ0 = 0.149766) over n = 5, 10, 15, 25, 50, 75, 200, 350 and 500 iterations, respectively. Again, except for some initial set of points not belonging to the corresponding attractor of 110 , the loci of the Lameray diagram is seen to converge to the forward time-1 map of 110 in Table 2 of [Chua et al., 2005].

N=

110

φ 0 = 0.40653

,

0

63 1

0

φn

n 10

0.5

20

0

30

0

606

n=5

0.5

φ n-1 1

0.5

φ n-1 1

1

φn

40

50

0.5

60 0 Fig. 13.

10

20

30

40

50

60

i

0 0

n = 63

(a) Dynamic pattern of 110 from φ0 = 0.40653. (b) Lameray diagram for first five iterations. (c) Lameray diagram over 63 iterations.


N = 110 ,

φ 0 = 0.149766

1

0

500

1

1

φn

φn

φn

0.5

0.5

0.5

0 0

n=5

0.5

0

φ n-1 1

0

1

n = 10

0.5

φ n-1 1

1

0 0

φn

φn

0.5

0.5

0.5

0

n = 25

0.5

0

φ n-1 1

0

1

n = 50

0.5

φ n-1 1

1

0

φn

0.5

0.5

0.5

n = 200

0.5 Fig. 14.

φ n-1 1

φ n-1 1

n = 75

0.5

φ n-1 1

0.5

φ n-1 1

1

φn

0

0.5

0

φn

0

n = 15

1

φn

0

607

0 0

n = 350

0.5

φ n-1 1

0 0

n = 500


2.7. Lameray diagram of 30 The dynamic pattern of the first 65 iterations of rule 30 (with complexity index κ = 2) starting from φ0 = 0.895314 is shown in Fig. 15(a). The first ten iterations of the Lameray diagram constructed from the characteristic function χ130 are

shown in Fig. 15(b), and the continued iterations up to n = 63 are shown in Fig. 15(c). A comparison of the loci in Fig. 15(c) with the forward time-1 map of 30 in Table 2 (p. 1071) of [Chua et al., 2005] shows that except for some initial set of points (which do not fall on the attractor associated with the loci but belong to its basin of

N=

30

φ 0 = 0.895314

,

0

63 1

0

φn

n 10

0.5

20

0

30

0

608

n = 10

0.5

φ n-1 1

0.5

φ n-1 1

1

φn

40

50

0.5

60 0 Fig. 15.

10

20

30

40

50

60

i

0 0

n = 63



attraction), all other points of the loci fall on the time-1 map of 30 associated with the corresponding attractor. To demonstrate this result holds for any other initial point belonging to the same basin of attraction, Fig. 16 shows the Lameray diagram of 30 (starting from a different initial point φ0 =

N = 30 ,

0.149766) over n = 7, 14, 21, 50, 75, 100, 200, 300 and 500 iterations, respectively. Again except for some initial set of points not belonging to the corresponding attractor of 30 , the loci of the Lameray diagram is seen to converge to the forward time-1 map of 30 in Table 2 of [Chua et al., 2005].

φ 0 = 0.149766

1

0

500

1

1

φn

φn

φn

0.5

0.5

0.5

0 0

n=7

0.5

φ n-1 1

1

0 0

n = 14

0.5

φ n-1 1

1

0 0

φn

φn

0.5

0.5

0.5

0

n = 50

0.5

φ n-1 1

1

0 0

n = 75

0.5

φ n-1 1

1

0

φn

0.5

0.5

0.5

n = 200

0.5 Fig. 16.

φ n-1 1

φ n-1 1

n = 100

0.5

φ n-1 1

0.5

φ n-1 1

1

φn

0

0.5

0

φn

0

n = 21

1

φn

0

609

0 0

n = 300

0.5

φ n-1 1

0 0

n = 500


610


3. Characteristic Functions are Fractals A careful examination of the characteristic functions χ1N in Table 2 would reveal that each graph of χ1N is composed of many subpatterns which are self-similar in the sense that each can be rescaled by appropriate horizontal and vertical scaling factors so that it coincides with a part of the composite pattern. Let us illustrate this fractal geometry [Barnsley, 1988] with some examples. Example 3.1 [Characteristic function χ12 ]. Let us

repeat the characteristic function χ12 from Table 2 in Fig. 17(a) by passing a curve through the small red and blue squares while deleting the red and blue vertical bars to avoid clutter. We will henceforth refer to this curve as the graph of χ1N . Let us examine two subpatterns of this graph. Subpattern 1 covers the “cyan” rectangular area bounded by φ 2 ∈ [0.5, 1.0] and χ12 ∈ [0, 0.2]. To show that this subpattern can be used as a template, which upon rescaling by appropriate horizontal and vertical scaling factors, can reproduce exactly corresponding portions of the graph at arbitrarily small scales, we have enlarged it as 1 in Fig. 17(a), by a factor of 2 in both subpattern horizontal and vertical directions. Let us observe next that the smaller “cyan” rectangle “2” bounded by φ 2 ∈ [0.25, 0.5] and χ12 ∈ [0.5, 0.6] can be enlarged by a horizontal scale = 22 and a vertical scale = 22 to obtain the subpattern 2 which is identical to the “template” 1 directly above it. Let us examine next the smaller cyan rect2 bounded by φ 2 ∈ angle “3” in subpattern [0.28125, 0.3125] and χ12 ∈ [0.5625, 0.575], and 1 observe that it too coincides with the template upon enlarging it horizontally by 25 and vertically by 25 . The above process can be repeated any number of times with appropriate choice of scaling factors, commensurate with the computer word length. Repeating the above scaling process to the three small cyan rectangles labeled “4”, “5”, and “6” in Fig. 17(b), we obtain the three subpatterns 4 5 6 of Fig. 17(b), each one is , and shown in , again found to be identical to the original template 1 in Fig. 17(a). We conclude therefore that the graph of χ12 near the origin consists of infinitely many scaled 1 In other infinitesimal subpatterns of template .

words, the graph of χ12 exhibits a fractal geometry in the sense that it is composed of infinitely many self-similar subpatterns. Example 3.2 [Characteristic function χ13 ]. The

graph of characteristic function χ13 replotted from Table 2 is shown in Fig. 18. By rescaling the three cyan rectangles labeled “1”, “2”, and “3” by appropriate scaling factors, we obtain the corresponding 1 2 and 3 in Fig. 18. Observe that subpatterns , these three subpatterns are all identical. Continuing this process, we found the graph of χ13 is composed 1 of infinitely many scaled copies of template . Example 3.3 [Characteristic function χ110 ]. The

graph of characteristic function χ110 replotted from Table 2 is shown in Fig. 19. Observe the three cyan areas labeled “1”, “2”, and “3” are identical after 1 appropriate rescaling, as shown in subpatterns , 2 and 3 in Fig. 19.

Example 3.4 [Characteristic function χ111 ]. The

graph of characteristic function χ111 is replotted from Table 2 in Fig. 20. Observe the cyan area 1 contains labeled “1” and rescaled as template infinitely many scaled copies of itself, as illustrated 2 in Fig. 20. in subpattern


graph of characteristic function χ117 replotted from Table 2 is shown in Fig. 21. The fractal geometry of χ117 is obvious from the rescaled 1 and . 2 patterns


graph of characteristic function χ1110 replotted from 1 Table 2 is shown in Fig. 22. The two subpatterns 1 2 reveal the fractal geometry of χ . and 110


graph of characteristic function χ1124 replotted from 1 Table 2 is shown in Fig. 23. The two subpatterns 1 2 illustrate the fractal geometry of χ . and 124






χ1 1

χ1

2

1

0.9

0.9

0.8

0.8

0.7

2

0.7

2

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

1

0.2

4

0.2

0.1

5

0.1 6

0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

χ1 0.2

1

φ

2

0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

χ1

2

0.3

1

φ

2

φ

2

φ

2

φ

2

2

4

1 0.1

0.275

0 0.5

χ1 0.6

0.6

0.7

0.8

0.9

1

φ

2

0.25 0.125

χ1

2

0.15

3

0.15

0.175

0.2

0.225

0.25

2

2

5

0.55

0.1375

0.5 0.25

χ 0.575

611

0.3

0.35

0.4

0.45

0.5

φ

2

0.125 0.0625

χ

1 2

0.075

0.075

0.0875

0.1

0.1125

0.125

1 2

6

3 0.56875

0.06875

0.5625 0.28125 0.2875

0.29375

0.3

φ

0.30625 0.3125

1 horizontal scaling = 21 , (a) subpattern : vertical scaling = 21 2 2 horizontal scaling = 2 , subpattern : 2 vertical scaling = 2 5 3 horizontal scaling = 2 , subpattern : vertical scaling = 25

Fig. 17.

2

0.0625 0.03125 0.0375

0.04375

0.05

0.05625 0.0625

4 horizontal scaling = 23 , (b) subpattern : vertical scaling = 23 4 5 horizontal scaling = 2 , subpattern : 4 vertical scaling = 2 5 6 horizontal scaling = 2 , subpattern : vertical scaling = 25

Fractal compositions of χ12 .

612


χ1 1

10

0.9

χ

1 3

0.8

1 3

0.9

0.7 2

0.8

0.6

0.7

0.5

0.6

1

0.4

1

0.5

0.3

0.4

2

0.2

0.3

0.1 3

0.2

0 0

0.1 0

χ1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

χ1

3

0.4

1

φ

10

φ

10

φ

10

10

3

1 0.55

1

0.3875

0.525

0.5 0.375

χ

0.4

0.425

0.45

0.475

0.5

φ

0.375 0.1885

χ1

3

1

0.2

3 0.775

0.2

0.2125

0.225

0.2375

0.25

10

2 2 0.19375

0.7625

χ1

0.75 0.1875

0.2

0.2125

0.225

φ

0.2375

0.25

3

3 0.8875

0.1875 0.09375

χ1 0.01

0.1

0.10625 0.1125

0.11875

0.125

10

3

3 0.09688

0.88125

0.875 0.09735

0.1

Fig. 18.

0.10625 0.1125

0.11875

φ

0.125


3 1 horizontal scaling = 2 , subpattern : 3 vertical scaling = 2 4 2 horizontal scaling = 2 , subpattern : 4 vertical scaling = 2 3 horizontal scaling = 25 , subpattern : vertical scaling = 25

3

0.09375 0.04685

0.05

Fig. 19.

φ

0.05311 0.05524 0.05937 0.0625 10


1 horizontal scaling = 24 , subpattern : vertical scaling = 24 5 2 horizontal scaling = 2 , subpattern : 5 vertical scaling = 2 6 3 horizontal scaling = 2 , subpattern : 6 vertical scaling = 2


χ1

11

χ1

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

1

0 0

χ1

17

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0.06

0

χ1

1

11

1

0

11

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

φ

17

φ

17

φ

17

1

0.05

0.04

0.04

0.03

0.03

0.02

0.02 0.01 2

0 0.625

χ1

1

0.06 17

0.05

0.01

613

0.6375

0.65

0.6625

2

φ

0.6875 11

0.675

χ1

2

11

0 0.6875

0.002

0.002

0.001

0.001

Fig. 20.

φ


4 1 horizontal scaling = 2 , subpattern : vertical scaling = 24 8 2 horizontal scaling = 2 , subpattern : 8 vertical scaling = 2

11

0.725

0.7375

0.75

2

0.003

0.668

0.7125

17

0.003

0 0.66409 0.66488 0.66566 0.66644 0.66722

0.7

0 0.731

Fig. 21.

0.732

0.733

0.734



614

χ1


χ1

1 110 0.9 0.8 0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

110

0.75

0

χ1

1

110

1

0.8

1

0

χ1

1 124 0.9

1

φ

124

φ

124

φ

124

0.875

1

124

0.7

0.85

0.65

0.825

0.6

0.8

0.55

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.775 2

2

0.5 0.5 0.53125 χ1 2 110 0.525

0.525

0.55

0.575

0.6

φ

0.625

110

0.75 0.5 0.765625 χ1 2 124 0.7625

0.51875

0.759375

0.5125

0.75625

0.50625

0.753125

φ

0.5 0.5

0.503125 0.50625 0.509375 0.5125 0.515625

Fig. 22.



110

0.525

0.55

0.575

0.6

0.625

0.75 0.5

0.503125 0.50625 0.509375 0.5125 0.515625

Fig. 23.




χ1

χ1

1 137 0.9

1 193 0.9

1

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 0

χ1

615

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

1

137

0

137

1

0

χ1

2

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

193

φ

193

φ

193

0.25

193

0.975

0.2375

0.95

0.225

0.925

0.2125

0.9

0.2

1

2

0.875 0.9375 1

0.95

χ1

0.9625

0.975

0.9875

1

φ

137

2

137 0.998438

0.1875 0.4375 0.203124 χ1 2 193 0.202343

0.996875

0.201562

0.995313

0.200781

0.99375

0.199999

0.992188 0.996094

Fig. 24.

φ 0.998047

1



137

0.45

0.199218 0.464844

Fig. 25.

0.4625

0.475

0.466797

0.4875

0.5

0.46875



616



χ1

graph of characteristic function 30 replotted from 1 Table 2 is shown in Fig. 26. The two subpatterns 1 . 2 reveal the fractal geometry of χ and 30

χ1

30


χ1

0.9

1 1 135 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

1

0.1

0.1

1

0 0

χ1


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

30

0.12

0 1

χ1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

φ

135

φ

135

φ

135

2

1

135

30

1

0.9875

0.1 0.08

0.975

1 0.06 0.9625 0.04 0.95

0.02 2 0 0

χ1

0.01

0.02

0.03

0.04

0.05

0.06

φ

0.9375

30

0

χ1

0.01

0.02

0.03

0.04

0.05

1

2

135

30

0.06

0.998438

0.006 0.005

0.996875

2

0.004

0.995313

0.003 0.002

0.99375 0.001 0 0

0.001

Fig. 26.

0.002

0.003



φ

30

0.992188 0

Fig. 27.

0.002

0.004

0.006





χ1

graph of characteristic function 90 replotted from 1 Table 2 is shown in Fig. 28. The two subpatterns 1 . 2 reveal the fractal geometry of χ and 90

χ1

90


graph of characteristic function χ1150 replotted from 1 Table 2 is shown in Fig. 29. The two subpatterns 1 . 2 reveal the fractal geometry of χ and 150

χ1

1 0.9

1 150 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

1

0 0

χ1

617

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

90

0.12

1

0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

150

φ

150

φ

150

χ1

90

150

0.1

0.006

0.08

0.005

1

0.004

0.06

1

0.003 0.04 0.002 0.02

0.001

2 0 0

χ1

0.01

0.02

0.03

0.04

0.05

0.06

φ

2

0

90

0

0.001

0.002

0.003

χ1

90

150 0.0004

0.006 0.005

0.0003

2

0.004

2 0.0002

0.003 0.002

0.0001 0.001 0 0

0.001

Fig. 28.

0.002

0.003



φ

90

0 0

Fig. 29.

0.0001

0.0002



618


4. Predicting the Fractal Structures

bits

The fractal structure of each characteristic function χ1N in Table 2 can be analyzed and predicted using the properties of characteristic functions to be presented below, and from the coefficient b1 in the explicit formulas from Table 1.

4.1. Two-level fractal stratifications Let us partition the unit interval [0,1] into four ∆

∆

∆

subintervals Φ1 = [0, 0.25), Φ2 = [0.25, 0.5), Φ3 = ∆

[0.5, 0.75), and Φ4 = [0.75, 1.0]. It follows from the binary-to-decimal conversion formula {x0 , x1 , x2 , . . . , xI } → φ =

I

2−(i+1) xi

(15)

i=0

that each binary string belonging to these four subintervals must have the following form:  {0, 0, x2 , x3 , x4 , . . . , xI } ∈ Φ1    {0, 1, x2 , x3 , x4 , . . . , xI } ∈ Φ2  (16) {1, 0, x2 , x3 , x4 , . . . , xI } ∈ Φ3     {1, 1, x2 , x3 , x4 , . . . , xI } ∈ Φ4 Let {y0 , y1 , y2 , . . . , yI } denote the image of {x0 , x1 , x2 , . . . , xI } under local rule N : TN

{x0 , x1 , x2 , . . . , xI } −→{y0 , y1 , y2 , . . . , yI }

(17)

where {y0 , y1 , y2 , . . . , yI } → χ1N (φ)

(18)

and φ is the decimal representation in Eq. (15), namely, χ1N (φ) =

I

2−(i+1) yi

(19)

i=0

Observe that if we let the binary string {x0 , x1 , x2 , . . . , xI } assume all binary combinations from {0, 0, 0, . . . , 0} to {1, 1, 1, . . . , 1} in Eq. (17), and calculate the corresponding decimal value from Eq. (19), we would obtain the coordinates (φ N , χ1N ) for plotting the characteristic functions in Table 2. Observe also that the first binary bit “y0 ” of Eq. (17) is given by the first step function (i = 0) in Table 1, as defined by Eq. (7). The value of y0 is therefore determined by the three binary 2

(x−1 , x0 , x1 ) = (xI , x0 , x1 )

(20)

in view of the periodic boundary condition x−1 = xI in Fig. 1(a). In other words, we have: The first binary bit y0 of χ1N of Eq. (18) depends in general on the last binary bit xI of the input binary string of Eq. (15). Property 4.1.

Since the first binary bit y0 of Eq. (18) contributes the largest component 2−1 = 0.5 (if y0 = 1), and since the last binary bit xI alternates between “0” and “1” as we increase φ N from φ N = 0 to φ N = 1 in Table 2, it follows that, depending on the formula in Table 1 for local rule N , the characteristic function χ1N may exhibit a discontinuous jump in χ1N equal to ∆χ1N = 0.5. This implies that certain subintervals of χ1N may exhibit vertical jumps equal to 0.5 between adjacent red and blue bars in Table 2, resulting in a two-level stratification of χ1N . In view of Property 4.1, the characteristic function χ1N in Table 2 may exhibit a discontinuous jump by an amount equal to ∆χ1N = 0.5 over different subintervals Φ1 , Φ2 , Φ3 and Φ4 , respectively. An examination of the graph of each characteristic function χ1N in Table 2 shows that within each subinterval Φi , one of the following groupings of small red squares (resp. blue squares) on top of each thin red bar (resp. blue bar) apply to all small squares located within the same subinterval Φi , i = 1, 2, 3, 4 (this information is coded in color red, blue or violet, in Table 5): Group 1. All small red squares have χ1N ≥ 0.5 and all small blue squares (in the case where there are two-level stratifications) have χ1N < 0.5. In this case, we will paint the upper rectangle of N in column Φi of Table 5 in red, and the rectangle below it will be painted blue. Group 2. The small red and blue squares are located opposite to those of Group 1. In this case, the upper rectangle of Table 5 will be painted blue, and the lower rectangle will be painted red. Group 3. There is no stratification and all small adjacent red and blue squares have χ1N ≥ 0.5. In this case, only the upper rectangle is painted in violet color,2 while the lower rectangle is left blank.

We have chosen the violet color as the nearest approximation of the color combination between the red and blue colors.


619

Table 5. Stratification of characteristic functions over subintervals Φ1 ∈ [0, 0.25), Φ2 ∈ [0.25, 0.5), Φ3 ∈ [0.5, 0.75), and Φ4 ∈ [0.75, 1].

N

Φ1

Φ2

Φ3

Φ4

N

0

32

1

33

2

34

3

35

4

36

5

37

6

38

7

39

8

40

9

41

10

42

11

43

12

44

13

45

14

46

15

47

16

48

17

49

18

50

19

51

20

52

21

53

22

54

23

55

24

56

25

57

26

58

27

59

28

60

29

61

30

62

31

63

Φ1

Φ2

Φ3

Φ4

620


N

Φ1

Φ2

Φ3

Φ4

(Continued )

N

64

96

65

97

66

98

67

99

68

100

69

101

70

102

71

103

72

104

73

105

74

106

75

107

76

108

77

109

78

110

79

111

80

112

81

113

82

114

83

115

84

116

85

117

86

118

87

119

88

120

89

121

90

122

91

123

92

124

93

125

94

126

95

127

Φ1

Φ2

Φ3

Φ4


N

Φ1

Φ2

Φ3

Φ4

(Continued )

N

128

160

129

161

130

162

131

163

132

164

133

165

134

166

135

167

136

168

137

169

138

170

139

171

140

172

141

173

142

174

143

175

144

176

145

177

146

178

147

179

148

180

149

181

150

182

151

183

152

184

153

185

154

186

155

187

156

188

157

189

158

190

159

191

Φ1

Φ2

Φ3

Φ4

621

622


N

Φ1

Φ2

Φ3

Φ4

(Continued )

N

192

224

193

225

194

226

195

227

196

228

197

229

198

230

199

231

200

232

201

233

202

234

203

235

204

236

205

237

206

238

207

239

208

240

209

241

210

242

211

243

212

244

213

245

214

246

215

247

216

248

217

249

218

250

219

251

220

252

221

253

222

254

223

255

Φ1

Φ2

Φ3

Φ4


Group 4. The small red and blue squares are located opposite to those of Group 3. In this case, only the lower rectangle is painted in violet color, while the upper rectangle is left blank.

4.1.1. Stratification prediction procedure The color painted in each of the two rectangles located in each subinterval Φi of the characteristic function χ1N of Table 5 is obtained by inspection of the graph of χ1N in Table 2. We will now show how these colors can be predicted directly from the “firing patterns” [Chua et al., 2003] of each rule N ; namely, from the “three-bit pattern” associated with each red vertex of the Boolean cube representing N in Fig. 1(e). Our goal is to determine the color of the first output bit y0 from these firing patterns. In particular, y0 depends on the three binary bits {x−1 , x0 and, x1 }, as shown in Fig. 30, where x−1 = xI in view of the periodic boundary condition in Fig. 1(a). The “color” of x0 and x1 in Fig. 30 is chosen from the subinterval Φi as defined by the first two columns of Eq. (16); namely,

2. x−1 = 1,

if

xI = 1

This corresponds to the case where the rightmost bar of χ1N in Table 2 is a blue bar. Since both x−1 and y0 can assume a “0” or a “1”, there are four possible outcomes: Outcome 1 : x−1 = 0 , y0 = 0 In this case, all bars belonging to subinterval Φi are colored red and lie below χ1N = 0.5. Outcome 2 : x−1 = 0 , y0 = 1 In this case, all bars belonging to subinterval Φi are colored red and lie above χ1N = 0.5. Outcome 3 : x−1 = 1 , y0 = 0 In this case, all bars belonging to subinterval Φi are colored blue and lie below χ1N = 0.5. Outcome 4 : x−1 = 1 , y0 = 1

x0

x1

In this case, all bars belonging to subinterval Φi are colored blue and lie above χ1N = 0.5.

Φ1 Φ2 Φ3 Φ4

0 0 1 1

0 1 0 1

4.1.2. Examples illustrating stratification prediction procedure Let us illustrate the above “stratification prediction” procedure with some examples.

if

xI = 0

Using the bar-coloring scheme described in p. 1049 of [Chua et al., 2005], this corresponds to

x −1

the case where the right-most bar of χ1N in Table 2 is a red bar.

Φi

The color of x−1 in Fig. 30 is chosen as follows: 1. x−1 = 0,

623

x0

x1

TN y0 Fig. 30. The left-most output bit y0 of χ1N is “1” if {x−1 , x0 , x1 } = {xI , x0 , x1 } is a firing pattern of N . Otherwise, y0 = 0.

Example 4.1 [Predicting Stratification for χ116 ].

The Boolean cube for rule 16 is reproduced from Table 1 of [Chua et al., 2003] in Fig. 31(a). Since 4 is painted red in this Boolean cube, only vertex → 1 0 there is only one firing pattern 0 , as shown in Fig. 31(b). For each subinterval Φi , i = 1, 2, 3, 4, we insert the corresponding x0 and x1 as specified in the preceding table. We then assign x−1 = 0 (for red bars) on the left column and x−1 = 1 (for blue bars) on the right column for each Φi . The color of y0 in each case is then determined from the firing patterns in Fig. 31(b). In this case, y0 = 1 only in (row 1, column 2). Since y0 = 0 in the left column of Φ1 and y0 = 1 in the right column of Φ1 , it follows that χ116 has two stratifications over the subinterval Φ1 , where the red bars are below χ116 = 0.5 and the blue bars are above χ116 = 0.5.

624


16 2

3 1

0

Φ1 Φ2 Φ3 Φ4

T 16

7

6 4

Firing Pattern

5

x −1 0

x0 0

x1 0

x −1 1

T 16

0 y0 x −1 0

x0 0

x1 1

x −1 1

T 16

x0 1

x −1 1

0 y0

0 y0

x0 1

x1 0

T 16

T 16

T 16

x1 1

0 y0 x1 0

x0 1

x0 0 T 16

0 y0

x −1 0

x1 0

1 y0

T 16

x −1 0

x0 0

0 y0 x1 1

x −1 1

x0 1

x1 1

T 16

0 y0

Fig. 31. Predicting stratification for rule 16 . (a) Boolean cube for 16 . (b) Firing patterns for 16 . (c) Stratification determination data.


lie below χ14 = 0.5, because y0 = 0. Again, this prediction is consistent with Table 2.

Since y0 = 0 for all other cases in Fig. 31(c), it follows that there are no stratifications in subinterval Φ2 , Φ3 and Φ4 , and all red and blue bars in these three subintervals are located below χ116 = 0.5, as is indeed the case in Table 2.


The Boolean cube for rule 110 is shown in Fig. 34(a). It has five firing patterns, → 0 0 1, → 0 1 0, → 0 1 1, → 1 0 1, → 1 1 0 , as shown in Fig. 34(b). The stratification determination data in Fig. 34(c) shows only subinterval Φ4 has a stratification where all red bars must lie above χ1110 = 0.5, and all blue bars must lie below it. Since y0 = 0 in both columns of Φ1 , it follows that all red and blue bars must lie below χ1110 = 0.5. On the other hand, since y0 = 1 in both columns of Φ2 and Φ3 , it follows that all red and blue bars in subintervals Φ2 and Φ3 must lie above χ1110 = 0.5. All of these predictions are consistent with Table 2, as expected.


The Boolean cube for rule 2 is shown in Fig. 32(a). → 0 It has only one firing pattern 0 1 , as shown in Fig. 32(b). The corresponding stratification determination data is shown in Fig. 32(c). From these data, we conclude that only subinterval Φ2 has a stratification in χ12 where all red bars lie above, and all blue bars lie below χ12 = 0.5. Since y0 = 0 in the other three subintervals Φ1 , Φ3 and Φ4 , all red and blue bars must lie below χ12 = 0.5. Observe that this prediction is consistent with Table 2, as expected. Example 4.3 [Predicting Stratification for χ14 ].

4.1.3. {Φ1 , Φ2 , Φ3 , Φ4 } — stratified families

The Boolean cube for rule 4 is shown in Fig. 33(a). → 0 1 0, It has only one firing pattern as shown in Fig. 33(b). In this case, only subinterval Φ3 in Fig. 33(c) has a stratification in χ14 where all red bars must lie above χ14 = 0.5, and all blue bars must lie below χ14 = 0.5. All red and blue bars in the other three subintervals Φ1 , Φ2 and Φ4 must

If we examine the color of the rectangles of the four subintervals Φ1 , Φ2 , Φ3 and Φ4 in Table 5 and code each violet rectangle by a “0” binary bit, and all other rectangles by a “1” binary bit, we will discover the first 16 rules follow the four-bit binary number system as shown below, henceforth called the stratified family “0”. Equivalent Decimal Number

N

ψ

Φ1

Φ2

Φ3

Φ4

0

ψ0

0

0

0

0

0

1

ψ1

1

0

0

0

1

2

ψ

0

1

0

0

2

3

ψ3

1

1

0

0

3

4

ψ

0

0

1

0

4

2

4

5

ψ5

1

0

1

0

5

6

0

1

1

0

6

Family

ψ6

7

ψ7

1

1

1

0

7

“0”

8

ψ8

0

0

0

1

8

9

ψ9

1

0

0

1

9

10

1

0

1

10

Stratified

625

ψ 10

0

11

ψ 11

1

1

0

1

11

12

ψ 12

0

0

1

1

12

13

ψ 13

1

0

1

1

13

14

ψ 14

0

1

1

1

14

15

ψ 15

1

1

1

1

15

Observe that the color background of Table 1 is determined from this coding scheme, where “0” is coded “light blue”, while “1” is coded “light pink” in Table 1.

626


2 2

3 1

0

Φ1 Φ2 Φ3 Φ4

T2

7

6 4

Firing Pattern

5

x −1 0

x0 0

x1 0

x −1 1

T2

0 y0

x −1 0

x0 0

x1 0 T2

0 y0

x0 0

x1 1

x −1 1

x0 0 T2

T2

0 y0

1 y0

x −1 0

x0 1

x1 0

x −1 1

x0 1

x1 0 T2

T2

0 y0

x −1 0

x1 1

0 y0

x0 1

x1 1

x −1 1

x0 1 T2

T2

0 y0

x1 1

0 y0

Fig. 32. Predicting stratification for rule 2 . (a) Boolean cube for 2 . (b) Firing patterns for 2 . (c) Stratification determination data .


4 2

3

Φ1 Φ2 Φ3 Φ4

T4

1

0 4

Firing Pattern

7

6

627

5

x −1 0

x0 0

x1 0

x −1 1

T4

0 y0

x −1 0

x0 0

x1 0 T4

0 y0

x0 0

x1 1

x −1 1

x0 0 T4

T4

0 y0

x −1 0

0 y0

x0 1

x1 0

x −1 1

x0 1

x1 0 T4

T4

0 y0

1 y0

x −1 0

x1 1

x0 1

x1 1

x −1 1

x0 1 T4

T4

0 y0

x1 1

0 y0


628


110 2

3

T 110

7

6

1

0 4

Firing Patterns T 110

T 110

T 110

T 110

5

Φ1 Φ2 Φ3 Φ4

x −1 0

x0 0

x1 0

x −1 1

T 110

x0 0

0 y0

x1 1

x −1 1

T 110

x0 1

x1 0

x −1 1

T 110

1 y0

x0 1

x1 0

T 110

1 y0

x0 1

x1 1

1 y0

T 110

x −1 0

x0 0 T 110

1 y0

x −1 0

x1 0

T 110

0 y0

x −1 0

x0 0

1 y0 x1 1

x −1 1

x0 1

x1 1

T 110

0 y0



Observe that no two members of the stratified family “0” have identical four-bit binary patterns. Since there are only 16 distinct combinations of four binary bits, the stratified family “0” had already consumed all of them and it is clear that the remaining rules in Table 5 must consist of repetitions

Stratified Family “1”

of these patterns. A careful analysis of the next 16 rules reveals, however, that the four-bit binary patterns of the next 16 rules {16, 17, . . . , 31} are mere permutations of the stratified family “0”, as illustrated below (henceforth called stratified family “1”):

N

ψ

Φ1

Φ2

Φ3

Φ4

Equivalent Decimal Number

16

ψ0

1

0

0

0

1

17

ψ1

0

0

0

0

0

18

ψ

1

1

0

0

3

19

ψ3

0

1

0

0

2

20

ψ

1

0

1

0

5

21

ψ5

0

0

1

0

4

22

ψ6

1

1

1

0

7

23

ψ7

0

1

1

0

6

24

ψ8

1

0

0

1

9

25

ψ9

0

0

0

1

8

26

ψ 10

1

1

0

1

11

27

ψ 11

0

1

0

1

10

28

ψ 12

1

0

1

1

13

29

ψ 13

0

0

1

1

12

30

ψ 14

1

1

1

1

15

31

ψ 15

0

1

1

1

14

2

4

629

Observe that the 4-bit code ψ = {Φ1 , Φ2 , Φ3 , Φ4 } of stratified family “1” and stratified family “0” are related by a 16 × 16 permutation matrix; namely,   0 ψ0 (1)     ψ1 (1)       ψ2 (1)   0     ψ (1)   0    3     ψ4 (1)   0     ψ5 (1)   0        ψ6 (1)   0     ψ7 (1)   0    =   ψ8 (1)   0     ψ9 (1)   0     ψ (1)   0  10       ψ11 (1)   0     ψ (1)   0  12       ψ13 (1)   0     ψ14 (1)   0    0 ψ15 (1) 

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 M1,0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

  0 ψ0 (0)   0   ψ1 (0)      0   ψ2 (0)    0   ψ3 (0)    0   ψ4 (0)      0   ψ5 (0)    0   ψ6 (0)     ψ7 (0)  0     0   ψ8 (0)      0   ψ9 (0)    0   ψ10 (0)       0   ψ11 (0)     0   ψ12 (0)    0   ψ13 (0)      ψ14 (0)    0 ψ15 (0)

630


It turns out that the remaining rules can be partitioned in a similar way into a stratified family “2”, “3”, . . . ,“15”. Together, these 16 families correspond exactly to the 16 rules listed in the 16 pages of Table 1, respectively. Let us summarize this remarkable organization as follows:

corresponding Φi of 255 , 231 , 135 and 35 , respectively.

Property 4.2. The 16 rules listed in each of the

Since the occurrence of two-level stratifications comes from the term b1 x−1 [i = 0 in Eq. (7)] in Table 1, it follows that no such stratification can occur in χ1N if b1 = 0 for rule N . An examination of Table 2 shows that there are only 16 rules with b1 = 0. These 16 xI -insensitive rules are listed in Table 7. Observe that these 16 rules can be generated from the following formula:

16 pages of Table 1 form a stratified family whose four-bit binary patterns ψ(k), k = 0, 1, 2, . . . , 15, are related to each other by a 16×16 permutation matrix ψ(k) = Mk,l ψ(l)

Table 6 lists 16 matrices which transform ψ(0) from stratified family “0” into ψ(k) of stratified family “k”, k = 0, 1, 2, . . . , 15. The four-bit binary patterns of the last eight rules of each stratified family “k” can be obtained by copying the four-bit binary patterns of the first eight rules and then complementing the fourth bit corresponding to Φ4 . Observe that, in Table 5, the four color codes of all Φi ’s of each local rule N is equivalent to the vertically-switched color codes of all Φi ’s of the corresponding local rule 255 − N as illustrated below:

, For example, the color code of each Φi of 0 , 24 , 120 and 220 in Table 5 can be obtained by switching vertically the color code of the

Single-Level F ractals

 

0, 0, . . . , 0 , x , x , . . . , xI → φ   k k+1 first k entries I i=0

2−(i+1) xi

β = 0, 1, 2, . . . , 15

This relationship can be derived from Table 6. In particular, since the right-most bit xI -insensitive rules corresponds to ψi = {Φ1 , Φ2 , Φ3 , Φ4 } = {0, 0, 0, 0}, it follows from Tables 1 and 6 that ψi = {0, 0, 0, 0} is repeated after integer multiplies of 17, and hence all xI -insensitive rules N must satisfy N = 17k, k = 0, 1, 2, . . . , 15. These 16 rules are composed of the corresponding (k + 1)th members in the stratified family “k”, k = 0, 1, 2, . . . , 15. For example, 0 is the first member of the stratified family “0”, 17 is the second member of the stratified family “1”, . . . , and 225 is the sixteenth member of the stratified family “15”. Property 4.3. The following 16 characteristic func-

tions have no stratifications:

χ1136 , χ1153 , χ1170 , χ1187 , χ1204 , χ1221 , χ1238 , and χ1255

We will now uncover a basic mechanism responsible for the presence of fractals in the characteristic functions shown in Table 2. Let us examine an infinitesimally small neighborhood of the origin of χ1N by considering the input string

=

N = 17 β ,

χ10 , χ117 , χ134 , χ151 χ168 , χ185 , χ1102 , χ1119 ,

4.3. Origin of the fractal structures

 

4.2. Rules having no fractal stratifications

(21)

If we multiply φ by 2k in Eq. (21), we would obtain φ • 2k =

I i=k

2−(i+1)+k xi =

I−k

2−(j+1) xj

(22)

j=0

where j (i−k). Observe that, as I → ∞, Eq. (22) converges to the binary string in Eq. (15). It follows that no matter how close the binary string in Eq. (21) is to the origin, we can always rescale it to recover the original string in Eq. (15). Moreover, except possibly for the (k + 1)th term xk = 1, all other terms xk+1 , xk+2 , . . . , xI in Eq. (21) will


631

Table 6. List of 16 permutation matrices Mk,l which transform ψ(0) from stratified family “0” into ψ(k) of stratified family “k”, k = 0, 1, 2, . . . , 15.

632


(Continued )

Chapter 5: Fractals Everywhere List of 16 right-most bit xI -insensitive local rules.

Table 7.

0

17

34

51

68

85

102

119

the form     0, 0, . . . , 0, yk−1 , yk , yk+1 , . . . , yI → χ1N   first k terms I −(i+1)

=

136

153

170

187

204

221

238

255

633

2

yi

(25)

i=k−1

Multiplying the right side of Eq. (25) by 2(k−1) we obtain I 1 k−1 2−(i+1)+(k−1) yi χ N (2 ) = i=k−1

generate the same corresponding output binary bit yi , i = k + 1, k + 2, . . . , I, because the same local rule N is used. Hence, by rescaling χ1N accordingly, we can recover the corresponding original graph. To determine the appropriate scaling factor, it is necessary to examine the output binary bit yi , i = 0, 1, 2, . . . , k − 1 because it may not be zero for two reasons: (i) yk−1 = 1 if TN

(xk−2 , xk−1 , xk ) = (0, 0, 1) −→ 1

(23)

is a firing pattern [Chua In other words, if et al., 2003] of Rule N , or equivalently, if vertex 1 in the Boolean cube in Fig. 1(e) is painted in red color.

TN

(xi−1 , xi , xi+1 ) = (0, 0, 0) −→ 1

j=0

=

I

2−(i+1) yi ,

as I → ∞ (26)

i=0

Hence, by multiplying the output string in Eq. (25) by the factor 2(k−1) , we obtain Eq. (26), which is identical to Eq. (18) as I → ∞. In other words, in Scenario 1, the characteristic function χ1N arbitrarily near the origin can be rescaled to obtain the corresponding original graph by multiplying φ N by 2k and χ1N by 2(k−1) . Scenario 2. (0, 0, 0) is not a firing pattern but

(1, 0, 0) is a firing pattern of N , and xI = 1 with b1 = 0.

pattern 3

(0, 0, 1) is a firing pattern of N , and xI = 0.

{1, 0, 0, . . . , 0, yk−1 , yk , yk+1 , . . . , yI }

(24)

for i = 0, 1, 2, . . . , k − 2. In other words, if is a firing pattern of Rule N , or 0 in the Boolean cube in equivalently, if vertex Fig. 1(e) is painted in red color. Let us examine now the consequences of the occurrence of one or both of these two cases. There are four possible scenarios: but

An example satisfying scenario 1 is Rule 2 , 0 is blue but vertex 1 is red. In this where vertex case, yk−1 = 1 and the output binary pattern has 3

2−(j+1) yj

In this case, y0 = 1 in view of Eq. (19), and the output binary string assumes the form:

(ii) yi = 1 if

Scenario 1. (0, 0, 0) is not a firing

=

I−k+1

→

I 1 + 2−(i+1) yi 2

(27)

i=k−1

It follows from Eq. (27) that 1

1

˜ N χN χ

I 1 − = 2−(i+1) yi 2 i=k−1

=

I

2−(i+1) yi ,

as I → ∞

(28)

i=0

as in Eq. (26). Hence, apart from a translation by ∆χ1N = 1/2, the infinitesimal pattern near the origin can be rescaled to coincide with the corresponding original pattern.

An examination of the Boolean cubes in Table 1 of [Chua et al., 2003] shows (0, 0, 0) is not a firing pattern for rules with an even number: i.e. N = 2n, n = 0, 1, 2, . . . , 128.

634


Scenario 3. (0, 0, 0) is a firing pattern of N , and

xI = 0.

first k−1 terms

2−(i+1) +

χ1N =

k−2

I

2−(i+1) +

i=0

2−(i+1) yi

i=k−1 −(k−1)

=1−2

+

I

2−(i+1) yi

I

2−(i+1) yi

(31)

i=k−1

Multiplying both sides of Eq. (31) by the scaling factor 2(k−1) , we obtain Ñ •2 χ

=1−

I

=1−

2−(i+1) yi ,

as I → ∞

(32)

i=0

as in Eq. (26). Since Eq. (32) is an affine transformation of Eq. (26), it follows that the fractal pattern arbitrarily near the origin is preserved upon scaling, apart from an affine transformation. Scenario 4. (0, 0, 0) is a firing pattern of N , but (1, 0, 0) is not a firing pattern of N , and xI = 1 with b1 = 0.

In this case, y0 = 0, in view of Eq. (18); namely,     0, 1, 1, . . . , 1, yk−1 , yk , yk+1 , . . . , yI → χ1N (33)   first k terms

where χ1N =

I i=1

I 1 − 2−(k−1) + 2−(i+1) yi 2

(34)

i=k−1

It follows from Eq. (34) that

= 2−(k−1) −

I

2−(i+1) yi ,

as I → ∞

(35)

2−(i+1) yi

as in Eq. (26). Since Eq. (34) is an affine transformation of Eq. (26), it follows that the fractal pattern arbitrarily near the origin is preserved upon scaling, apart from an affine transformation.

5. Gardens of Eden

2−(i+1)+(k−1) yi

i=k−1 I

i=k−1

i=0

It follows from Eq. (30) that

(k−1)

2−(i+1) yi

i=k−1

(30)

i=k−1

˜ 1N 1 − χ1N = 2−(k−1) − χ

=

I

1 ˜ 1N 1 − χ1N − χ 2 I 2−(i+1) yi = 2−(k−1) −

where

4

k−2 i=1

In this case, yi , i = 0, 1, 2, . . . , k − 2; namely,     1, 1, . . . , 1, yk−1 , yk , yk+1 , . . . , yI → χ1N (29)  

1

=

A cursory inspection of the graphs of the characteristic functions χ1N displayed in Table 2 reveals that most of these graphs exhibit a discontinuous jump − 1 over a finite range ∆χ1N = χ1N (φ+ jump )−χ N (φjump ) at various discrete points φjump ∈ [0, 1] where all − 1 points within the interval (χ1N (φ+ jump ), χ N (φjump )) have no preimage under χ1N . Such a characteristic function is therefore not a surjective (onto) function. Points within such intervals are special cases of the following interesting class of initial bit-string configurations. Definition 5.1. Garden of Eden of N . An (I + 1)-bit binary string {x0 , x1 , x2 , . . . , xI } is said to be a garden of Eden of a local rule N iff it does not have a predecessor under the local rule transformation T N . It follows from Definition 5.1 that a garden of Eden φ0 Ii=0 2−(i+1) xi of N can never occur as a point on an orbit of N arising from some initial bit-string configuration whose decimal equivalent is different from φ0 . Hence, a garden of Eden has no past, but only present and future.4

The name garden of Eden was first introduced by Moore in the context of von Neumann’s self-reproducing automata [Moore, 1962].


χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1 χ1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

A compendium of all gardens of Eden (colored in green) over the range [0, 1] of χ1N .

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

635

636



64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95

(Continued )

=

χ1

=

χ1

=

χ1

=

χ1

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

96 97 98 99

100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127

= = = =



128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159

(Continued )

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191

637

638



192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223

(Continued )

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

=

χ1

=

224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255


A compendium of period-n isles of Eden of N . Table 9-1(a).

n

Period-1 Isle of Eden

Period-1 isles of Eden.

Rules Endowed with Isle of Eden 134

140

142

148

150

156

158

162

166

170

172

174

176

180

182

184

186

188

190

196

198

204

206

212

214

220

222

226

228

230

234

236

238

240

242

244

246

248

250

252

254

128

130

132

134

136

138

140

142

144

146

148

150

152

154

156

158

160

162

168

170

176

184

186

192

194

196

198

200

202

204

206

208

210

212

214

216

220

224

226

240

242

4

6

7

12

14

15

20

21

22

23

30

31

68

84

85

86

87

132

134

135

140

142

143

148

149

150

151

158

159

196

204

206

207

212

213

214

215

220

221

222

223

72

73

76

104

105

108

109

200

201

204

205

233

236

237

5

72

76

200

204

236

6

200

204

205

236

237

7*

45

101

173

204

205

8*

74

75

88

89

204

1

2

3

4

* Here I +1 must be divisible by 3.

229

639

640

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 9-1(b).

n


Additional period-1 isles of Eden.

Rules Endowed with Isle of Eden

9

4

204

10

4

204

4

204

12

204

223

13

204

223

204

223

15

200

204

16

200

204

200

204

18

204

236

19

204

236

204

236

11

14

17

20 21

any combination of with blue string for any I :

any combination of with red string for any I :

any combination of with blue string for any I :

any combination of with red string for any I :

any pattern and any I

204


Table 9-2.

n


1

2

3 4

any pattern and any I

641



15

27

29

35

41

43

49

51

57

59

69

71

77

79

85

93

97

99

105

107

113

115

121

32

33

34

35

40

41

42

43

48

49

51

59

96

97

104

105

106

107

112

113

115

120

121

123

168

169

170

171

187

224

225

232

233

234

235

240

241

243

248

249

251

18

19

22

23

50

51

54

55

146

147

150

151

178

179

182

182

51

Remarks : For n = 1, Period-2 Isles of Eden can exist for any I. For n = 2, Period-2 Isles of Eden can exist for any I + 1 that is divisible by 2. For n = 3, Period-2 Isles of Eden can exist for any I + 1 that is divisible by 4.

642


Table 9-3.

n 1

2

3

4




3

42

43

75

99

106

107

130

131

170

171

202

203

226

227

234

235

16

17

57

89

112

113

120

121

144

145

184

185

216

217

240

241

248

249

40

41

42

43

44

45

56

57

62

63

168

169

170

171

172

184

190

191

96

97

98

99

100

101

112

113

118

119

224

225

226

228

240

241

246

247

Period-3 Isles of Eden can exist for any I + 1 that is divisible by 3.


Table 9-4.

n




1

10

15

170

175

2

80

85

240

245

3

15

4

85


643

644

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 9-5(a).

n




1

6

2

20

3

159

4

215

5

26

27

31

58

58

63

6

82

83

87

114

115

119

7

3

7

35

39

163

167

Chapter 5: Fractals Everywhere Table 9-5(b).

n




8

17

21

49

53

117

181

9

40

41

42

43

56

170

10

96

97

98

112

113

240

11

43

107

170

171

227

235

12

113

121

185

240

241

249

13

15

43

14

85

113

645

646


Table 9-6.

n




1

14

15

142

143

2

84

85

212

213

170

240

3

any pattern at I = 5


Chapter 5: Fractals Everywhere Table 9-7.

n




1

40

41

42

43

169

170

2

96

97

112

113

225

240

3

43

106

107

170

171

235

4

113

120

121

240

241

249


647

648

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 9-8(a).

n




1

40

41

42

43

56

169

170

2

96

97

98

112

113

225

240

3

43

106

107

170

171

227

235

4

113

120

121

185

240

241

249


Table 9-8(b).

n




5

14

15

142

143

6

84

85

212

213


649

650

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 9-9.

n




1

40

41

42

43

169

170

2

96

97

112

113

225

240

3

43

106

107

170

171

235

4

113

120

121

240

241

249



Table 9-10(a).

n




1

40

41

42

43

169

170

2

96

97

112

113

225

240

3

43

106

107

170

171

235

651

652


n




4

113

120

121

240

5

14

15

142

143

6

84

85

212

213

241


249

Chapter 5: Fractals Everywhere Table 9-11(a).

n

1

2

3




40

41

42

43

169

170

96

97

112

113

225

240

43

106

107

170

171

235

653

654


n

4

5

6




113

120

121

240

241

249

40

41

42

43

169

170

96

97

112

113

225

240

Chapter 5: Fractals Everywhere Table 9-11(c).

n



7

8

655


43

106

107

170

171

235

113

120

121

240

241

249

Period-11 Isles of Eden can exist for any I + 1 that is divisible by 11. Observe that any binary string {x0 , x1 , x2 , . . . , xI } → φ0 =

I

2−(i+1) xi

(36)

i=0

which has no preimage under χ1N (i.e. φ0 does not belong to the range of χ1N ) is a garden of Eden of N . In other words, φ0 ∈ [0, 1] is a garden of Eden of N if there does not exist a φ−1 ∈ [0, 1] such that φ0 = χ1N (φ−1 ), where φ−1 = φ0 . Note that this property is only a sufficient condition for φ0 to be a garden of Eden. In the next section, we will see that there exist some rather special points which

violate this property, but is nevertheless a garden of Eden because it satisfies Definition 5.1. A compendium of all gardens of Eden belonging to each local rule N is listed in Table 8. Each point φ0 ∈ [0, 1] printed in green along the χ1N axis does not have a preimage under χ1N and is therefore a garden of Eden.

6. Isle of Eden For most rules N , there exist some special period-1 points which have no predecessors in the sense that no orbits from other initial bit-string

656


Table 10.

Rules without period-k isles of Eden, k < 12.

0

1

5

8

9

11

24

25

28

36

37

38

46

47

52

60

61

64

65

66

67

70

78

81

90

91

92

94

95

102

103

110

111

116

117

122

124

125

126

127

129

133

137

139

141

153

155

157

161

164

165

189

193

195

197

199

209

211

218

219

231

239

253

255

configurations can converge to such points, and hence they are also gardens of Eden in view of Definition 5.1. Moreover, since each of these points is a period-1 point, they have a preimage under χ1N ; namely, itself. Such special points (obtained by exhaustive computer search) are henceforth called isles of Eden of N . Observe that an isle of Eden has no past, and no future (in the poetic sense that time stood still)! A compendium of all period-1 isles of Eden of N for binary strings of various length I +1 is listed in Tables 9-1(a) and 9-1(b). Let us now generalize the concept of period-1 isle of Eden for “time-n” characteristic functions χnN . Since such special period-n points also have no predecessors under the “nth iterated” map χnN , they are henceforth called period-n isles of Eden. Table 9-k lists all period-k isles of Eden, k = 1, 2, 3, . . . , 11, obtained by an exhaustive computer search over all binary bit-strings of length I + 1 = 8, 9, 10, 11, respectively. In most cases, the same bit-string configurations can be extended to arbitrarily large I, provided I + 1 is divisible by some integer specified in the table.

Finally, we remark that based on exhaustive computer search, there are 64 local rules that do not have period-k isles of Eden, at least for k < 12. They are listed in Table 10.

7. Concluding Remarks We have demonstrated that except for the eight affine (mod 1 ) rules listed in Table 3, the graph of all characteristic functions χ1N in Table 2 are endowed with a fractal structure. Indeed, even the graph of the eight affine (mod 1) rules can be considered to exhibit a degenerate form of fractal structure since arbitrarily short segments of the graph can be made to coincide with corresponding portions of the original graphs by appropriate affine transformations. We have traced the origin of these fractals to the decimal representation of binary bitstrings in Eq. (2), as well as to the local rule, which must apply to any bit string, regardless of the number of “zeros” in front of it. The explicit formulas in Table 1 for converting any bit string {x0 , x1 , x2 , . . . , xI } into a real number χ1N ∈ [0, 1] is remarkable for its scope of potential applications. Indeed, these formulas can be interpreted as a digital-to-analog converter in closed form. Such formulas could not have been derived without the explicit “universal” formula derived in [Chua et al., 2003] for all local rules. The widespread presence of period-k “isles of Eden” came as a surprise since they certainly have no counter part in hyperbolic differential equations. To dramatize this phenomenon, we end Part V with our following poetic interpretation of the above new phenomenon: Hidden within the “garden of Eden”, which has no past, one finds immortality in an “isle of Eden”, which has neither past nor future.

♦

Chapter 6 FROM TIME-REVERSIBLE ATTRACTORS TO THE ARROW OF TIME ♦

Ë This paper proves, via an analytical approach, that 170 (out of 256) Boolean CA rules in a onedimensional cellular automata (CA) are time-reversible in a generalized sense. The dynamics on each attractor of a time-reversible rule N is exactly mirrored, in both space and time, by its bilateral twin rule N † . In particular, all 69 period-1 rules, 17 (out of 25) period-2 rules, and 84 (out of 112) Bernoulli rules are time-reversible. The remaining 86 CA rules are time-irreversible in the sense that N and N † mirror their dynamics only in space, but not in time. In this case, each attractor of N defines a unique arrow of time. A simple “time-reversal test” is given for testing whether an attractor of a CA rule is timereversible or time-irreversible. For a time-reversible attractor of a CA rule N the past can be uniquely recovered from the future of N † , and vice versa. This remarkable property provides 170 concrete examples of CA time machines where time travel can be routinely achieved by merely hopping from one attractor to its bilateral twin attractor, and vice versa. Moreover, the time-reversal property of some local rules can be programmed to mimic the matter –antimatter “annihilation” or “pair-production” phenomenon from high-energy physics, as well as to mimic the “contraction” or “expansion” scenarios associated with the Big Bang from cosmology. Unlike the conventional laws of physics, which are based on a unique universe, most CA rules have multiple universes (i.e. attractors), each blessed with its own laws. Moreover, some CA rules are endowed with both time-reversible attractors and time-irreversible attractors. Using an analytical approach, the time-τ return map of each Bernoulli στ -shift attractor of all 112 Bernoulli rules are shown to obey an ultra-compact formula in closed form, namely,

or its inverse map. 657

658


These maps completely characterize the time-asymptotic (steady state) behavior of the nonlinear dynamics on the attractors. In-depth analysis of all but 18 global equivalence classes of CA rules have been derived, along with their basins of attraction, which characterize their transient regimes. Above all, this paper provides a rigorous nonlinear dynamics foundation for a paradigm shift from an empirical-based approach ` a la Wolfram to an attractor-based analytical theory of cellular automata.

1. Recap on Time-τ Characteristic Functions and Return Maps

Observe that

To each (I + 1)-bit forward Boolean string ∆

x = [x0

x1

x2

···

xI−2

xI−1 xI ], xi ∈ {0, 1}

φ=

2−(i+1) xi

(1)

(2)

i=0

∆

xI−1

xI−2

···

x2

x1

x0 ]

=

I

2−(I+1)+i xi

where the equality sign on the left is attained when xi = 0, i = 0, 1, 2, . . . , I, and where the equality sign on the right is attained when xi = 1, i = 0, 1, 2, . . . , I, where I → ∞. Let ∆

τ xn+τ = T N (xn ),

(4)

τ φn+τ = T N (φn ),



0  0  † ∆  .. T =.  0 1

1

0 0 .. . 1 0

··· 0 ··· 1 .. .. . . ··· 0 ··· 0

(5)  1  0 ..  .   0 0

τ = 1, 2, . . . , M

χτN : [0, 1] → [0, 1]

The forward and backward Boolean strings are related by1 where

τ = 1, 2, . . . , M

(9)

(6)

(10)

denote the bit-string sequence resulting from applying the local rule N to the initial configuration xn a total of τ times. The graph

i=0

= T† (x)

(8)

∆

(3)

we associate a unique real number φ†

0 ≤ φ† ≤ 1

and its decimal representation2

Conversely, to each backward Boolean string = [xI

(7)

and

of the one-dimensional cellular automata shown in Fig. 1(a), we associate a unique real number I

0≤φ≤1

(11)

defined by χτN

φn → φn+τ

(12)

where φn ∈ [0, 1] is defined in Eq. (2), and I → ∞, is called the time-τ characteristic function of N . In the case where I is finite, the domain and range of the characteristic function χτN is a finite subset of [0, 1], as depicted by the 201 thin red and blue bars in Table 2 of [Chua et al., 2005a], and in Table A-1 of the Appendix.

We have abused our notation by choosing, for convenience, the same symbol defined differently in [Chua et al., 2004] as an element of the Vierergruppe. The choice will be clear from the context. 2 The italic “T τ ” in Eqs. (9) and (10) denotes the τ th iterate of xn under the Boolean function N , and is not related to “T† ” in Eq. (6).

Chapter 6: From Time-Reversible Attractors to the Arrow of Time

0 0

0

1

1

0

1

1

Cell (I-2) Cell Cell (I-1) I

1

0

1

1

1 1

0

1

0

0

0

1

0

0

1

1

Symbolic Truth Table

output uit +1

Local Rule N

uit +1

=

N (uit−1 , uit , uit+1 )

22 =

26 =

64

24 = 16

0

0

0

1

0

0

1

2

0

1

0

3

0

1

1

4

1

0

0

5

1

0

1

6

1

1

0

7

1

1

1

(-1,1,-1) 4 0 1 0 2

6 20 = 1

1 1 0

0

(1,1,-1) 0 0 0

u

t i −1

4

(1,-1,-1)

0

Cell (i+1)

β0 β1 β2 β3 β4 β5 β6 β7

uit−1 uit uit+1 uit +1 0

-1

-1

-1

1

-1

-1

1

2

-1

1

-1

3

-1

1

1

4

1

-1

-1

5

1

-1

1

6

1

1

-1

7

1

1

1

γ0 γ1 γ2 γ3 γ4 γ5 γ6 γ7

u it (-1,1,1)

(1,1,1) 7 1 1

1

0 (-1,-1,-1)

(1,-1,1) 1 0 0

1

Numeric Truth Table

uit−1 uit uit+1 uit +1 input u it−1 u it u it+1

1

1

0

Cell Cell (i-1) i

Cell Cell Cell 2 0 1

0

659

5

1 0 1

3

0 1 1

23 = 8

27 = 128 1 0 0 1 21 = 2 (-1,-1,1)

u it+1

25 = 32


660


For each attractor Λi , and for each nonattracting invariant orbit Λi , of a local rule N , we define an associated forward time-τ return map ρτN : φn−τ → φn

(13)

where ∆

ρτN (φn−τ ) = χτN (φn−τ )

(14)

Similarly, we define an associated backward time-τ return map3 τ

ρ†N : φ†n−τ → φ†n

(15)

where τ

∆

ρ†N (φ†n−τ ) = χτN (φ†n−τ )

(16)

The forward and backward time-1 return maps associated with the robust attractors of 250 local rules, as well as with the nonattracting invariant orbits of the six remaining local rules 15 , 51 , 85 , 170 , 204 , and 240 , are listed in Table 2 of [Chua et al., 2005a].

2. Rule 62 Has Four Distinct Topological Dynamics An inspection of Table 9 (p. 1137) and Table 13-9 (p. 1150) of [Chua et al., 2005a] reveals that the local rule 62 is endowed with at least three types of invariant orbits with distinct topological dynamics. Our goal in this section is to conduct an indepth study of the qualitatively distinct asymptotic (steady state) behaviors of the evolution dynamics of 62 . Let us begin by characterizing the properties of period-k orbits, k = 1, 2, . . . , 6, by examining the six time-τ characteristic functions χτ62 of 62 shown in Fig. 2.4 A careful examination of the six graphs in Fig. 2 shows that there are no square icons lying on the 3

main diagonal in Figs. 2(a), 2(b), 2(d) and 2(e). It follows therefore that 62 has no period-1, period-2, period-4, and period-5 orbits among the 201 points φi ∈ [0, 1], i = 1, 2, . . . , 201 shown in Fig. 2. A more careful analysis in Sec. 2.1, however, will prove that the origin (φ = 0), which corresponds to a bit string made of only blue pixels, is in fact a period-1 orbit of 62 . It follows that φ = 0 is a period-k orbit of 62 for all k = 1, 2, . . ..5 The topological characteristics of this rather trivial period-1 orbit Λ1 ( 62 ) are summarized in the first part of Table 1A. Column 1 of this table lists the five firing patterns of 62 [Chua et al., 2003]. Column 2 of Table 1A decrees the “law governing the attractor bit-strings”,6 which in this case is trivial. Column 3 of Table 1A shows the forward time-1 return map ρ162 of Λ1 ( 62 ) on top, and a directed graph 1 ( 62 ) (below) which provides a simple algorithm for generating all orbits inhabiting the attractor Λ1 ( 62 ). Column 4 lists the other local rules which are globally equivalent [Chua et al., 2004] to 62 , along with their firing patterns. A cursory inspection of Figs. 2(c) and 2(f) shows that there are many icons lying on the main diagonal line of χ362 and χ662 . It follows that 62 has many period-3 points, and hence also period-6 points, and period-3k points, k = 3, 4, . . .. We will prove in Sec. 2.2 that some of these period-3 points are in fact period-3 isles of Eden [Chua et al., 2005b] in the sense that there are no bit strings different from these three points which converge to them, i.e. they have an empty basin of attraction. The characterization of a generic7 “isle of Eden” invariant orbit Λ2 ( 62 ) is summarized in the second part of Table 1A. The bit string Λ2 ( 62 ) can be generated trivially from the graph 2 ( 62 ). The remaining period-3 points all exhibit the same topological dynamics; namely, each has a

We have modified the return map notations in [Chua et al., 2005a] slightly for better clarity. Each characteristic function χτ62 is defined by the coordinates of 201 blue or red small square icons, each one sitting on top of a thin blue or red bar. The alternating blue and red bars are separated from each other by ∆φ = 0.005 as in [Chua et al., 2005a]. The first bar is a blue bar located at φ = 0.000 . . . 001, and the last bar is a red bar located at φ = 0.9999999, as specified in Table A-1 of the Appendix. 5 No square icon appears at the origin φ = 0.00 . . . 0 of the six characteristic functions shown in Fig. 2 because the first bar plotted in Fig. 2 starts with φ1 = 0.000 + 2−I , where I = 63. 6 We abuse our language here by using the term “attractor” in the column heading of Table 1, and elsewhere, in favor of the technically more precise term “invariant orbits”. 7 We will show in Fig. 5 that 62 can have several distinct “period-3 isles of Eden”. Since their qualitative dynamics are the same, we exhibit only one period-3 isle of Eden in Table 1A, as implied by the heading “generic” attractor bit-strings. 4

Chapter 6: From Time-Reversible Attractors to the Arrow of Time 1 62

1 0.9

0.9 0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.2

0.4

0.6

0.8

1

φ

1

0 4 62

0.9

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.4

0.6

0.8

1

φ

1

0 6 62

0.9

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.4 Fig. 2.

0.6

0.8

1

φ

0.8

0.2

0.4

0.6

0.8

1

φ

0.2

0.4

0.6

0.8

1

φ

0.9 0.8

0.2

0.6

1

0.8

0

0.4

0.9 0.8

0.2

1

φ

0.2

1

0.8

0 5 62

1

0.8

0 3 62

2 62

661

0

Six time-τ characteristic functions χτ

62

of 62 , τ = 1, 2, 3, 4, 5, 6.

Table 1A.

Characterization of bit strings inhabiting the four topologically distinct classes of invariant orbits Λi ( 62 ), i = 1, 2, 3, 4, of local rule 62 .

Firing Patterns

Laws Governing Generic Attractor Bit-Strings

( )

Period-1Attractor : Λ1 62

62 All pixels are blue.

Attractor Bit-String Generator (N)

( )

( )

Attractor : Λ1 62

662

( )

1 2 3 4 5

2 adjacent red bits separated from each other by one blue bit:

xi = 1 ⇒ xi −1 = 0 and xi +1 = 1 , or xi −1 = 1 and xi +1 = 0

( )

0

145

7 1

⇓

2

118

5

( )

φn 1

4

( )

Time-1 Return map

0

⇓

T† 62

T 62

4 6 0

1

⇓ 7

131

1

( )

T* 62

Example : 011011011011011011011011011011011

( 62 )

Isle of Eden : Λ 2 62

ρ1[62] : φn −1

1

7

( )

1

62

0

T* 62

Isle of Eden : Λ 2 62

6

131

000000000000000000000000000000000

5

4

⇓

0

Example :

118

5

T 62

2

4

⇓

2

( )

φn

3

1

T† 62

Time-1 Return map

ρ1[62] : φn −1

1

Equivalent Rules and Firing Patterns

2

( ) 62

0

⇓

4

145

7

Table 1A.

Firing Patterns


Attractor Bit-String Generator (N

3 4

1. One red bit followed by

1 2

5

)

( )

( )

62


Attractor : Λ 3 62

Period-3 Attractor : Λ 3 62 One, two, four or five adjacent red bits separated from each other by at most four blue bits subject to restrictions:

(Continued )

( )

Time-1 Return map

ρ 1[62] :φ n−1

1

T† 62 ⇓

2

118

5

( )

φn

T 62

4 6 0

⇓ 7

131

1

1 or 2 blue bits.

1

1

1

663

0

0

0

0

1

0

0

1

1

1

1

0

0

0

1

0

1

( )

T* 62

2. Two red bits followed by 1, 3, or 4 blue bits. 3. Four red bits followed by 1, 3, or, 4 blue bits. 4. Five red bits followed by 1, 3, or 4 blue bits.

1

1

0

1

Example : 01001111100011011111011011 3

( 62 )

0

⇓

4

145

7

Table 1A.

Firing Patterns

(Continued )



( )

Bernoulli Shift Attractor : Λ 4 62

62 1

664

2 3 4 5

Two or three adjacent red bits separated from each other by one or two blue bits: xi = 1 ⇒ xi −1 = 0 and xi +1 = 1 , xi −1 = 1 and xi +1 = 0 , xi −1 = 1 and xi +1 = 1 , or xi −1 = 1 and xi +1 = 0 Example :

011011100110011011011001101100111


( ) ⇓

2

118

5

( )

T 62

4 6 0

1

⇓ 7

131

0

0

1

1

1 4

( 62 )

( )

T* 62

0

⇓

4

145

7

Remark. We exclude period-3 “isle of Eden” Λ 2 62 , such as bit strings made of repeated triads “011”, etc., from Λ 3 62

( )

1

T† 62

( )

( )

and Λ 4 62

.

Table 1B.

N =

62

Transient regime of a period-1 attractor Λ1 ( 62 ) and a typical period-3 noninvertible attractor Λ3 ( 62 ) of local rule 62 .

Firing Patterns

1

2

3

4

5

(

)

Transient T Λ 62 = 2 Period T∆ ∆ 1

n=0

T∆

n=1 n=2 n=3

665

Remark The transient period T defect cited in the following page depends on the length of the interval occupied by “defect ” pixels in the initial bit string which move towards each other until they are annihilated upon collision. In general, 1 < Tdefect ≤ 1 ( I + 1) where the equality sign is attained 2 only if the defects included the boundary pixels at i = 0 and i = I.

Table 1B.

N = n=0 n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=8 n=9

666

T∆

n=10 n=11 n=12 n=13 n=14 n=15 n=16 n=17 n=18 n=19 n=20 n=21 n=22 n=23

62

Firing Patterns

1

2

3

(Continued )

4

5

(

)

Transient T Λ 62 = Tdefect Period T∆ ∆ 3


nonempty basin of attraction. Their characterization is summarized in the third part of Table 1A, and identified as attractor Λ3 ( 62 ). The bit-string generator graph 3 ( 62 ) shows many distinct combinations, each one corresponding to a different period-3 time-1 return map. The period-3 attractor shown in Table 1 represents only a generic group. There is a fourth robust class of bit strings which converges to a subset of the Bernoulli στ -shift map, with σ = −1 and τ = 2 [Chua et al., 2005a], as shown in the fourth part of Table 1A.

χ162

i=0

It follows from Eq. (19) that φ = 0 is a fixed point of χ162 for all values of I. Observe that the above conclusion could also have been obtained by observing any bit string made of all zeros (blue bits) is a nonfiring (i.e. quenching) pattern of 62 [Chua et al., 2003], and that the periodic boundary conditions give rise to a circular “blue” bit string. However, we will see later that for more complicated properties, the closed-form expression in Table 1 of [Chua et al., 2005b] for the characteristic function χ1N is essential in deriving a rigorous proof.8 Definition 2.1. Basin of attraction of Λi ( 62 ).

8

2.1. Period-1 attractor Λ1 ( 62 ) and its basin tree I+1 [Λ1 ( 62 )] To prove analytically that the origin φ = 0 is a period-1 point of 62 , let us examine the time-1 characteristic function

(17)

i=0

Substituting xi−1 = 0, xi = 0, and xi+1 = 0 into Eq. (17), we obtain I 3 1 − 2 2−(i+1) = 0 (19) χ 62 (φ = 0) = 2

I+1 [Λi (

The bit-string generator graph 4 ( 62 ) in Table 1A shows many distinct combinations. Each bit string generated from 4 ( 62 ) gives rise to a different time-1 return map for finite I. However, in Sec. 2.4, it will be clear that each one converges to a subset of an invariant Bernoulli time-2 map obtained with I → ∞.

I 3 − |(−2xi−1 − 2xi − xi+1 + 2)| 2−(i+1) (φ) = 2

from Table 1 of [Chua et al., 2005b], where [x0 , x1 , x2 , . . . , xI ] → φ, and the step function (·) is defined by

(w) = 1, if w > 0 (18) = 0, if w ≤ 0

The set

667

62 )] of all (I + 1)-bit initial states

((I + 1)-bit string configurations) which converge to the attractor Λi ( 62 ) of 62 is called the basin of attraction of Λi ( 62 ). By brute-force computer simulation of χ162 of Eq. (17) over all combinations of initial bit strings {x0 x1 x2 . . . xI }, we have found the basin of attraction of the period-1 point φ = 0 of rule 62 for I = 2, 3, 4, . . . , 8. They are summarized, for each I, in the form of a tree, henceforth called a basin tree, in Figs. 3(a), 3(b), . . . , 3(g). The 13 initial states {x0

x1

x2

...

x8 } → φ =

8

2−(i+1) xi

(20)

i=0

belonging to the nine-bit basin tree 9 [Λ1 ( 62 )] in Fig. 3(g) are listed in Fig. 4(a). These 13 points φ1 , φ2 , . . . , φ13 are identified by a small green dot in Fig. 4(b). Observe that, except for φ13 , which converges to φ = 0 in one iteration as shown in Table 1B, all other members of the basin tree 9 [Λ1 ( 62 )] converge to φ = 0 in exactly two iterations, as depicted in the Lameray diagrams shown in Fig. 4(b) through φ1 (in green) and φ11 (in brown), respectively.

It follows similarly from Table 1 of [Chua et al., 2005b] that the origin is a period-1 point of all even-numbered local rules N , N = 0, 2, 4, . . . , 2k.

2 φ = 0.250

1

5

4

φ = 0.125

7

φ = 0.500

10

φ = 0.3125

φ = 0.875

φ = 0.9375

0

0

φ = 0.000

φ = 0.000

668

21

10 φ = 0.3125

9

18

φ = 0.28125

31

5 φ = 0.15625

φ = 0.625

15

φ = 0.5625

42 φ = 0.140625

φ = 0.000

φ = 0.5625

63

9 φ = 0.625

0

36

φ = 0.28125

20

φ = 0.96875

φ = 0.328125

18

φ = 0.984375

0

φ = 0.65625 φ = 0.000

Fig. 3. Basin trees I [Λ1 ( 62 )] of the period-1 attractor φ = 0 of 62 for I + 1 = 3, 4, . . . , 9. Each bit string is coded in the usual way (blue for “0” and red for “1”). The decimal equivalent of these (I + 1)-bit binary words is enclosed inside a circle for a more compact representation.

37

φ = 0.3203125

41

φ = 0.328125

42

74

φ = 0.578125

82

φ = 0.2890625

φ = 0.640625

21

84

127 φ = 0.9921875

φ = 0.1640625

φ = 0.65625

0 φ = 0.000

669

φ = 0.3203125 φ = 0.2890625 φ = 0.28515625

74

85

82

φ = 0.33203125

146

φ = 0.5703125

148

73 255

41

φ = 0.578125

164 φ = 0.640625

φ = 0.16015625

37

φ = 0.99609375

170

0 φ = 0.000

φ = 0.14453125

Fig. 3.

(Continued )

φ = 0.6640625

170

169

φ = 0.33203125

φ = 0.330078125

292

165

φ = 0.5703125

φ = 0.322265625

298

149

φ = 0.58203125

φ = 0.291015625

330

146

φ = 0.64453125

φ = 0.28515625 670

338

85

φ = 0.66015625

φ = 0.166015625

340

73 511

φ = 0.142578125

φ = 0.6640625 φ = 0.998046875

0 φ = 0.000

Fig. 3.

(Continued )

Chapter 6: From Time-Reversible Attractors to the Arrow of Time 8

Initial state

φk = ∑ 2− (i +1) xi

{ x0 x1 x2 x3 x4 x5 x6 x7 x8 } 1

{0 0 1 0 0 1 0 0 1}

2

{0 0 1 0 1 0 1 0 1}

3

{01 0 0 1 0 0 1 0}

4

{0 1 0 0 1 0 1 0 1}

5

{0 1 0 1 0 0 1 0 1}

6

{0 1 0 1 0 1 0 0 1}

7

{0 1 0 1 0 1 0 1 0}

8

{1 0 0 1 0 0 1 0 0}

9

{1 0 0 1 0 1 0 1 0}

10

{1 0 1 0 0 1 0 1 0}

11

{1 0 1 0 1 0 0 1 0}

12

{1 0 1 0 1 0 1 0 0}

13

{1 1 1 1 1 1 1 1 1}

1 62

671

i =0

φ1 = 1•2 + 1•2 + 1•2−9 = 0.142578125 φ2 = 1•2−3 + 1•2−5 + 1•2−7 + 1•2−9 = 0.166015625 φ3 = 1•2−2 + 1•2−5 + 1•2−8 = 0.28515625 φ4 = 1•2−2 + 1•2−5 + 1•2−7 + 1•2−9 = 0.291015625 φ5 = 1•2−2 + 1•2−4 + 1•2−7 + 1•2−9 = 0.322265625 φ6 = 1•2−2 + 1•2−4 + 1•2−6 + 1•2−9 = 0.330078125 φ7 = 1•2−2 + 1•2−4 + 1•2−6 + 1•2−8 = 0.33203125 φ8 = 1•2−1 + 1•2−4 + 1•2−7 = 0.5703125 φ9 = 1•2−1 + 1•2−4 + 1•2−6 + 1•2−8 = 0.58203125 φ10 = 1•2−1 + 1•2−3 + 1•2−6 + 1•2−8 = 0.64453125 φ11 = 1•2−1 + 1•2−3 + 1•2−5 + 1•2−8 = 0.66015625 φ12 = 1•2−1 + 1•2−3 + 1•2−5 + 1•2−7 = 0.6640625 φ13 = 1•2−1 + 1•2−2 + ... + 1•2−9 = 0.998046875 −3

−6

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.1

⋅⋅

0.2

⋅⋅ ⋅⋅⋅ 0.3

0.4

φ1 φ 2 φ 3 φ 4 φ 5 φ 6 φ 7

0.5

⋅⋅ ⋅ ⋅

φ8 φ9

0.6

0.7

φ10 φ11 φ12

0.8

0.9

φ13

⋅

1

φ 62

Fig. 4. (a) Locations of 13 initial states which converge to the period-1 point φ = 0 of 62 . (b) Lameray (cobweb) diagram from two initial states φ1 and φ11 .

672


3

3 φ = 0.09375

φ = 0.375

62

62 6

5

φ = 0.75

φ = 0.625

(a) I + 1 = 3 :

3

22

29

φ = 0.6875

φ = 0.90625

( 62 )

(i)

6

11 φ = 0.34375

φ = 0.1875

62

62 13

27

φ = 0.40625

φ = 0.84375

30

17

φ = 0.9375

φ = 0.53125

(ii)

(iii)

12

15 φ = 0.46875

φ = 0.375

62

62 26

23

φ = 0.8125

φ = 0.71875

24

21

φ = 0.75

φ = 0.65625

(iv)

(b) I + 1 = 5 :

5

( 62 )

(v)

27 φ = 0.421875

62 45

54 φ = 0.84375

φ = 0.703125

(c) I + 1 = 6 : Fig. 5.

6

( 62 )

State transition diagrams of all isles of Eden Λ2 ( 62 ) for 62 for I + 1 = 3, 5, 6, 7, 8, and 9.


φ = 0.0859375

11

φ = 0.1171875

62

62 φ = 0.8828125

94

113

φ = 0.734375

φ = 0.9140625

22

φ = 0.234375

φ = 0.4765625 φ = 0.8359375

44

φ = 0.3671875

φ = 0.953125

φ = 0.5390625

(v)

φ = 0.46875

60

98

87

120

69 (vi)

62 φ = 0.6796875

47 62

122

71

φ = 0.3828125

(iv)

62 φ = 0.5546875

49

107

(iii)

φ = 0.34375

30 62

61

99

φ = 0.6875

(ii)

62 φ = 0.7734375

88

117

(i)

φ = 0.171875

15

φ = 0.765625

(vii)

(d) I + 1 = 7 : Fig. 5.

7

( 62 )

(Continued )

φ = 0.9375

673

674


φ = 0.05859375

15

φ = 0.07421875

62

62 152

245

φ = 0.7421875

φ = 0.87890625

(i)

(ii)

30

φ = 0.1484375

49

φ = 0.48828125

φ = 0.76171875

(iii)

(iv)

60

φ = 0.296875

98

φ = 0.9765625

φ = 0.52734375

(v)

(vi)

95

φ = 0.46875

240

175 φ = 0.9375

φ = 0.53515625

120 62

62 137

250

135 φ = 0.3828125

φ = 0.83984375

φ = 0.37109375

76 62

62 215

125

195 φ = 0.19140625

φ = 0.91796875

φ = 0.234375

38 62

62 235

190

225 φ = 0.59375

φ = 0.95703125

φ = 0.1171875

19

φ = 0.765625

φ = 0.68359375

(vii)

(e) I + 1 = 8 : Fig. 5.

196

8

( ) 62

(Continued )

(viii)


φ = 0.427734375

219

φ = 0.091796875

62

φ = 0.734375

φ = 0.884765625

(ii)

(i)

94

φ = 0.3671875

62

482

279 φ = 0.470703125

φ = 0.771484375

188 62

241

395

376

453 φ = 0.85546875

φ = 0.712890625

φ = 0.18359375

47 62

438

365

675

(iii)

(f) I + 1 = 9 : Fig. 5.

Observe that the 12 “pink” bit strings φ1 , φ2 , . . . , φ12 of the basin tree 9 [Λ1 ( 62 )] in Fig. 3(g) are all “Gardens-of-Eden” [Chua et al., 2005b] since they have no predecessors. Similarly, all pink bit strings on the outer periphery of the basin trees 3 [Λ1 ( 62 )], 4 [Λ1 ( 62 )], . . . , 8 [Λ1 ( 62 )] in Figs. 3(a)–3(f) are also gardens of Eden.

2.2. Period-3 “isle of Eden” orbits Λ2 ( 62 ) By brute-force computer simulations, we have identified the “isles of Eden” [Chua et al., 2005b] among all period-3 points (those lying on the main diagonal of the time-3 characteristic function χ362 in Fig. 2(c) for I = 2, 3, 4, . . . , 8. These period-3 orbits are isles of Eden because they have an empty basin of attraction. In other words, none of the three-member triads has a predecessor except themselves. Such period-3 “isles of Eden” can be represented precisely by an isolated

φ = 0.94140625

φ = 0.544921875

9

( 62 )

(iv)

(Continued )

single-loop state transition diagram, as shown in Figs. 5(a)–5(f).

2.3. Period-3 attractors Λ3 ( 62 ) and their basin trees I+1 [Λ3 ( 62 )] Observe that we call the period-3 “isle of Eden” in Fig. 5 “period-3 orbits”, and not “period-3 attractors” because these period-3 triads are invariant sets, and not attractors. These period-3 orbits are all singular in the sense that their basins of attraction are empty. They are clearly not robust, and cannot be detected from the random bit-string test used to generate the time-1 return maps in Table 3 of [Chua et al., 2005a]. The local rule 62 , however, has numerous robust period-3 attractors, as depicted by the violet clusters of period-3 attractors in Table 2 (p. 1079) of [Chua et al., 2005a], where only one of them is identified by three red dots, for the readers’ convenience. They all lie on a generic Lameray diagram

676


phenomenon results from the periodic boundary condition in Fig. 1(a), and the observation that any one of the I + 1 cells can be designated as cell x0 .

similar to that shown in Fig. 11(a) of [Chua et al., 2005a]. There are no period-3 attractors for “3-bit”, “4bit”, or “5-bit” initial bit strings. There are six distinct period-3 attractors with I + 1 = 6 bit-strings. Their basin trees 6 [Λ3 ( 62 )] are shown in Figs. 6a-(i), 6a-(ii), . . . , 6a-(vi). There are no period-3 attractors with I + 1 = 7 bit strings. There are 16 period-3 attractors with I + 1 = 8 bit strings. Their basin trees 8 [Λ3 ( 62 )] are shown in Figs. 6(b)-(i), 6(b)-(ii), . . . , 6(b)-(xvi). There are 18 period-3 attractors with I + 1 = 9 bit strings. Their basin trees are shown in Figs. 6(c)-(i), 6(c)-(ii), . . . , 6(c)-(xviii). Observe that the period-3 attractors in Fig. 6 are organized into several different subgroups where the graphs depicting their basins of attraction all have an identical dendrite-like topology. This

2.4. Bernoulli στ -shift attractors Λ4 ( 62 ) and their basin trees I+1 [Λ4 ( 62 )] By brute force computer simulations, we found the remaining attractors for I + 1 = 3, 4, . . . , 9 bit strings are “Bernoulli στ -Shift” attractors with σ = −1 and τ = 2 [Chua et al., 2005a], as exhibited in Figs. 7–14. This implies that the dynamics on the attractor consists of shifting each bit string on the attractor one bit to the right every two iterations. The basin trees associated with those Bernoulli στ -shift attractors which are not isles of Eden are shown in Figs. 8, 9, 11–13 and 14(a).

46

23 φ = 0.71875

φ = 0.359375

60

57

1

φ = 0.9375

2

φ = 0.890625

φ = 0.015625

φ = 0.03125

7

35

φ = 0.109375

φ = 0.546875

40

17

22

φ = 0.625

44

φ = 0.265625

φ = 0.34375

φ = 0.6875

59

61

φ = 0.921875

φ = 0.953125

3

38

φ = 0.046875

φ = 0.59375

6

13

φ = 0.09375

φ = 0.203125

(ii)

(i) (a) I + 1 = 6 :

6

( )

 Λ 3 62   

Fig. 6. Basin trees I+1 [Λ3 ( 62 )] of the period-3 attractor Λ3 ( 62 ) of 62 , for I + 1 = 6, 8, 9. (a) I + 1 = 6: There are six basins; i = (i), (ii), . . . , (vi), each containing six bit strings. (b) I + 1 = 8: There are 16 basins; i = (i), (ii), . . . , (xvi), each containing eight bit strings. (c) I + 1 = 9: There are 18 basins; i = (i), (ii), . . . , (xviii), each containing nine bit strings.


43

29 φ = 0.671875

φ = 0.453125

51

30

4

φ = 0.796875

32

φ = 0.46875

φ = 0.0625

φ = 0.5

49

14

φ = 0.765625

φ = 0.21875

34

20

25

φ = 0.53125

11

φ = 0.3125

φ = 0.390625

φ = 0.171875

62

55

φ = 0.96875

φ = 0.859375

12

26

φ = 0.1875

φ = 0.40625

33

19

φ = 0.515625

φ = 0.296875

(iv)

(iii) (a) I + 1 = 6 :

6

( )

 Λ 3 62   

53

58 φ = 0.828125

φ = 0.90625

39

15

8

φ = 0.609375

16

φ = 0.234375

φ = 0.125

φ = 0.25

56

28

φ = 0.875

φ = 0.4375

5

10

50

φ = 0.078125

37

φ = 0.15625

φ = 0.78125

φ = 0.578125

31

47

φ = 0.484375

φ = 0.734375

24

52

φ = 0.375

φ = 0.8125

48

41

φ = 0.75

φ = 0.640625

(vi)

(v) (a) I + 1 = 6 : Fig. 6.

6

( )

 Λ 3 62   

(Continued )

677

678


φ = 0.54296875

φ = 0.625

φ = 0.08984375

φ = 0.25390625

139

160

23

65 188

94 φ = 0.734375

φ = 0.3671875

241

227 φ = 0.94140625

φ = 0.88671875 φ = 0.23828125

φ = 0.6171875

11

158

22

61

φ = 0.04296875 (i)

φ = 0.0859375

(ii)

φ = 0.71875

φ = 0.0390625

φ = 0.1796875

φ = 0.5078125

184

10

46

130 121

229 φ = 0.47265625

φ = 0.89453125

199

31

φ = 0.77734375

φ = 0.12109375 φ = 0.91015625

φ = 0.4765625

176 φ = 0.6875

233

44 φ = 0.171875

122 (iv)

(iii)

(b) I + 1 = 8 : Fig. 6.

8

( )

 Λ 3 62   

(Continued )


φ = 0.44140625

φ = 0.078125

φ = 0.76953125

φ = 0.3125

113

20

197

80

679

47

203 φ = 0.18359375

φ = 0.79296875

248

62

φ = 0.96875

φ = 0.2421875 φ = 0.30859375

φ = 0.82421875

97

211

133

79

φ = 0.37890625 (v)

φ = 0.51953125

(vi)

φ = 0.359375

φ = 0.01953125

φ = 0.8828125

φ = 0.15625

92

5

226

40 151

242 φ = 0.58984375

φ = 0.9453125

124

143

φ = 0.484375

φ = 0.55859375 φ = 0.65234375 φ = 0.953125

88 φ = 0.34375

244 (vii)

(b) I + 1 = 8 : Fig. 6.

194

167

8

( )

 Λ 3 62   

(Continued )

(viii)

φ = 0.7578125

680


φ = 0.15234375 φ = 0.28125 φ = 0.33984375 φ = 0.3046875

39

72

1

252

87

φ = 0.5625

φ = 0.6796875

78

144

174

2

249

φ = 0.984375 φ = 0.00390625

φ = 0.97265625

131

φ = 0.0078125

φ = 0.51171875

70 φ = 0.92578125

237

φ = 0.7109375

φ = 0.02734375

140 φ = 0.2734375 φ = 0.85546875

27

182

7

φ = 0.10546875

(ix)

219

54

109 φ = 0.42578125

φ = 0.546875

(x)

φ = 0.140625 φ = 0.57421875 φ = 0.66796875 φ = 0.03515625 φ = 0.890625

36

147

128

126

171

9

228

32

159

φ = 0.4921875

φ = 0.4375

35 φ = 0.9609375

φ = 0.35546875

200 φ = 0.13671875 φ = 0.73828125

141

91 (xi)

234

112

φ = 0.125

φ = 0.75390625

246

φ = 0.9140625

φ = 0.62109375

193

φ = 0.5

φ = 0.2109375

189

99

214

φ = 0.55078125 φ = 0.8359375

(b) I + 1 = 8 : Fig. 6.

8

( )

 Λ 3 62   

(Continued )

φ = 0.78125

(xii)

φ = 0.38671875


681

φ = 0.4453125 φ = 0.45703125 φ = 0.515625 φ = 0.12890625 φ = 0.36328125 φ = 0.609375

114

117

16

207

132

33

93

4

243

φ = 0.80859375

φ = 0.94921875

56

φ = 0.0625

14

φ = 0.015625

φ = 0.21875

φ = 0.0546875

100 φ = 0.8671875

222

φ = 0.41796875

25 φ = 0.390625 φ = 0.71484375

177

107 (xiii)

156

183

108

218

φ = 0.69140625 φ = 0.8515625

φ = 0.09765625

φ = 0.421875

(xiv)

φ = 0.22265625 φ = 0.2578125 φ = 0.7265625 φ = 0.0703125 φ = 0.78515625 φ = 0.83203125

57

66

8

231

186

18

201

64

63

φ = 0.90234375

φ = 0.24609375

28

φ = 0.03125

224

φ = 0.25

φ = 0.109375

φ = 0.875

50 φ = 0.43359375

111

φ = 0.70703125

145 φ = 0.1953125 φ = 0.48046875

216

181

213

φ = 0.67578125

(xv)

(b) I + 1 = 8 :

8

( )

 Λ 3 62   

(Continued )

φ = 0.56640625

198

173

φ = 0.84375

Fig. 6.

123

(xvi)

φ = 0.7734375

682


275

320 φ = 0.625

φ = 0.537109375

190 481

φ = 0.37109375

φ = 0.939453125

19

318

φ = 0.037109375

φ = 0.62109375

(i)

312

10 φ = 0.01953125

φ = 0.609375

485 31

φ = 0.947265625

φ = 0.060546875

304

489

φ = 0.59375

φ = 0.955078125

(ii)

39

129

φ = 0.076171875

φ = 0.251953125

380

451

φ = 0.7421875

φ = 0.880859375

38

125

φ = 0.07421875

φ = 0.244140625

(iii)

(c) I + 1 = 9 : Fig. 6.

9

( )

 Λ 3 62   

(Continued )


20

113

φ = 0.0390625

φ = 0.220703125

459

62

φ = 0.896484375

φ = 0.12109375

97

467

φ = 0.189453125

φ = 0.912109375

(iv)

78

258

φ = 0.15234375

φ = 0.50390625

249

391

φ = 0.486328125

φ = 0.763671875

76

250

φ = 0.1484375

φ = 0.48828125

(v)

226

40

φ = 0.44140625

φ = 0.078125

407

124

φ = 0.794921875

φ = 0.2421875

194

423

φ = 0.37890625

φ = 0.826171875

(vi)

(c) I + 1 = 9 : Fig. 6.

9

( )

 Λ 3 62   

(Continued )

683

684


156

5 φ = 0.009765625

φ = 0.3046875

498

271

φ = 0.97265625

φ = 0.529296875

152

500

φ = 0.296875

φ = 0.9765625

(vii)

452

80

φ = 0.8828125

φ = 0.15625

303

248

φ = 0.591796875

φ = 0.484375

388

335

φ = 0.7578125

φ = 0.654296875

(viii)

393

160

φ = 0.767578125

φ = 0.3125

95

496

φ = 0.185546875

φ = 0.96875

265

159

φ = 0.517578125

φ = 0.310546875

(ix)

(c)

I +1 = 9 : Fig. 6.

9

( )

 Λ 3 62   

(Continued )


486

187

φ = 0.953125 φ = 0.94921875

φ = 0.365234375

67 273

φ = 0.130859375

φ = 0.533203125

φ = 0.34375

488

385 φ = 0.751953125

φ = 0.9375

φ = 0.560546875

176

287

285

490 φ = 0.556640625

480

685

9 484

179 φ = 0.349609375

φ = 0.95703125

φ = 0.017578125 φ = 0.9453125

254

φ = 0.373046875

φ = 0.49609375

191

494

184

φ = 0.96484375

256 277

φ = 0.5

180

φ = 0.359375

281 φ = 0.548828125

φ = 0.3515625

φ = 0.5859375

84

φ = 0.6484375

300

332

183

φ = 0.541015625 φ = 0.1640625

φ = 0.357421875

299

71 492

φ = 0.9609375

φ = 0.283203125

φ = 0.337890625

145

173

φ = 0.138671875

φ = 0.583984375

331

13

283

6

507

φ = 0.01171875

φ = 0.990234375

φ = 0.552734375 φ = 0.646484375

339

φ = 0.025390625

182

81

φ = 0.35546875

φ = 0.662109375

77 493

φ = 0.962890625

φ = 0.158203125 φ = 0.150390625

326 (x) φ = 0.63671875

φ = 0.60546875

310

27 φ = 0.052734375

Fig. 6.

(Continued )

(c)

I +1 = 9 :

9

( )

 Λ 3 62   

686


461

374

φ = 0.908203125 φ = 0.900390625

φ = 0.73046875

134 35

φ = 0.26171875

φ = 0.068359375

φ = 0.6875

465

352

449

φ = 0.505859375

φ = 0.876953125

63

59

469 φ = 0.115234375

259

φ = 0.123046875

18 457

358 φ = 0.69921875

φ = 0.916015625

φ = 0.03515625 φ = 0.892578125

508 382

φ = 0.74609375

φ = 0.9921875

477

368

φ = 0.931640125

1 43

360

φ = 0.001953125

φ = 0.71875

51 φ = 0.099609375

φ = 0.703125

φ = 0.56640625

87

φ = 0.67578125

290

346

366

φ = 0.083984375 φ = 0.169921875

φ = 0.71484375

151

142 473

φ = 0.923828125

φ = 0.30078125

φ = 0.31640625

154

162

φ = 0.27734375

φ = 0.294921875

167

26

55

12

503

φ = 0.0234375

φ = 0.982421875

φ = 0.107421875 φ = 0.326171875

168

φ = 0.05078125

364

153

φ = 0.7109375

φ = 0.328125

89 475

φ = 0.927734375

φ = 0.298828125 φ = 0.173828125

141 (xi) φ = 0.275390625

φ = 0.212890625

109

54 φ = 0.10546875

Fig. 6.

(Continued )

(c)

I +1 = 9 :

9

( )

 Λ 3 62   


411

237

φ = 0.818359375 φ = 0.802734375

φ = 0.462890625

268 70

φ = 0.5234375

φ = 0.13671875

7 φ = 0.013671875

φ = 0.755859375

φ = 0.376953125

419

φ = 0.24609375

193

126

118

427 φ = 0.23046875

387

687

36 403

205 φ = 0.400390625

φ = 0.833984375

φ = 0.0703125 φ = 0.787109375

507

φ = 0.494140625

φ = 0.990234375

253

443

225

φ = 0.865234375

2 86

209

φ = 0.00390625

φ = 0.439453125

102 φ = 0.19921875

φ = 0.408203125

φ = 0.6015625

174

φ = 0.6328125

308

324

221

φ = 0.16796875 φ = 0.33984375

φ = 0.431640625

302

284 435

φ = 0.849609375

φ = 0.353515625

φ = 0.59765625

181

306

φ = 0.5546875849609375

φ = 0.58984375

334

52

110

24

495

φ = 0.046875

φ = 0.966796875

φ = 0.21484375 φ = 0.65234375

336

φ = 0.1015625

217

178

φ = 0.423828125

φ = 0.65625

69 439

φ = 0.857421875

φ = 0.34765625 φ = 0.134765625

282 (xii) φ = 0.55078125

φ = 0.42578125

218

108 φ = 0.2109375

Fig. 6.

(Continued )

(c)

I +1 = 9 :

9

( )

 Λ 3 62   

688


243

349

φ = 0.4765625 φ = 0.474609375

φ = 0.681640625

289 392

φ = 0.564453125

φ = 0.765625

φ = 0.171875

244

88

240

φ = 0.875

φ = 0.46875

399

398

260 φ = 0.77734375

448

φ = 0.779296875

245 242

345 φ = 0.673828125

φ = 0.5078125

φ = 0.478515625 φ = 0.47265625

127 351

φ = 0.685546875

φ = 0.248046875

247

92

φ = 0.482421875

128 394

φ = 0.25

90

φ = 0.1796875

396 φ = 0.7734375

φ = 0.17578125

42

φ = 0.640625

φ = 0.66796875

328

342

347

φ = 0.76953125 φ = 0.08203125

φ = 0.677734375

405

291 246

φ = 0.48046875

φ = 0.57421875

φ = 0.578125

294

296

φ = 0.568359375

φ = 0.791015625

421

397

262

3

509

φ = 0.005859375

φ = 0.994140625

φ = 0.775390625 φ = 0.822265625

425

φ = 0.51171875

91

166

φ = 0.177734375

φ = 0.830078125

150 502

φ = 0.98046875

φ = 0.32421875 φ = 0.29296875

163 (xiii) φ = 0.318359375

φ = 0.302734375

155

269 φ = 0.525390625

Fig. 6.

(Continued )

(c)

I +1 = 9 :

9

( )

 Λ 3 62   


444

215

φ = 0.119140625 φ = 0.8671875

φ = 0.419921875

200 98

φ = 0.390625

φ = 0.19140627

φ = 0.04296875

61

112 φ = 0.21875

φ = 0.1171875

φ = 0.943359375

22

483

355

189 φ = 0.693359375

60

689

65 316

214 φ = 0.41796875

φ = 0.369140625

φ = 0.126953125 φ = 0.6171875

471

415

φ = 0.919921875

φ = 0.62109375

445

32 354

φ = 0.0625

23

φ = 0.869140625

278

φ = 0.044921875

99 φ = 0.193359375

φ = 0.54296875

φ = 0.642578125

229

φ = 0.666015625

329

341

470

φ = 0.69140625 φ = 0.447265625

φ = 0.91796875

233

456 317

φ = 0.619140625

φ = 0.572265625

φ = 0.580078125

293

297

φ = 0.890625

φ = 0.455078125

234

227

321

384

255

φ = 0.75

φ = 0.498046875

φ = 0.443359375 φ = 0.45703125

266

φ = 0.626953125

406

82

φ = 0.79296875

φ = 0.51953125

74 381

φ = 0.744140625

φ = 0.16015625 φ = 0.14453125

424 (xiv) φ = 0.828125

φ = 0.82421875

422

195 φ = 0.380859375

Fig. 6.

(Continued )

(c)

I +1 = 9 :

9

( )

 Λ 3 62   

690


311

474 φ = 0.607421875

φ = 0.92578125

25 140

φ = 0.75390625

φ = 0.638671875

φ = 0.048828125

φ = 0.2734375

327

386

263

φ = 0.02734375

φ = 0.513671875

252

236

343 φ = 0.4609375

14

φ = 0.4921875

72 295

410 φ = 0.80078125

φ = 0.669921875

φ = 0.140625 φ = 0.576171875

499 506

φ = 0.98828125

φ = 0.974609375

375

450

φ = 0.732421875

4 172

418

φ = 0.0078125

φ = 0.87890625

204 φ = 0.3984375

φ = 0.81640625

93

φ = 0.6953125

φ = 0.70703125

356

362

442

φ = 0.3359375 φ = 0.181640625

φ = 0.86328125

157

57 359

φ = 0.701171875

φ = 0.267578125

φ = 0.26953125

137

138

φ = 0.111328125

φ = 0.306640625

161

104

220

48

479

φ = 0.093753125

φ = 0.935546875

φ = 0.4296875 φ = 0.314453125

348

φ = 0.203125

434

105

φ = 0.84765625

φ = 0.6796875

101 367

φ = 0.716796875

φ = 0.205078125 φ = 0.197265625

53 (xv) φ = 0.103515625

φ = 0.8515625

436

216 φ = 0.421875

Fig. 6.

(Continued )

(c)

I +1 = 9 :

9

( )

 Λ 3 62   


111

437

φ = 0.279296875 φ = 0.216796875

φ = 0.853515625

50 280

φ = 0.09765625

φ = 0.546875

28 φ = 0.0546875

φ = 0.029296875

φ = 0.509765625

143

φ = 0.984375

261

504

472

175 φ = 0.921875

15

691

144 79

309 φ = 0.603515625

φ = 0.341796875

φ = 0.28125 φ = 0.154296875

487 501

φ = 0.978515625

φ = 0.951171875

239

8 344

389

φ = 0.466796875

325

φ = 0.015625

φ = 0.759765625

408 φ = 0.796875

φ = 0.634765625

185

φ = 0.53515625

φ = 0.5390625

274

φ = 0.671875

276

373 φ = 0.361328125

φ = 0.728515625

186

114 207

φ = 0.404296875

φ = 0.41015625

φ = 0.416015625

210

213

φ = 0.22265625

φ = 0.36328125

314

440

208

96

447

φ = 0.1875

φ = 0.873046875

φ = 0.859375 φ = 0.61328125

322

φ = 0.40625

357

202

φ = 0.697265625

φ = 0.62109375

201 223

φ = 0.435546875

φ = 0.39453125 φ = 0.392578125

106 (xvi) φ = 0.20703125

φ = 0.705078125

361

432 φ = 0.84375

Fig. 6.

(Continued )

(c)

I +1 = 9 :

9

( )

 Λ 3 62   

692


377

430

φ = 0.23828125 φ = 0.736328125

φ = 0.83984375

400 196

φ = 0.78125

φ = 0.3828125

φ = 0.0859375

122

44

120

φ = 0.4375

φ = 0.234375

455

199

378 φ = 0.388671875

224

φ = 0.888671875

130 121

428 φ = 0.8359375

φ = 0.73828125

φ = 0.25390625 φ = 0.236328125

319 431

φ = 0.841796875

φ = 0.623046875

379

46

φ = 0.740234375

64 197

φ = 0.125

45

φ = 0.08984375

198 φ = 0.38671875

φ = 0.087890625

21

φ = 0.3203125

φ = 0.333984375

164

171

429

φ = 0.384765625 φ = 0.041015625

φ = 0.837890625

458

401 123

φ = 0.240234375

φ = 0.287109375

φ = 0.2890625

147

148

φ = 0.783203125

φ = 0.89453125

466

454

131

257

510

φ = 0.501953125

φ = 0.99609375

φ = 0.88671875 φ = 0.9105625

468

φ = 0.255859375

301

83

φ = 0.587890625

φ = 0.9140625

75 251

φ = 0.490234375

φ = 0.162109375 φ = 0.146484375

337 (xvii) φ = 0.658203125

φ = 0.650390625

333

390 φ = 0.76171875

Fig. 6.

(Continued )

(c)

I +1 = 9 :

9

( )

 Λ 3 62   


222

363

φ = 0.55859375 φ = 0.708984375

56 30

φ = 0.109375

φ = 0.05859375

φ = 0.970703125

11

497

433 φ = 0.1953125

φ = 0.095703125

φ = 0.021484375

286

φ = 0.43359375

100 49

693

350

φ = 0.845703125

107

288 158

φ = 0.208984375

φ = 0.68359375

φ = 0.5625 φ = 0.30859375

463

φ = 0.958984375

φ = 0.904296875

491

478

267

φ = 0.93359375

16 177

φ = 0.03125

139

φ = 0.521484375

305 φ = 0.271484375

φ = 0.595703125

φ = 0.8203125

117

φ = 0.83203125

420

426

235

φ = 0.345703125 φ = 0.228515625

φ = 0.458984375

133

228 414

φ = 0.80859375

φ = 0.7890625

φ = 0.78515625

404

402 φ = 0.4453125

φ = 0.259765625

370

416

369

192

383

φ = 0.375

φ = 0.748046875

φ = 0.720703125 φ = 0.72265625

372

φ = 0.8125

203

41

φ = 0.396484375

φ = 0.7265625

37 446

φ = 0.87109375

φ = 0.080078125 φ = 0.072265625

212 (xviii)

φ = 0.4140625 φ = 0.412109375

211 Fig. 6.

353 φ = 0.689453125

(Continued )

(c)

I +1 = 9 :

9

( )

 Λ 3 62   

694


φ = 0.375

3

6

5

φ = 0.750

φ = 0.625

Fig. 7. Basin tree 3 [Λ4 ( 62 )] of Bernoulli στ -shift attractor Λ4 ( 62 ), where σ = 1, τ = 1 (left shift on time-1 map), or σ = −1, τ = 2 (right shift on time-2 map).

4

φ = 0.1875

8 φ = 0.5

φ = 0.25

3 13

14

φ = 0.8125

9

6

φ = 0.5625

φ = 0.375

7

11 φ = 0.6875

φ = 0.0625 Fig. 8.

Basin tree

4 [Λ4 (

1

φ = 0.875

φ = 0.4375

12 φ = 0.75

2 φ = 0.125

62 )] of Bernoulli στ -shift attractor Λ4 ( 62 ), where σ = −1, τ = 2 (right shift on time-2 map).


695

2 φ = 0.0625

7 8

16 φ = 0.5

φ = 0.25

φ = 0.21875

25

28

φ = 0.78125

14

φ = 0.125 Fig. 9.

Basin tree

5 [Λ4 (

φ = 0.875

φ = 0.4375

19

φ = 0.59375

4

1 φ = 0.03125


φ = 0.421875

27

45 φ = 0.703125

54 φ = 0.84375

Fig. 10. Basin tree 6 [Λ4 ( 62 )] of Bernoulli στ -shift attractor Λ4 ( 62 ), where σ = 1, τ = 1 (left shift on time-1 map), or σ = −1, τ = 2 (right shift on time-2 map).

696


φ = 0.4453125

φ = 0.453125

φ = 0.515625

φ = 0.2578125

φ = 0.71875

φ = 0.03125

φ = 0.359375

φ = 0.609375

φ = 0.625

57

58

66

33

92

4

46

78

80

103 φ = 0.2265625

φ = 0.8046875

121

14

29

φ = 0.9453125 φ = 0.109375

7

25

28 8

φ = 0.21875

115 φ = 0.0625

50

φ = 0.8984375

φ = 0.8671875

φ = 0.4140625

φ = 0.1875

φ = 0.4296875

79 114

φ = 0.0390625

φ = 0.890625

φ = 0.6171875

116

φ = 0.046875

φ = 0.203125

φ = 0.40625

13 φ = 0.1015625

108

55

φ = 0.84375

89 56

6

φ = 0.1328125

52

5

φ = 0.9609375

17

26

φ = 0.265625

123

φ = 0.3515625 φ = 0.09375

34

53

24

φ = 0.59375

45

12

φ = 0.703125

φ = 0.53125

76

φ = 0.015625 φ = 0.9296875

90

68

2

119

φ = 0.390625

111

φ = 0.0546875 φ = 0.1953125

91 φ = 0.7109375

φ = 0.6953125

φ = 0.4375 φ = 0.90625

110

100

φ = 0.421875

φ = 0.859375

φ = 0.78125

16

95

φ = 0.125

φ = 0.7421875

9 φ = 0.375

106

109

93

27

φ = 0.8515625

102

φ = 0.8125

φ = 0.546875

118

3

φ = 0.921875

φ = 0.796875

81

59

77

φ = 0.6328125

φ = 0.0234375

125 φ = 0.9765625

φ = 0.6015625

φ = 0.4609375

96 φ = 0.140625

63 φ = 0.4921875

φ = 0.6640625

32

φ = 0.5703125

112

φ = 0.25

φ = 0.28125

φ = 0.96875

1 23

φ = 0.0078125

62

φ = 0.1796875

φ = 0.484375

97

φ = 0.2421875

124

φ = 0.671875

φ = 0.3359375

19 φ = 0.1484375

31

φ = 0.5234375

86

43

φ = 0.984375

φ = 0.875

67

φ = 0.5625

36

φ = 0.5078125

126

φ = 0.296875

72

φ = 0.2734375

65

85 73

38

35

18

φ = 0.75

φ = 0.7578125

10

101

φ = 0.078125

Fig. 11.

70

φ = 0.2109375

φ = 0.7265625

104

φ = 0.828125

51 φ = 0.3984375

48

φ = 0.0703125

54

Basin tree

φ = 0.7890625

7 [Λ4 (

105 φ = 0.8203125

64 φ = 0.5

20 φ = 0.15625

75 φ = 0.5859375

83 φ = 0.6484375

40 φ = 0.3125

39 φ = 0.3046875



68

φ = 0.19921875

136 φ = 0.53125

697

φ = 0.265625

51 221

238 φ = 0.9296875

φ = 0.86328125

153

102 φ = 0.3984375

φ = 0.59765625 φ = 0.46484375

119

187 φ = 0.73046875

204 34

17 φ = 0.796875

φ = 0.06640625 Fig. 12.

Basin tree

8a [Λ4 (

φ = 0.1328125


698


φ = 0.26953125

φ = 0.6953125

69

φ = 0.703125

178

180

φ = 0.34765625

89

φ = 0.5546875

24

φ = 0.203125

φ = 0.11328125

φ = 0.4140625

58 φ = 0.2265625

101

134

3

253

φ = 0.80078125

φ = 0.98828125

φ = 0.5234375

59 φ = 0.23046875

166

230

209 φ = 0.8984375

φ = 0.72265625

φ = 0.6484375

206

157

φ = 0.81640625

168

φ = 0.61328125

115

96

232

φ = 0.44921875

116 φ = 0.453125 φ = 0.74609375

127 202 φ = 0.7890625

Basin tree

21 φ = 0.08203125

8b [Λ4 (

169 φ = 0.66015625

φ = 0.26171875

φ = 0.16796875

129

192 φ = 0.75

212

43

67 φ = 0.62890625

191 φ = 0.828125

φ = 0.65625 φ = 0.90625

161

φ = 0.375

Fig. 13.

150

φ = 0.01171875

φ = 0.40234375

φ = 0.8125

φ = 0.8203125

φ = 0.5859375

φ = 0.63671875

φ = 0.60546875

φ = 0.859375

208

210

86

φ = 0.3359375

163

φ = 0.8046875

φ = 0.39453125

172

205

185 φ = 0.41015625

φ = 0.31640625

φ = 0.27734375

φ = 0.4609375

103 105

81

13

φ = 0.921875

φ = 0.69921875

220 φ = 0.40625

71

φ = 0.31640625

8

118

179

φ = 0.87109375

φ = 0.05078125

155

110

φ = 0.1875

104

φ = 0.1015625

φ = 0.84765625

φ = 0.4296875

48

φ = 0.03125

217 236

29

106

φ = 0.046875

55

φ = 0.21484375

77

251

26

138

φ = 0.30078125

45

φ = 0.98046875

12

52

223

162

φ = 0.96484375

142

φ = 0.09375

φ = 0.5390625

φ = 0.17578125

247 φ = 0.93359375

φ = 0.20703125

φ = 0.6328125

154

90

239

53

φ = 0.6015625

φ = 0.3515625

75

φ = 0.50390625

254 φ = 0.49609375

165 φ = 0.64453125

149 φ = 0.58203125

φ = 0.29296875

φ = 0.9921875

42 φ = 0.1640625

84 φ = 0.328125

83 φ = 0.32421875



135 φ = 0.06640625

17

φ = 0.232421875

φ = 0.615234375

409 φ = 0.1328125

φ = 0.44921875

238 29

φ = 0.814453125

φ = 0.224609375

462

476

132 φ = 0.2578125 φ = 0.90234375

φ = 0.9296875

313

103 φ = 0.011328125

φ = 0.515625

417

115

φ = 0.599609375

φ = 0.265625

264

φ = 0.806640625

307

136

φ = 0.630859375

413

φ = 0.46484375 φ = 0.056640625

323

230

φ = 0.798828125

68

φ = 0.033203125

φ = 0.8984375

315

119

270

φ = 0.52734375

φ = 0.263671875

34 460

699

464

58 φ = 0.201171875

231

441

66

272 φ = 0.53125

φ = 0.90625

φ = 0.611328125

φ = 0.861328125

412

206

φ = 0.12890625

φ = 0.451171875

232 116 φ = 0.2265625

φ = 0.40234375

371

φ = 0.8046875

φ = 0.453125

φ = 0.724609375

33 φ = 0.064453125

(a) Fig. 14.

Basin tree

9 [Λ4 (


700


φ = 0.712890625

365

219

438

φ = 0.427734375

φ = 0.855468750

(b) Fig. 14.

3. Concept of a Time-Reversible Attractor An examination of Figs. 3–14 shows that, except for period-k “isle of Eden” bit strings, all attractors of the local rule 62 have a nonempty basin of attraction containing two or more gardens of Eden (pink) bit strings. It follows that there are many initial bit strings which converge to the same attractor. Hence, given any bit string on an attractor, it is impossible to retrace its dynamics in backward time to find where it had originated in the transient regime. In fact, unlike in ordinary differential equations (ODE), it is even impossible, for some rules, to retrace its past history on the attractor. The following four examples illustrate two possible scenarios. Example 3.1. A Time-Reversible Period-3 Isle of Eden of 62 . The top pattern of Fig. 15 shows

the evolution under rule 62 from an initial bit string {110110110 . . . 110110110} on a period-3 isle 63 bits

of Eden corresponding to the nine-bit “isle of Eden” shown in the upper-left corner of Fig. 5(f). Observe that there is no transient in the evolution dynamics since the first three rows of Fig. 15 are repeated periodically. In other words, we have a period-3 orbit. The fact that these three rows constitute an “isle of Eden” follows from observing that the subsequence “110” comes from the nine-bit period-3 isle of Eden shown in Figs. 5(f)–5(i). 9

(Continued )

Now suppose we choose the last row of the top pattern in Fig. 15 (identified as row 0) as the initial state of rule 118 = T † ( 62 ), henceforth called the bilateral twin of 62 , where T † is the left-right transformation defined in [Chua et al., 2004].9 Iterating this bit string 19 times, we obtain the second pattern shown in Fig. 15. Observe that this 20-row pattern, identified on the right margin from row 0, −1, −2, . . . , −19, is a mirror image of the last 20 rows of the upper pattern, identified by the corresponding row numbers 0, 1, 2, . . . , 19! To verify rigorously that these two 20-row patterns are exact mirror images of each other, and not based on a superficial inspection, let us reproduce the last 20 rows of the upper pattern in the bottom of Fig. 15. Let us then “rotate” the 20-row pattern of the middle pattern about row 0 by 180◦ , and then superimpose the resulting “flip” pattern at the bottom in such a way that row “0” of each pattern are aligned. An examination of the resulting “time reversal comparison pattern” in Fig. 15 shows no difference from the original 20-row pattern from the top pattern.10 In other words, we have demonstrated, via this example, that for certain rules N , and certain invariant orbits, it is possible to apply the bilateral twin rule 118 of 62 to a point on this orbit, and successfully recover its past. Just in case the cautious reader suspects the above example demonstrating a “time-reversible”

The left-right transformation T † is a member of the four-element Klein Vierergruppe presented in [Chua et al., 2004]. A fundamental property of T † is that any rule N and its bilateral twin T † ( N ), are globally equivalent to each other, and hence have identical dynamic behaviors. 10 Our choice of “20” is completely arbitrary.


invariant set is a fluke due to our contrived choice of an “isle of Eden”, consider the next example. Example 3.2. A Time-Irreversible Period-3 Isle of Eden of 62 . Let us repeat the above exercise to

another period-3 “isle of Eden” of 62 . Consider the upper pattern shown in Fig. 16 showing the evolution under rule 62 from the initial bit string {000101111000101111 . . . 000101111} on a period-3 63 bits

“isle of Eden” corresponding to the nine-bit “isle of Eden” shown in the upper-right corner of Fig. 5(f). Just as in Example 3.1, rule 62 spawns a period-3 invariant set from this initial bit string, without any transients. Let us repeat the same “time reversal test” from Example 3.1 by choosing the last 20 rows of the top pattern in Fig. 16 for setting up the “time reversal comparison pattern”. Applying rule 118 to the last row (row “0”) of the top pattern, we obtain the second pattern in Fig. 16. Copying the last 20 rows of the top pattern to the bottom of Fig. 16 and superimposing the “flip” pattern of the second 20-row pattern generated by rule 118 , we find the results do not coincide with each other, as is evident from the appearance of “white” pixels.11 It follows that this particular period-3 “isle of Eden” is not time-reversible in the sense that the past evolution of the invariant set under rule 62 cannot be retrieved via its bilateral twin rule 118 . To assure the curious reader that the concept of a time-reversible invariant set is not restricted to the rather specialized steady state dubbed isle of Eden, consider the next example. Example 3.3. A Attractor of 62 .

Time-Irreversible

Period-3

Let us choose an initial state of 62 which belongs to the “basin of attraction” of a period-3 attractor,12 and use it to behold a nonempty transient stage which converges towards an attractor; composed of the last 15 rows of the upper evolution pattern shown in Fig. 17. Let us repeat the time-reversal test from Example 3.1 by applying the bilateral twin rule 118 to the last row (row 0) of the top pattern of Fig. 17 to obtain the corresponding evolution pattern shown in the second part of Fig. 17. Since we are only interested in testing the time-reversibility of 11

701

attractors, we would need to iterate only 15 times since only 15 rows of the attractor are available for comparison in the top evolution pattern. For pedagogical reasons, we actually generated 20 rows, five more than needed in the middle pattern of Fig. 17. Copying next the last 20 rows of the top evolution pattern to the bottom of Fig. 17, and superimposing the “flip” pattern from the second part, we obtain the “time reversal comparison pattern” shown in the bottom of Fig. 17. Examining the last 15 rows of this superimposed pattern, we find they are not identical by virtue of the presence of “white” pixels between rows “0” to “14”. Hence this period3 attractor is not time-reversible. Before rushing to conclude that time-reversible attractors are rare happenstances, consider the next example. Example 3.4. A Time-Reversible Bernoulli σ τ Shift Attractor. Recall from Table 13-9 of

(p. 1150) of [Chua et al., 2004], as well as from Figs. 7–14 that rule 62 has a robust Bernoulli στ shift attractor Λ4 ( 62 ) with σ = −1, and τ = 2, in addition to the more numerous period-3 attractors Λ3 ( 62 ). Let us choose an initial bit string belonging to the basin of attraction of one of these Bernoulli στ -shift attractors and apply rule 62 to generate the upper evolution pattern shown in Fig. 18. Observe that after an initial transient stage, the initial bit string finally converges to an attractor Λ4 ( 62 ) (last 16 rows of Fig. 18). To test whether this Bernoulli στ -shift attractor is time-reversible or not, let us apply the same “time reversal test” from Example 3.1 and iterate the last row (row “0”) of the top pattern via rule 118 to obtain the second evolution pattern shown in Fig. 18. Again, although we need only to iterate 16 times since only 16 rows of the attractor being tested are available for comparison, we display 20 rows, four more than necessary, for pedagogical reasons. Let us copy next the last 20 rows from the top evolution pattern to the bottom of Fig. 18, and superimpose the “flip” pattern of 118 by aligning the corresponding row “0”, as shown in the bottom of Fig. 18. Finally, let us examine the “time reversal comparison pattern”. Observe that the first 16 rows

A white pixel denotes a contradiction in the color of the corresponding pixels in the top and middle patterns. Since attractors of rule 62 are dominated by period-3 attractors, as is evident from Fig. 6, it is a cinch to pick a bit string which converges under 62 to some period-3 attractor.

12

702


62

Row 69

Row 19

Row 0

118

Row 0

Row -19

62

118 Fig. 15.

Row 19

-19

Row 0

0

An example demonstrating the invariant set generated by the initial bit string {110110110 .{z . . 110110110}} via rule | 63 bits

62 satisfies the “time reversal test” by virtue of the fact that the bottom “time reversal comparison pattern” between corresponding evolutions of 62 and 118 , coincides with the last k rows of the top pattern, where k = 20 in this example.

in the superimposed patterns in Fig. 18 are identical! It follows that this Bernoulli στ -shift attractor of 62 is time-reversible in the sense that we can retrieve the last 16 rows of the top evolution pattern

in Fig. 18 by simply iterating the last row (row “0”) via its bilateral twin rule 118 a total of 16 times. As a confirmation that the Bernoulli στ -shift attractor in the top evolution pattern of 62 consists


703

Row 69

62

Row 19 Row 14

Row 0 Row 0

118

Row -19

62

118 Fig. 16.

Row 19

-19

Row 0

0

An example demonstrating the invariant set {000101111000101111 | {z . . . 000101111}} of rule 62 violates the “time 63 bits

reversal test” by virtue of the fact that the bottom “time reversal comparison pattern” is not identical to the last k rows of the top pattern, where k = 20 in this example.

of only the last 16 rows, observe our “comparison test” fails in the rows above row 15, where white pixels begin to emerge. Hence, iterating a few more rows than necessary allows us to reconfirm the end

of the transient regime, or equivalently, the beginning of the “attractor” regime. This example is highly significant because it demonstrates that a time-reversible attractor need

704


62

Row 69

Row 19 Row 14

Row 0

118

Row 0

Row -14 Row -19

62

118

Row 19

-19

Row 14

-14

Row 0

0

Fig. 17. An example demonstrating a time-irreversible period-3 attractor of rule 62 which fails the “time reversal test” in view of the emergence of white pixels in the “time reversal comparison pattern”.

not be periodic, assuming I → ∞. In fact, we will show in Table 4 that there are 170 local rules, out of 256, which are endowed with time-reversible attractors. Since the remaining 86 rules do not have robust time-reversible attractors, it is impossible to

retrieve the past of any bit string lying on their attractors. Inspired by the classic notion of “direction of time” from irreversible thermodynamics, we can assert that each time-irreversible attractor defines an arrow of time [Sachs, 1987;


62

705

Row 69

Row 19 Row 15

Row 0

118

Row 0

Row -15 Row -19

62

118

Row 19

-19

Row 15

-15

Row 0

0

Fig. 18. An example demonstrating a time-reversible Bernoulli στ -shift attractor Λ4 ( 62 ) with σ = −1 and τ = 2, of rule 62 which satisfies the “time-reversal test”.

Mackey, 1992] of the dynamic evolution on the attractor.

4. Time-Reversible Rules We have seen examples from Sec. 3 where a local rule N can be endowed with two distinct

types of attractors Λi ( N ), or invariant orbits Λi ( N ); namely, a time-reversible type and a timeirreversible type. The “past” of any bit string originating from Λi ( N ) can be completely retrieved by a repeated application of the associated bilat∆ eral twin rule N † = T † (N ) of N if, and only if, Λi ( N ) is time-reversible. Our goal in this section is

706


to define the above concept formally, and then identify all rules that are endowed with time-reversible attractors. Definition 4.1. Time-Reversible Λi ( N ).

An attractor Λi ( N ), or an invariant orbit Λi ( N ), is said to be time-reversible iff any “k” consecutive bit strings {x1 , x2 , . . . , xk } belonging to Λi ( N ) can be completely retrieved by applying ∆ the bilateral twin rule N † = T † (N ) of N to the last bit string xk , henceforth called the “present string”, a total of k iterations. It follows from Definition 4.1 that  xk−1 = N † (xk )      xk−2 = N † (N † (xk ))    † † †  xk−3 = N (N ((N (xk )))) .. .     † † †  x1 = N N (N · · · (xk )) · · ·    

ρ1 [N ] = [ρ†1 [N ]]−1

(22)

then these time-1 return maps are said to be invertible.15 Geometrically, this means the attractor vignette on the left, and its corresponding attractor vignette on the right in Table 2 of [Chua et al., 2005a] are mirror images of each other with respect to a mirror placed along the main diagonal of each return map. There are 146 local rules that are endowed with at least one invertible attractor; they are identified by a cross “ x” in column 7 of Table 19 in [Chua et al., 2005a]. Our next theorem guarantees that all of these 146 local rules have time-reversible attractors. Theorem 1. Invertible implies time-reversible.

(21)

k−2 times

Observe that Eq. (21) is satisfied if, and only if, the time reversal test described in Sec. 3 is satisfied. Definition 4.2.

Time-Irreversible Λi ( N ). An attractor Λi ( N ), or an invariant orbit Λi ( N ), is said to be time-irreversible iff Λi ( N ) is not timereversible. Definition 4.3. Time-Reversible Rules. A local

rule N is said to be time-reversible iff all attractors Λi ( N ), or all invariant orbits Λi ( N ), of N are time-reversible.

Definition 4.4. Time-Irreversible Rules. A local

rule N is said to be time-irreversible iff all robust13 attractors Λi ( N ) of are time-irreversible.

4.1. Relationship between invertible and time-reversible attractors Recall the 64-page gallery of robust attractors of all 256 local rules given in Table 2 of [Chua et al., 2005a]. Each attractor is characterized by a forward time-1 return map ρ1 [N ] on the left, and a backward time-1 return map ρ†1 [N ] on the right.14 By definition, each return map must have an inverse. If, in addition, the forward return map ρ1 [N ] and the backward return map ρ†1 [N ] are symmetrical with 13

respect to the main diagonal in the sense that

An attractor Λi ( N ), or an invariant orbit Λi ( N ), of local rule N is time-reversible if its associated forward (resp. backward) time-1 return map is invertible.

Proof. Since Λi ( N ) is invertible, by hypothesis, Eq. (22) holds. Since the backward time-1 return map ρ†1 [N ] of N is identical to the forward time-1 return map ρ1 [N † ] of N † (recall the “time-1 map property 1” on p. 1128 of [Chua et al., 2005a]), we have

ρ†1 [N ] = ρ1 [N † ]

(23)

Substituting Eq. (23) into Eq. (22), we obtain ρ1 [N ] = [ρ†1 [N ]]−1 = [ρ1 [N † ]]−1

(24)

Geometrically, Eq. (24) means that the left vignette of the forward time-1 return map ρ1 [N ] of N in Table 2 of [Chua et al., 2005a] and the left vignette of the forward time-1 return map ρ1 [N † ] of N † are mirror images of each other with respect to the main diagonal. It follows from Eq. (24) that ρ1 [N † ] = [ρ1 [N ]]−1

(25)

Since Eq. (25) is nothing more than a compact representation of Eq. (21), it follows that Λi ( N ) is time-reversible.

4.2. Time reversible does not imply invertible The “invertible” hypothesis in Theorem 1 is a sufficient but not necessary condition for an attractor

An attractor Λi ( N ) is said to be robust iff it can be observed by applying some random initial bit strings. The forward (resp. backward) time-1 return map is defined in Table 1 of [Chua et al., 2005a], and in Eq. (14) (resp. Eq. (16)). 15 See Definition 3 and Eq. (34) in [Chua et al., 2005a]. 14


Λi ( N ) to be time-reversible. For example, the 24 period-1 rules listed in Table 4 (p. 1132) of [Chua et al., 2005a] are not invertible. Yet, an exhaustive computer simulations show that they are nevertheless time-reversible. This result follows of course directly from the period-1 nature of the attractor. To uncover the mechanism and hence prove that all period-1 fixed points of χ1N are also fixed points of χ1 † , let us examine the firing patterns of N

the six noninvertible rules 12 , 13 , 44 , 78 , 140 and 170 .16 (1) Consider the first noninvertible rules N = 12 and its bilateral twin N † = 68 . Both rules share a . In addition, common bilateral firing pattern 12 has a nonbilateral firing pattern while 68 has a nonbilateral firing pattern . An analysis of the firing patterns of 12 shows that only the is active in any period-1 bilateral pattern fixed point of 12 or 68 . In other words, if φ0 → {x0 x1 x2 · · · xI−2 xI−1 xI }

xi−1

xi

xi+1

··· (26)

is a period-1 fixed point of 12 , then T † φ0 → {xI xI−1 xI−2 · · · xi+1 xi−1 · · · x2 x1 x0 }

xi (27)

must also be a fixed point of 12 if only the is active in both (26) bilateral pattern and (27). Since both 12 and 68 have the same active firing pattern, it follows that φ0 is also a period-1 fixed point of χ1 † . Hence, 12 is timereversible.

N

(2) Consider next N = 13 and its bilateral twin 69 . By the same reasoning, we find only the two and common bilateral firing patterns are active in any period-1 bit string of 13 and 69 . It follows that 13 is time-reversible. (3) Consider next N = 44 and its bilateral twin 100 . In this case, only the two common bilateral and are active in any firing patterns period-1 bit string of 44 and 100 . It follows that 44 is time-reversible. (4) Consider next N = 78 and its bilateral twin 92 . In this case, only three out of four firing patterns are active in any period-1 bit string of 78 and 92 ; namely, , and . Observe 16

707

that even though the last two firing patterns are not bilateral, together, they complement each other bilaterally so that the same reasonings in the preceding cases remain valid here. Hence, 78 is timereversible. (5) Consider next N = 140 and its bilateral twin 196 . In this case, only the two common bilateral and are active in any firing patterns period-1 bit string. It follows by the same reasoning that 140 is time-reversible. (6) Finally, consider N = 172 and its bilateral twin 228 . In this case, only the three common , and are bilateral firing patterns active in any period-1 bit string. It follows by the same reasoning that 172 is time-reversible.

4.3. Time-reversible implies invertible if it is not period-1 The preceding section shows that a time-reversible rule does not have to be invertible. There are eight noninvertible period-2 rules listed in Table 8 of [Chua et al., 2005a]. Could these rules also be time-reversible? The answer is no, as demonstrated by the “time-reversal test” shown in Figs. 19–21, respectively, for rules 28 , 29 , and 156 . Since each of the other five rules is globally equivalent to one of these three rules, they are also not timereversible. In fact, an exhaustive “time-reversal test” carried out for the remaining 78 noninvertible rules identified from Table 19 of [Chua et al., 2005a] shows that they are all time-irreversible. Let us combine this result with Theorem 1 as follows: An attractor, or an invariant orbit, of rule N which is not period-1, is time-reversible if, and only if, it is invertible. Corollary to Theorem 1.

4.4. Table of time-reversible rules All rules N that possess at least one time-reversible attractor, or invariant orbit, are listed in Table 2. Except for the four globally-equivalent rules 62 , 118 , 131 , and 145 , which are endowed with both a period-3 attractor and a Bernoulli στ -shift attractor, all other rules listed are endowed with only one or the other type; as indicated by a cross “ x” under the appropriate heading.

It suffices to analyze only these six rules since any of the other 18 rules in Table 4 of [Chua et al., 2005b] is globally equivalent to one of these six rules.

708


28

Row 69

Row 19

Row 0

70

Row 0

Row -19

28

70

Row 19

-19

Row 0

0

Fig. 19. Time-reversal test for noninvertible period-2 rules 28 shows it is not time-reversible, in view of the presence of white pixels in the time reversal comparison pattern.

Whether an attractor, or invariant orbit, is invertible or not is identified by a similar cross in the “pink” columns. Each rule in Table 2 endowed with only periodk orbits (k = 1, 2, 3), can be either bilateral (i.e.

T † ( N ) = N ), or nonbilateral. However, all rules endowed with a Bernoulli στ -shift attractor or invariant orbit are nonbilateral. Each of the 256 rules listed in Table 3 can be either bilateral or nonbilateral, but not both. A


29

709

Row 69

Row 19

Row 0

71

Row 0

Row -19

29

71

Row 19

-19

Row 0

0

Fig. 20. Time reversal test for noninvertible period-2 rules 29 shows it is not time-reversible, in view of the presence of white pixels in the time reversal comparison pattern.

breakdown of the 256 rules into different categories is given in Table 4. There are 170 time-reversible rules,17 almost twice as many as the number of 17

time-irreversible rules, of which there are 86. There are 146 invertible rules and 110 noninvertible rules. Observe that there are only 64 bilateral rules, less

Each rule endowed with at least one time-reversible attractor, or invariant orbit, is counted as a time-reversible rule in Table 4, although they are not strictly-time-reversible in the sense that Definition 4.2 is only partially satisfied.

710


156

Row 69

Row 19

Row 0

198

Row 0

Row -19

156

198

Row 19

-19

Row 0

0

Fig. 21. Time reversal test for noninvertible period-2 rules 156 shows it is not time-reversible, in view of the presence of white pixels in the time reversal comparison pattern.

than one-third of the number of nonbilateral rules, of which there are 192. Observe that all 64 bilateral rules are segregated in the left column, and all 192 nonbilateral rules are segregated in the right

column. Observe also that all 146 invertible rules are segregated in the upper two horizontal blocks, and all 110 noninvertible rules are segregated in the blocks below them. Finally, observe that


711

Table 2. List of 170 rules endowed with at least one time-reversible attractor, or an invariant orbit. Among them, 104 rules have complexity index κ = 1 (coded in red), 48 rules have κ = 2 (blue) and 18 rules have κ = 3 (green).

Invertible

Period k

Bernoulli-στ-shift

N Yes

No

1

2

3

σ

τ

Invertible

Period k


N Yes

No

1

2

3

x

σ

τ

-1, 1

1, 1

0

x

1

x

2

x

1

1

46

x

1

1

3

x

-1

2

47

x

-1, 1

1, 1

4

x

48

x

-1

1

5

x

49

x

1, -1

2, 1

7

x

50

x

x

8

x

51

x

x

10

x

1

1

55

x

x

11

x

-1, 1

1, 1

56

x

-1, 1

1, 1

x

43

x

x

44

x x -1

2

x

x

12

x

x

57

x

-1, 1

1, 1

13

x

x

58

x

1, -1

1, 2

-1, 1

2, 1

-1, 1

2, 1

-1

2

1

1

14

x

1, -1

1, 1

59

x

15

x

-1

1

62

x

16

x

-1

1

63

x

17

x

1

2

64

x

19

x

66

x

21

x

23

x

24

x

-1

1

72

x

x

31

x

-1

2

76

x

x

32

x

77

x

x

33

x

34

x

1

35

x

-1, 1

36

x

37

x

40

x

42

x

x 1

2

x

x

x

x

68

x

x

69

x

x

78

x

x

1

79

x

x

2, 1

80

x

-1

1

81

x

1, -1

1, 1

84

x

-1, 1

1, 1

85

x

1

1

87

x

1

2

x

x x x 1

x

1

712


Invertible

Period k


N Yes 91

No

1

x

2

3

σ

τ

x

(Continued )

Invertible

Period k


N No

1

140

x

x

x

x

Yes

2

3

σ

τ

92

x

x

141

93

x

x

142

x

1, -1

1, 1

x

143

x

1, -1

1, 1

144

x

-1

1

145

x

1, -1

2, 1

-1

1

94

x

95

x

96

x

98

x

1, -1

1, 1

152

x

99

x

1, -1

1, 1

160

x

x

162

x

1

1

x

163

x

1, -1

1, 2

164

x

x x

x x

x

100

x

x

x

104

x

108

x

112

x

-1

1

168

x

113

x

1, -1

1, 1

170

x

1

1

114

x

-1, 1

1, 2

171

x

1

1

115

x

1, -1

2, 1

172

116

x

-1

1

174

x

1

1

117

x

1, -1

1, 1

175

x

1

1

118

x

1, -1

2, 1

176

x

-1

1

119

x

1

2

177

x

-1, 1

1, 2

123

x

x

178

x

x

127

x

x

179

x

x

128

x

184

x

-1, 1

1, 1

130

x

131

x

132

x

133

x

x

x

x

x

x

1

1

185

x

1, -1

1, 1

-1, 1

2, 1

186

x

1

1

x

187

x

1

1

x

x

188

x

1

1

136

x

x

189

x

1

1

138

x

1

1

190

x

1

1

139

x

1

1

191

x

1

1

x

x

Chapter 6: From Time-Reversible Attractors to the Arrow of Time Table 2.

Invertible

Period k


N Yes 192

x

194

x

No

1

2

3

σ

τ

x 1

1

(Continued )

Invertible

Period k

Yes

No

1

231

x

232

x

x

196

x

x

233

x

x

197

x

x

234

x

x

x

235

x

x

236

x

x

200

x

201

x

x


N

202

x

x

237

x

x

203

x

x

238

x

x x

2

3

σ

τ

-1

1

204

x

x

239

x

205

x

x

240

x

-1

1

206

x

x

241

x

-1

1

207

x

x

242

x

-1

1

208

x

-1

1

243

x

-1

1

209

x

-1

1

244

x

-1

1

212

x

-1, 1

1, 1

245

x

-1

1

213

x

-1, 1

1, 1

246

x

-1

1

-1

1

216

x

x

247

x

217

x

x

248

x

x

218

x

x

249

x

x

219

x

x

250

x

x

220

x

x

251

x

x

221

x

x

252

x

x

222

x

x

253

x

x

223

x

x

254

x

x

224

x

x

255

x

x

226

x

1, -1

1, 1

227

x

-1, 1

1, 1

-1

1

x

228 230

x

x

713

714


N 0 1

Classification of each rule N ∈ {0, 1, 2, . . . , 255} into a bilateral rule where N † = N , or a nonbilateral rule.

Bilateral Yes

x x

3

5

10 11 14 15 16 17 19

23

x

32 33

x x

35 36 37

43 46 47 48 49 50 51 55 56 57

84 85 87

94

98

108

x x x

115 116 117 118 119 123 127 128

x x

132 133

x x

143 144

x

163

x

170

176 177 178

x x

185 186 187 188 189 190 191 192 194 200 201 204 205 208 209

Yes

213 218 219 222 223

x x x x x x x x x

224 226

230 231

233

x x x x

234 235 236 237

x x x x x x x x x x x x x x

238 239

241

x x x x

242 243 244 245 246 247 248 249 250 251

x x x x

252 253 254 255

x x

No

x x

240

x x x x x x x x x x

184

Bilateral

212

232

x x x x x x x

168

N

227

x x

162

179

x x

131

142

175

x x x

130

139

174

x x x x x x x x

114

138

164

No

x x x x x x x x

171

x x

112

Yes

136

160

x x x

96

Bilateral

152

x x x

113

x x x x x x x

42

81

N

145

x x x x x

80

104

x x

40

x x x

99

x x

34

77

95

x x

31

76

91

x

24

63

66

x

21

62

72

No

x x x x x x

64

x x x x x x x x

8

Yes

59

x x

7

Bilateral

58

x x

2

4

No

N

x x

Chapter 6: From Time-Reversible Attractors to the Arrow of Time Table 4. Partitioning of the 256 rules into invertible, noninvertible, time-reversible, time-irreversible, bilateral and nonbilateral rules. The Table number pertains to the table in [Chua et al., 2005a].

146 invertible rules

64 bilateral rules

192 non-bilateral rules

29 invertible bilateral period-1 rules (Table 3)

16 invertible non-bilateral period-1 rules (Table 3)

17 invertible bilateral period-2 rules (Table 7)

84 invertible non-bilateral Bernoulli στ-shift rules (Table 10)

170 timereversible rules

24 non-invertible non-bilateral period-1 rules (Table 4) 8 non-invertible non-bilateral period-2 rules (Table 8)

110 noninvertible rules

28 non-invertible non-bilateral Bernoulli στ-shift rules (Tables 11, 12) 18 non-invertible bilateral rules (Table 17)

86 timeirreversible rules

32 non-invertible non-bilateral rules (Table 18)

Fig. 22. Venn diagram depicting time-reversible rules intersecting with invertible, and nonbilateral rules, respectively. 18

715

all 170 time-reversible rules are segregated in the upper three horizontal blocks, and all 86 timeirreversible rules are segregated in the blocks below them. The Venn diagram shown in Fig. 22 depicts the intersections among the three dominant groups of rules; namely, time-reversible, invertible, and non bilateral rules.

5. There are 84 Time-Reversible Bernoulli στ -Shift Rules Our objective in this section is to verify that all 84 Bernoulli rules listed in Table 10 of [Chua et al., 2005a] are time-reversible, by applying the “timereversal test” to one member of each equivalent class listed in these tables. We will organize these tests into four logically similar groups.

5.1. There are 42 time-reversible Bernoulli στ -shift rules (with |σ| = 1, β = 2σ > 0, and τ = 1) having only one Bernoulli attractor There are 11 globally equivalent classes of Bernoulli στ -shift Rules with |σ| = 1, β = 2σ > 0, and τ = 1; namely, 2 , 10 , 24 , 34 , 42 , 46 , 130 , 138 , 152 , 162 , and 170 . The “time-reversal tests” for these rules are exhibited in Figs. 23(a), 23(b), . . . , 23(k), respectively. Observe that none of the “timereversal comparison patterns” at the bottom of these figures contains any “white” pixels. Since a 63-bit random bit string is chosen as the initial configuration, it follows that the 11 Bernoulli rules in Figs. 23(a)–23(k) are all time-reversible as predicted by Theorem 1 since all of these rules are invertible. Moreover, since the global equivalence class εκm of each of the first 10 rules in Fig. 23 has four distinct members,18 and since the global equivalence class ε134 of the last rule 170 in Fig. 23 has only two distinct members, namely, { 170 , 240 }, there are all together 42 time-reversible Bernoulli rules (with |σ| = 1, β = 2σ > 0, and τ = 1) endowed with only one Bernoulli attractor. Since rule 240 , the bilateral twin of rule 170 , is of fundamental importance, along with rules 170 , 85 , and 15 , in unifying the theory for all Bernoulli rules, we have exhibited also the time-reversal tests of rules 240 , 85 , and 15 in Figs. 23(l), (24), and (25), respectively.

See Tables 2–4 of [Chua et al., 2003] or Table 10 of [Chua et al., 2005a].

716


Row 69

2

Row 19

Row 0 Row 0

16

Row -19

2

16

Row 19

-19

Row 0

0

(a) Time reversal test for Bernoulli left-shift rule 2 . Fig. 23.

Time reversal tests for 12 generic Bernoulli στ -shift rules with |σ| = 1, β = 2σ > 0, and τ = 1.


Row 69

10

Row 19

Row 0 Row 0

80

Row -19

10

80 (b) Time reversal test for Bernoulli left-shift rule 10 . Fig. 23.

(Continued )

Row 19

-19

Row 0

0

717

718


Row 69

24

Row 19

Row 0 Row 0

66

Row -19

24

66 (c) Time reversal test for Bernoulli right-shift rule 24 . Fig. 23.

(Continued )

Row 19

-19

Row 0

0


Row 69

34

Row 19

Row 0 Row 0

48

Row -19

34

48 (d) Time reversal test for Bernoulli left-shift rule 34 . Fig. 23.

(Continued )

Row 19

-19

Row 0

0

719

720


Row 69

42

Row 19

Row 0 Row 0

112

Row -19

42

112 (e) Time reversal test for Bernoulli left-shift rule 42 . Fig. 23.

(Continued )

Row 19

-19

Row 0

0


Row 69

46

Row 19

Row 0 Row 0

116

Row -19

46

116 (f) Time reversal test for Bernoulli left-shift rule 46 . Fig. 23.

(Continued )

Row 19

-19

Row 0

0

721

722


Row 69

130

Row 19

Row 0 Row 0

144

Row -19

130

144 (g) Time reversal test for Bernoulli left-shift rule 130 . Fig. 23.

(Continued )

Row 19

-19

Row 0

0


Row 69

138

Row 19

Row 0 Row 0

208

Row -19

138

208 (h) Time reversal test for Bernoulli left-shift rule 138 . Fig. 23.

(Continued )

Row 19

-19

Row 0

0

723

724


Row 69

152

Row 19

Row 0 Row 0

194

Row -19

152

194 (i) Time reversal test for Bernoulli right-shift rule 152 . Fig. 23.

(Continued )

Row 19

-19

Row 0

0


Row 69

162

Row 19

Row 0 Row 0

176

Row -19

162

176 (j) Time reversal test for Bernoulli left-shift rule 162 . Fig. 23.

(Continued )

Row 19

-19

Row 0

0

725

726


Row 69

170

Row 19

Row 0 Row 0

240

Row -19

170

240 (k) Time reversal test for Bernoulli left-shift rule 170 . Fig. 23.

(Continued )

Row 19

-19

Row 0

0


Row 69

240

Row 19

Row 0 Row 0

170

Row -19

240

170 (l) Time reversal test for Bernoulli right-shift rule 240 . Fig. 23.

(Continued )

Row 19

-19

Row 0

0

727

728


Row 69

85

Row 19

Row 0 Row 0

15

Row -19

85

15 Fig. 24.

Time reversal test for complemented Bernoulli left-shift rule 85 .

Row 19

-19

Row 0

0


Row 69

15

Row 19

Row 0 Row 0

85

Row -19

15

85 Fig. 25.

Row 19

-19

Row 0

0

Time reversal test for complemented Bernoulli right-shift rule 15 .

729

1.0

1.0

φn

φn

0.5

0.5

730

0

0.5 2

Firing Patterns

3 7

6

1

0

0 red pixel :

4

3 red pixels :

3 7

Bernoulli Shift map ρ 170

5

0.5

= 2 (φn −1 − 0.5 ) , 0.5 ≤ φn −1 ≤ 1

170 Fig. 26.

2

Firing Patterns

3 7

6

1

0

0 red pixel :

φn = 2φn −1 , 0 ≤ φn −1 < 0.5

1

0

1.0

5

1 red pixel : 2 red pixels :

φn −1

4

3 red pixels :

5

Bernoulli Shift map ρ 240 1 2

4

2 red pixels :

1.0

Lower branch

5

1 red pixel :

φn −1

φn = φn −1 , 0 ≤ φn −1 ≤ 1

6

7

Upper branch

φn =

240

Graph and equation defining the two canonical Bernoulli shift maps ρ

170

and ρ

1 (φn−1 +1) , 0 ≤ 2

240

.

n−1

≤1

1.0

1.0

φn

φn

0.5

0.5

731

0

0.5 2

Firing Patterns

3 7

6

1

0

0 red pixel : 0

1 red pixel :

4

2

2 red pixels :

5

4

φn −1

Complemented Bernoulli Shift map ρ 85

φn = −2 (φn −1 − 0.5 ) ,

6

0 ≤ φn −1 < 0.5

3 red pixels :

= −2 (φn −1 − 1) ,

85

0.5 ≤ φn −1 ≤ 1 Fig. 27.

0

1.0

0.5 2

Firing Patterns

3 7

6

1

0

0 red pixel : 0

1 red pixel :

1

2 red pixels :

4

5

2

3

3 red pixels :

15

Graph and equation defining the two flipped canonical Bernoulli shift maps ρ

φn −1

1.0

Complemented Bernoulli Shift map ρ 15 Lower branch 1 φn = − (φn −1 − 1) , 0 ≤ φn−1 ≤ 1 2 Upper branch

φn = − 15

and ρ

1 (φn−1 − 2) , 0 ≤ φn−1 ≤ 1 2 85

.

732


5.2. Four canonical Bernoulli shift maps An examination of the Bernoulli στ -shift maps of the attractor of rules 85 and 15 in Table 13-2 (p. 1143) of [Chua et al., 2005a] shows that β = −2, σ = 1, τ = 1 for rule 85 , and β = −(1/2), σ = −1, τ = 1 for rule 15 . Since β < 0 for these two maps, they are not included in Sec. 5.1, which collects only time-reversible rules with |σ| = 1, τ = 1, and β > 0. Since the four Bernoulli στ -shift maps ρ 170 , ρ 270 , ρ 85 , and ρ 15 associated with rules 170 , 240 , 85 , and 15 , respectively, play a fundamental role as building blocks of all other Bernoulli στ -shift maps, they will henceforth be referred to as canonical Bernoulli shift maps. More precisely, the inverse pair of maps {ρ 170 , ρ 240 } will be called the canonical Bernoulli shift maps, while the inverse pair of maps {ρ 85 , ρ 15 } will be called the flipped canonical Bernoulli shift maps. The graphs and equations defining these maps are shown in Figs. 26 and 27, respectively. Observe that the graph of ρ 85 can be obtained by flipping the graph of ρ 170 along the horizontal line φn = 0.5. Similarly, the graph of ρ 15 can be obtained by flipping the graph of ρ 240 along the same line. For future reference the complete set of time-1 return maps of all 44 time-reversible Bernoulli στ -shift rules are collected in Table 5. They are ordered consecutively in accordance with the rule numbers.

5.3. There are eight time-reversible time-2 Bernoulli στ -shift rules (with |σ| = 1, β = 2σ > 0, and τ = 2) having only one Bernoulli attractor There are two globally equivalent classes of Bernoulli στ -shift rules with σ = −1, β = 2σ > 0, and τ = 2; namely, 3 and 7 .19 The “time reversal tests” for these rules are exhibited in Figs. 28(a) and 28(b), respectively. The absence of “white” pixels in the bottom of each figure implies that rules 3 and 7 are indeed time-reversible. Since rule 3 19

is a member of the global equivalence class ε14 = { 3 , 17 , 63 , 119 } and rule 7 is a member of the global equivalence class ε17 = { 7 , 21 , 31 , 87 }, it follows that there are eight time-reversible time-2 Bernoulli στ -shift rules under the above grouping. The time-1 and time-2 return maps of the above eight time-reversible time-2 Bernoulli rules are collected in Table 6.

5.4. There are 32 time-reversible Bernoulli στ -shift rules with two invertible attractors There are ten globally equivalent classes of Bernoulli στ -shift rules endowed with two invertible attractors; namely, 11 , 14 , 35 , 43 , 56 , 57 , 58 , 62 ,20 142 , and 184 . The time reversal tests for these rules are shown in Figs. 29(a1 ), 29(a2 ); 29(b1 ), 29(b2 ); . . . , 29(j1 ), 29(j2 ), respectively, one for each attractor. Again, observe that they are all time-reversible, as expected from Theorem 1, since all of these rules are invertible. An examination of Table 10 of [Chua et al., 2005a] shows that except for rules 43 , 57 , 142 , and 184 whose equivalence class has only two members each, all others have four members. All together, there are 32 time-reversible Bernoulli στ shift rules under the above grouping. Among the 32 rules, 20 are endowed with two time-1 return maps each. They are ordered in accordance with their rule numbers and collected in Table 7. The remaining 12 time-reversible Bernoulli rules are endowed with a time-1 Bernoulli attractor and a time-2 Bernoulli attractor. Their return maps are collected in Table 8.

5.5. Composition of 84 time-reversible Bernoulli rules Finally, let us add all time-reversible Bernoulli στ -shift rules found in the previous subsections; namely, 42 from Sec. 5.1, 2 from Sec. 5.2, 8 from Sec. 5.3, and 32 from Sec. 5.4. All together, there are 84 time-reversible Bernoulli στ -shift rules, thereby verifying the number listed in row 2, column 2 of Table 4.

See Table 10 of [Chua et al., 2005a]. The time-1 attractor of 62 is the “period-3” isle of Eden given already in Fig. 15. Consequently, both tests for this attractor are for the “time-2” Bernoulli-shift attractor φn−2 → φn , with different initial states. 20


Time-1 return maps of 44 time-reversible Bernoulli rules.

733

734


(Continued )


(Continued )

735

736


Row 69

3

Row 19

Row 0 Row 0

17

Row -19

3

17

Row 19

-19

Row 0

0

(a) Time reversal test for rule 3 . Fig. 28.

Time reversal tests for two generic Bernoulli στ -shift maps with σ = −1, β = 2σ > 0, and τ = 2.


Row 69

7

Row 19

Row 0 Row 0

21

Row -19

7

21 (b) Time reversal test for rule 7 . Fig. 28.

(Continued )

Row 19

-19

Row 0

0

737

Table 6.

1

0.5

1

1

φn

φn

Return maps of eight time-reversible time-2 Bernoulli rules with τ = 2.

0.5

0.5

0

0.5

φn −1

0

1

time-1 map

738

S12

0.5

φn −2

φn

φn

0.5

φn −1

1

φn −2

time-2 map

N = 17

κ=1

1

φn −2

1

κ=1

N=7 1

φn 0.5

0.5

0.5

0.5

time-2 map

φn

0

0

1

1

1

time-1 map

S12

φn −1

S13

0.5

0.5

0.5

time-1 map

κ=1

N=3

0.5

0

1

time-2 map

1

0

φn

φn

1

0

0.5

φn −1

1

0

time-1 map

S13

0.5

φn −2

time-2 map

N = 21

κ=1

1

Table 6.

1

φn

φn

1

1

0

0.5

φn −1

0

1

time-1 map

739

S15

0.5

0.5

φn −2

1

0.5

S16

κ=1

0.5

φn −1

1

time-1 map

S15

0

N = 63 φn

φn −2

time-2 map

N = 87

κ=1

1

φn −2

1

κ=1

1

0.5

0.5

0.5

0.5

time-2 map

φn 0.5

0

φn −1

0

1

1

1

φn

φn

0.5

time-1 map

time-2 map

N = 31

0.5

0

1

1

φn

φn

0.5

0.5

(Continued )

0

0.5

φn −1

1

0

time-1 map

S16

0.5

φn −2

time-2 map

N = 119

κ=1

1

740


Row 69

11

Row 19

Row 0 Row 0

81

Row -19

11

81

Row 19

-19

Row 0

0

(a1 ) Time reversal test for the first attractor of rule 11 . Fig. 29.

Time reversal tests for ten Bernoulli στ -shift maps with two invertible attractors.


Row 69

11

Row 19

Row 0 Row 0

81

Row -19

11

81 (a2 ) Time reversal test for the second attractor of rule 11 . Fig. 29.

(Continued )

Row 19

-19

Row 0

0

741

742


Row 69

14

Row 19

Row 0 Row 0

84

Row -19

14

84 (b1 ) Time reversal test for the first attractor of rule 14 . Fig. 29.

(Continued )

Row 19

-19

Row 0

0


Row 69

14

Row 19

Row 0 Row 0

84

Row -19

14

84 (b2 ) Time reversal test for the second attractor of rule 14 . Fig. 29.

(Continued )

Row 19

-19

Row 0

0

743

744


Row 69

35

Row 19

Row 0 Row 0

49

Row -19

35

49 (c1 ) Time reversal test for the first attractor of rule 35 . Fig. 29.

(Continued )

Row 19

-19

Row 0

0


Row 69

35

Row 19

Row 0 Row 0

49

Row -19

35

49 (c2 ) Time reversal test for the second attractor of rule 35 . Fig. 29.

(Continued )

Row 19

-19

Row 0

0

745

746


Row 69

43

Row 19

Row 0 Row 0

113

Row -19

43

113 (d1 ) Time reversal test for the first attractor of rule 43 . Fig. 29.

(Continued )

Row 19

-19

Row 0

0


Row 69

43

Row 19

Row 0 Row 0

113

Row -19

43

113 (d2 ) Time reversal test for the second attractor of rule 43 . Fig. 29.

(Continued )

Row 19

-19

Row 0

0

747

748


Row 69

56

Row 19

Row 0 Row 0

98

Row -19

56

98 (e1 ) Time reversal test for the first attractor of rule 56 . Fig. 29.

(Continued )

Row 19

-19

Row 0

0


Row 69

56

Row 19

Row 0 Row 0

98

Row -19

56

98 (e2 ) Time reversal test for the second attractor of rule 56 . Fig. 29.

(Continued )

Row 19

-19

Row 0

0

749

750


Row 69

57

Row 19

Row 0 Row 0

99

Row -19

57

99 (f1 ) Time reversal test for the first attractor of rule 57 . Fig. 29.

(Continued )

Row 19

-19

Row 0

0


Row 69

57

Row 19

Row 0 Row 0

99

Row -19

57

99 (f2 ) Time reversal test for the second attractor of rule 57 . Fig. 29.

(Continued )

Row 19

-19

Row 0

0

751

752


Row 69

58

Row 19

Row 0 Row 0

114

Row -19

58

114 (g1 ) Time reversal test for the first attractor of rule 58 . Fig. 29.

(Continued )

Row 19

-19

Row 0

0


Row 69

58

Row 19

Row 0 Row 0

114

Row -19

58

114 (g2 ) Time reversal test for the second attractor of rule 58 . Fig. 29.

(Continued )

Row 19

-19

Row 0

0

753

754


Row 69

62

Row 19

Row 0 Row 0

118

Row -19

62

118 (h1 ) Time reversal test for the second attractor of rule 62 . Fig. 29.

(Continued )

Row 19

-19

Row 0

0


Row 69

62

Row 19

Row 0 Row 0

118

Row -19

62

118

Row 19

-19

Row 0

0

(h2 ) Time reversal test for the second attractor of rule 62 with a different initial state. Fig. 29.

(Continued )

755

756


Row 69

142

Row 19

Row 0 Row 0

212

Row -19

142

212 (i1 ) Time reversal test for the first attractor of rule 142 . Fig. 29.

(Continued )

Row 19

-19

Row 0

0


Row 69

142

Row 19

Row 0 Row 0

212

Row -19

142

212 (i2 ) Time reversal test for the second attractor of rule 142 . Fig. 29.

(Continued )

Row 19

-19

Row 0

0

757

758


Row 69

184

Row 19

Row 0 Row 0

226

Row -19

184

226 (j1 ) Time reversal test for the first attractor of rule 184 . Fig. 29.

(Continued )

Row 19

-19

Row 0

0


Row 69

184

Row 19

Row 0 Row 0

226

Row -19

184

226 (j2 ) Time reversal test for the second attractor of rule 184 . Fig. 29.

(Continued )

Row 19

-19

Row 0

0

759

Table 7. Time-1 return maps of 20 time-reversible Bernoulli rules. Each rule has two attractors, identified in red and blue colors, respectively.

760

Table 7.

(Continued )

761

Table 8.

Time-1 and Time-2 return maps of 12 time-reversible rules with two Bernoulli return maps.

762

Table 8.

(Continued )

763

Table 8.

(Continued )

764


6. What Bit Strings Are Allowed in an Attractor or Invariant Orbit? The concept of an attractor, or an invariant orbit, Λ( N ) of a local rule N is a natural generalization of the classical concept of “steady state” behavior from linear systems. However, unlike linear dissipative systems where all initial states must converge to a unique steady state, each local rule N in a cellular automata can have multiple attractors Λi ( N ), i = 1, 2, . . ., m, each having its associated “basin of attraction” (Λi ( N )), as clearly illustrated in Fig. 6. It can also support multiple invariant orbits. Only certain bit strings, however, can inhabit an attractor, or an invariant orbit, of a particular rule N . Our goal in this section is to uncover the “laws” that each attractor, or invariant orbit, imposes upon its “citizens”, namely, the bit strings, for each rule N . An initial bit string x0 which violates the laws governing all attractors and invariant orbits of N is said to be a transient bit string, or transient state, and the number of iterations T∆ ( N ) it took x0 to converge to an attractor Λi ( N ) is called its transient period. If T∆ ( N ) = 0, then x0 is either already on an attractor, or on an invariant orbit. The longest transient period corresponds to the farthest “Garden of Eden” initial bit string x0 from Λi ( N ). For example, T∆ ( 62 ) = 4 for the six-bit string for rule 62 in Fig. 6(a) is the longest transient period for all period-3 attractors with six bits. In this section we will specify not only the “laws governing each attractor, or invariant orbit”, (following the same format as Table 1), but also an accompanying “table of transient regimes” which contains a selected sequence of iterations of a typical bit-string evolution towards an attractor, or an invariant orbit Λ( N ). Whenever possible, a formula of T∆ ( N ) will also be given in this table. For pedagogical reasons, a directed graph ( N ) will also be given along with the bit-string laws for each Λi ( N ), which is no more than a compact algorithm for generating most, if not all, bit strings inhabiting Λi ( N ). The forward time-1 return map associated with the bit strings is also given above ( N ) for the reader’s convenience.

6.1. Laws governing period-1 bit strings To illustrate how information is organized in the tables in this section, let us begin with the

765

“trivial” period-1 rules listed in Table 9A; namely, the eight equivalent classes of period-1 rules 0 , 8 , 32 , 40 , 128 , 136 , 160 , and 168 from Table 5 of [Chua et al., 2005a] which converge to a homogeneous “zero” bit string. The “bit-string laws” and their associated “directed graph” ( N ) in this case are trivial. The right-most column in this table provides a list of globally equivalent rules, along with their firing patterns. All together, Table 9A provides the laws governing 24 “homogeneous” period-1 rules, modulo a transformation T † , T , or T ∗ defined in [Chua et al., 2004]. A “table of transient regimes” associated with the rules listed in Table 9A is given in Table 9B. A formula for the “transient period ” T∆ ( N ) for each rule N listed is also given in terms of the three parameters C1 , C2 , and C3 defined at the end of this table. The laws governing 12 time-reversible and invertible nonhomogeneous period-1 rules extracted from Table 3 of [Chua et al., 2005a]; namely, 4 , 36 , 72 , 76 , 77 , 94 , 104 , 132 , 164 , 200 , 204 , and 232 are presented in Table 10A. It would be instructive for the reader to generate all possible period-1 bit strings inhabiting an attractor of 104 using the directed graph ( 104 ). Any bit string sequence obtained by following the one way “traffic signs” in ( 104 ) is a bona fide citizen of an attractor Λ( 104 ) of 104 . The self loop through “0” indicates that the “0” bit can be repeated as a subsequence any number of times commensurate with the length of the bit string. For convenience, the forward time-1 return map associated with the allowed period-1 bit strings is shown on top of ( N ). All together, there are 45 time-reversible period-1 rules whose allowed bit strings are specified in Tables 9A and 10A, modulo a transformation. The table of transient regimes associated with the 12 period-1 rules in Table 10A is given in Table 10B. The laws governing the six time-reversible and noninvertible nonhomogeneous period-1 rules 12 , 13 , 44 , 78 , 140 , and 172 are presented in Table 11A. The associated table of transient regimes is given in Table 11B. The time-1 characteristic functions χ1N associated with the 12 time-reversible, invertible nonhomogeneous period-1 rules listed in Table 10A, and the six time reversible, noninvertible nonhomogeneous period-1 rules listed in Table 11A are exhibited in a gallery in Table 12.

Table 9A.

Firing Patterns

Table of bit-string laws for eight homogeneous “0” period-1 rules.

Law Governing Attractor Bit Strings

Only blue bits can occur : 0 xi = 0 ,

i = 0, 1, 2, ··· I



( )

( )

Attractor : Λ 0

T† 0 ⇓

Time-1 Return map

ρ1[0] : φn −1

0

( )

φn

T 0

0

8

Only blue bits can occur :

766

xi = 0 ,

3

(0)

i = 0, 1, 2, ··· I

( )

255

5

6

7

0

1

⇓

2

3

4

255

5

6

7

( )

Time-1 Return map

64

ρ1[8] : φn −1

⇓ 6

( )

φn

T 8

0

⇓

2

3

239

5

6

( )

T* 8

(8)

000000000000000000000000

Only blue bits can occur : xi = 0 ,

4

T† 8

Example :

5

3

Attractor : Λ 8

0

32

2

T* 0

000000000000000000000000

i = 0, 1, 2, ··· I

7

0

⇓

3

4

253

5

6

7

( )

( )

T† 32

Time-1 Return map

32

⇓

5

φn

T 32

( )

0

1

⇓

3

4

0

251

6

7

Example : 000000000000000000000000

1

2

Attractor : Λ 32

ρ1[32] : φn −1

1

⇓

( )

Example :

0

( 32 )

5

T* 32

( )

0

1

⇓

3

4

6

7

251

5

Table 9A.

Firing Patterns



3

i = 0, 1, 2, ··· I

(Continued )



( )

( )

Attractor : Λ 40

T† 40 ⇓

Time-1 Return map

ρ1[40] : φn −1

96

( )

φn

5

T 40

( )


767

xi = 0 ,

7

( 40 )

i = 0, 1, 2, ··· I

(

Attractor : Λ 128

)

(

φn

T 128

3

xi = 0 ,

i = 0, 1, 2, ··· I

( 128 )

7

Attractor : Λ 136

) 7

)

1

⇓

254

5

6

7

⇓

2

3

4

254

5

6

7

6

7

T† 136 ⇓

)

1

)

192

(

φn

T 136

0

1

⇓

3

238

5

6

7

2

3

4

5

6

7

T* 136

( 136 )

) 2

(

Example : 000000000000000000000000

7

4

(

)

Time-1 Return map

ρ1[136] : φn −1

6

3

T* 128


4

2

(

Example :

(

5

3

128

000000000000000000000000

7

⇓

Time-1 Return map

ρ1[128] : φn −1

249

(

6 0

⇓

T† 128

0

136

5

T* 40

000000000000000000000000

1

3

235

Example :

6 0

⇓

0

128

5

⇓

252

)

Table 9A.

Firing Patterns



5

i = 0, 1, 2, ··· I

(Continued )


(

Attractor : Λ 160


)

(

⇓

Time-1 Return map

ρ1[160] : φn −1

7

160

(

φn

T 160

(

T* 160

768

( 160 )

000000000000000000000000

3

Only blue bits can occur : xi = 0 ,

i = 0, 1, 2, ··· I

5 7

Attractor : Λ 168

)

1

5

3

4

6

7

)

250

(

1 3

4

5

6

7

5

6

7

)

⇓

224

(

φn

T 168

)

1

⇓

3

234

0

(

Example : 000000000000000000000000

⇓

T† 168

Time-1 Return map

ρ1[168] : φn −1

7

)

250

Example :

168

5

⇓

0

(

)

T† 160

T* 168

( 168 )

5

7

3

4

6

7

)

⇓

248

6

5

Table 9B.

N = T∆

0

Firing Patterns

8

Firing Patterns

32

Firing Patterns

Table of transient regimes for eight homogeneous “0” period-1 rules.

( )

Transient Period T∆

T∆ 0 = 1

3


T∆ 8 = 2

5


n=0 n=1 n=2

N =

( )

n=0

T∆ n=1 n=2

769

n=3

N = n=0 n=1

T∆

n=2 n=3 n=4 n=5 n=6

( )

 C1   +1  2

T∆ 32 = int 

Table 9B.

N =

40

Firing Patterns

3

128

Firing Patterns

7

5

(Continued )


 2C2   +1  3 

( )

T∆ 40 = int 

n=0 n=1 n=2

T∆

n=3 n=4 n=5 n=6

770

n=7 n=8

N = n=0 n=1

T∆

n=2 n=3 n=4 n=5


(

)

 C3 + 1   2 

T∆ 128 = int 

Table 9B.

N = 136

(Continued )

(

Firing Patterns

3

7


Firing Patterns

5

7

Transient T 160 = int Period T∆ ∆

)

T∆ 136 = C3

n=0 n=1

T∆

n=2 n=3 n=4 n=5 n=6

771

N = n=0 n=1

T∆

n=2 n=3 n=4 n=5 n=6

160

(

)

C or C     2 

Table 9B.

N = 168

Firing Patterns

3

5

(Continued )

7


n=0 n=1 n=2 n=3

T∆

n=4 n=5 n=6

772

n=7 n=8 n=9

C1 = number of bits in the longest substring consisting of one red bit separated by exactly one blue bit. Example : C1 C2 = number of bits in the longest substring consisting of 2 red bits separated by exactly one blue bit. Example : C2 C3 = largest number of consecutive and adjacent red bits, i.e, no intervening blue bits. Example : C3 int (x)

integer portion of x (e.g., int (5/2) = 2).

(

)

T∆ 168 = C3

Table 10A.

Firing Patterns

4

Table of bit-string laws for 12 time-reversible and invertible nonhomogeneous period-1 rules.



Only single red bits separated from Attractor : Λ ( 4 ) Time-1 each other by one or more blue bits:

( )

T† 4 ⇓

ρ1[4] : φn −1

2

4

Return map

xi = 1 ⇒ xi −1 = 0 and xi +1 = 0

2


( )

φn

T 4 ⇓

2

223

1

0

*

Example :

(4)

010100100010001010010000010001001

36

Only single red bits separated from each other by at lease 2 blue bits:

773

xi = 1 ⇒ xi −2 = 0 , xi −1 = 0,

2 5

xi +1 = 0 , and xi + 2 = 0.

( )

1

6

7

0

1

⇓

3

4

219

6

7

( )

0

1

⇓

3

4

219

6

7

⇓

223 ⇓

2

36

5

T 36

1

Example :

( 36 )

000100100100001001000001001000001

6

xi = 1 ⇒ XOR( xi −1 , xi +1 ) = 1

( )

( )

Attractor : Λ 72

T† 72

Time-1 Return map

ρ1[72] : φn −1

3

0

φn 1

⇓

3

72

6

( )

0

T 72

1

⇓

2

3

237

5

6

( )

T* 72

Example : 001100011011000011000000110000011

7

4

T 36

72

6 0

*

Only adjacent pairs of red bits separated from each other by one or more blue bits:

4

3

2

( )

φn 0

3

T† 36

Time-1 Return map

0

1

( )

Attractor : Λ 36

ρ1[36] : φn −1

( )

T 4

0

( 72 )

7

0

⇓

2

3

237

5

6

7

Table 10A.

Firing Patterns

76


3 6


ρ1[76] : φn −1

⇓

774

xi = 1 ⇒ xi −1 = 0 and xi +1 = 0 , or xi −1 = 0 and xi +1 = 1 , or xi −1 = 1 and xi +1 = 0

0 2 3 6

Example :

T 76 ⇓

94 1 2 3

xi = 1 ⇒ xi −1 = 0 and xi +1 = 1, or xi −1 = 0 and xi +1 = 0, or xi −1 = 1 and xi +1 = 0

4

Example :

6

011010101011011011011011011010101

0 2

205

1

1

0

( )

( )

2

205 ⇓

( )

0 2

77

0

0

1

( )

( 77 )

⇓

φn 1

77

3 6

T† 94

Time-1 Return map

0

0 2

( )

( )

Attractor : Λ 94

ρ1[94] : φn −1

⇓

3 6

T* 77

1

3 6

T 77 ⇓

1 2

3

94

6

( )

0

T 94 ⇓

7

( )

T* 94

( 94 )

⇓

133

4

2

133

1

7

0 2

77

φn

3 6

T† 77

Time-1 Return map

7

0

( )

Attractor : Λ 77

ρ1[77] : φn −1

⇓

3 6

T* 76

( 76 )

3 6

( )

φn

011010010110011001101001101011001

Only single red bits or adjacent pairs of red bits separated from each other by only one blue bit:

2

76

010100110100011001000001000110001

Only single red bits or adjacent pairs of red bits separated from each other by one or two blue bits:

( )

T† 76

Example :

77


Attractor : Λ( 76 ) Only single red bits or adjacent Time-1 pairs of red bits separated from Return map each other by one or more blue bits:


2

(Continued )

0 2 7

Table 10A.

Firing Patterns

104


Only adjacent pairs of red bits separated from each other by at least 2 blue bits:

xi = 1 ⇒ xi −1 = 0 and xi +1 = 1

3 5

or xi −1 = 1 and xi +1 = 0

6

(Continued )


(

Attractor : Λ 104

)

Time-1 Return map

ρ1[104] : φn −1

(

φn

T 104

5

)

1

(

( 104 )

Only single red bits separated from Attractor : Λ ( 132 ) each other by one or more blue bits: Time-1

⇓

(

775

(

φn

T 132

( 132 )

010000010010100001000100100000101

xi = 1 ⇒ xi −2 = 0 , xi −1 = 0, xi +1 = 0 , and xi + 2 = 0

5 7

(

Attractor : Λ 164

0

001000100100010001000001000001001

)

1 2

(

T 164

1

4

6

7 1

2

3

4

6

7

) 2 5

7

)

1

⇓

3

4

218

6

7

(

( 164 )

3

)

164

φn 0

⇓

T* 164

Example :

7

222 T† 164

7

2

⇓

(

)

Time-1 Return map

ρ1[164] : φn −1

6

)

222

(

Example :

2

3

⇓

1

7

0

5

⇓

T* 132

164

)

6

132

0

Only single red bits separated from each other by at least 2 blue bits:

5

233 T† 132

Return map

ρ1[132] : φn −1

3

233

1

6 0

⇓

T* 104

xi = 1 ⇒ xi −1 = 0 and xi +1 = 0

7

3

104

0

0

)

⇓

001100110000110000011001100000011

2

(

T† 104

Example :

132


⇓

218

)

1 3

4

6

7

Table 10A.

Firing Patterns

200 3 6 7

(Continued )



Attractor : Λ( 200 ) At least 2 adjacent red bits separated Time-1 from each other by one or more blue Return map bits: xi = 1 ⇒ xi −1 = 0 , xi +1 = 1, and xi + 2 = 1, ρ [200] : φn −1 φn or xi −1 = 0 , xi +1 = 1, and xi + 2 = 0, 1 or xi −1 = 1 , xi +1 = 1, and xi + 2 = 0,

1

0

or xi −1 = 1 , xi +1 = 1, and xi + 2 = 1 Example :

204 776

2

xi ∈ {0, 1} , i = 1, 2, ··· , I

(

⇓

(

(

T 200

1

2

3

236

5

6

2

3

5

6

2

3

(

)

236

(

(

7

232 3 5 6 7

(

T* 204

xi = 1 ⇒ xi −1 = 0 , xi +1 = 1, and xi + 2 = 1, or xi −1 = 0 , xi +1 = 1, and xi + 2 = 0, or xi −1 = 1 , xi +1 = 1, and xi + 2 = 0, or xi −1 = 1 , xi +1 = 1, and xi + 2 = 1

Example : 001100011110011100001111100010011

( 204 )

(

Attractor : Λ 232

0

0

⇓

(

1

T 232

2

(

( 232 )

⇓

216

7

) 3 5

6

7

) 3

232 T* 232

3 6

⇓

1

7

)

232

φn

3 6

204 T† 232

Time-1 Return map

ρ1[232] : φn −1

2

⇓

(

)

7

)

204

010001100100111100001111010010101

At least 2 adjacent red bits separated from each other by at least 2 blue bits:

6

⇓

1

7

)

⇓

T 204

7

)

204

0

Example :

⇓

T† 204

φn

7

)

⇓

3 6

6

T* 200

Time-1 Return map

ρ1[204] : φn −1

3

200

( 200 ) Attractor : Λ 204

)

T† 200

011011110011101100011110011100111

Any bit string:


5

6

7

3

4

6

7

)

Table 10B.

N = T∆

Table of transient regimes for 12 time-reversible and invertible nonhomogeneous period-1 rules.

4

Firing Patterns

2

36

Firing Patterns

2

72

Firing Patterns

3


T∆ ( 4 ) = 1

5


T∆ ( 36 ) = 2

6


T∆ ( 72 ) = 2

n=0 n=1 n=2

N = n=0

T∆ n=1 n=2

777

n=3

N = n=0

T∆ n=1 n=2 n=3

Remark The transient period T defect depends on the length of the interval occupied by “defect” pixels in the initial bit string which move towards each other until they are annihilated upon collision. In general, 1 < Tdefect ≤ 1 ( I + 1) where the equality sign is attained only if the defects 2 included the boundary pixels at i = 0 and i = I.

Table 10B.

N = T∆

(Continued )

76

Firing Patterns

2

3

6

77

Firing Patterns

0

2

3

6

94

Firing Patterns

1

2

3

4


T∆ ( 76 ) = 1

n=0 n=1 n=2

N =

 C (or C4 ) − 1  Transient T∆ ( 77 ) = int  3  Period T∆   2

n=0

T∆

n=1 n=2

778

n=3 n=4

N = n=0 n=1

T∆

n=2 n=3 n=4 n=5 n=6

6

Transient T ( 94 ) = Tdefect Period T∆ ∆

Table 10B.

N = 104

Firing Patterns

3

5

Firing Patterns

2

7

6

(Continued )

 C + 1 Transient T∆ ( 104 ) = int  1  Period T∆  2 

n=0 n=1 n=2

T∆

n=3 n=4 n=5 n=6

779

n=7 n=8

N = n=0 n=1

T∆

n=2 n=3 n=4 n=5

132

 C3  Transient T ( 132 ) = int   Period T∆ ∆  2

Table 10B.

N = 164

(Continued )

Firing Patterns

2

5

7


Firing Patterns

3

6

7


 C3    2

T∆ ( 164 ) = int 

n=0 n=1 n=2

T∆ n=3 n=4

780

n=5 n=6 n=7

N = T∆

n=0 n=1 n=2

200

T∆ ( 200 ) = 1

Table 10B.

N = 204

(Continued )

Firing Patterns

2

3

6

7


T∆ ( 204 ) = 0

Firing Patterns

3

5

6

7


C  T∆ ( 232 ) = int  1   2 

n=0 n=1

N =

232

n=0 n=1

T∆

n=2

781

n=3 n=4 n=5 n=6

C1 = number of bits in the longest substring consisting of one red bit separated by exactly one blue bit. Example : C1 C3 = largest number of consecutive and adjacent red bits, i.e, no intervening blue bits. Example : C3 C4 = largest number of consecutive and adjacent blue bits, i.e., no intervening red bits. Example : C4 int (x)


Table 11A.

Firing Patterns

12

Table of bit-string laws for six time-reversible and noninvertible nonhomogeneous period-1 rules.



Only single red bits separated from Attractor : Λ ( 12 ) Time-1 each other by one or more blue bits:

( )

T† 12 ⇓

ρ1[12] : φn−1

2

68

Return map

xi = 1 ⇒ xi −1 = 0 and xi +1 = 0

2


6

( )

φn

T 12 ⇓

3

0 2

207

1

0

Example :

13 782

0

Only single red bits separated from each other by one or two blue bits:

xi = 1 ⇒ xi −1 = 0 , xi +1 = 0 , and xi + 2 = 1 , or xi −1 = 0 ,

( )

2

xi +1 = 0 , and xi + 2 = 0

3

( 12 )

( ) φn

2

xi = 1 ⇒ xi −2 = 0 , xi −1 = 0, xi +1 = 0 , and xi + 2 = 0.

3 5

Example : 001000100100001000100001

( )

0 2

0 2

93

3

4

6

T† 44

Time-1 Return map

⇓

2

100

5

( )

T 44

φn 0

3

( )

( )

Attractor : Λ 44

0

⇓

1

6

( )

ρ1[44] : φn −1

7

6

T* 13

( 13 )

6 0

79

1

4

69 ⇓

0

3

2

T 13

010100101001001010010101

44

221 ⇓

Time-1 Return map

0

2

T† 13

Example : Only single red bits separated from each other by at least 2 blue bits:

⇓

7

0

( )

Attractor : Λ 13

ρ1[13] : φn −1

6

T* 12

010100101000100100000101

1

3

1

0

⇓

3

203

6

1

7

( )

0

⇓

3

4

217

6

7

T* 44

( 44 )

6

Table 11A.

Firing Patterns

78 2




Only single red bits or adjacent Attractor : Λ ( 78 ) Time-1 pairs of red bits separated from Return map each other by only 1 blue bit:

xi = 1 ⇒ xi −1 = 0 or xi +1 = 0

3

(Continued )

ρ1[78] : φn−1

( )

T† 78 ⇓

φn

0

1

783

2

( 78 )

ρ1[140] : φn −1

3

xi +1 = 0 , and xi + 2 = 0.

5

(

φn

T 140

(

T* 140

( 140 )

0

(

φn 0

T 172

1

6

7

2

3

4

6

7

6

7

) 2 5

)

1

⇓

3

202

6

7

3

4

6

7

(

T* 172

( 172 )

3

)

228

Return map

ρ1[172] : φn −1

⇓

7 1

2

220

(

6

)

⇓

T† 172

7

2

⇓

1

6

)

206

Example : 001000100100001000100001

197

196

Only single red bits separated from Attractor : Λ ( 172 ) each other by at least 2 blue bits: Time-1

xi = 1 ⇒ xi −2 = 0 , xi −1 = 0,

2

⇓

010100010100100001001001

3

0

⇓

(

4

7

( )

T† 140

Example :

2

2

T 78

0

172

0

*

Return map

3

( )

141

1

Only single red bits separated from Attractor : Λ ( 140 ) each other by one or more blue bit: Time-1

xi = 1 ⇒ xi −1 = 0 and xi +1 = 0

0

6

⇓

010110101011010101011010

3

92 T 78

Example :

140

2

⇓

216

)

Table 11B.

N = T∆

Table of transient regimes for six time-reversible and noninvertible nonhomogeneous period-1 rules.

12

Firing Patterns

2

3

13

Firing Patterns

0

2

44

Firing Patterns

2

( )


T∆ 12 = 1


T∆ 13 = C3 − 1

n=0 n=1 n=2

N =

3

( )

n=0

T∆

n=1 n=2

784

n=3 n=4

N = n=0 n=1

T∆ n=2 n=3 n=4 n=5

3

5


( )

 C2   +1  2

T∆ 44 = int 

Table 11B.

N =

78

Firing Patterns

1

2

3

140

Firing Patterns

2

3

7

(Continued )

6

( )


T∆ 78 = C4 − 1


T∆ 140 = C3 − 1

n=0 n=1 n=2

T∆

n=3 n=4 n=5 n=6

785

n=7

N = n=0 n=1

T∆

n=2 n=3 n=4 n=5

(

)

Table 11B.

N =

172

Firing Patterns

2

3

5

(Continued )

7


n=0 n=1 n=2

T∆

n=3 n=4 n=5 n=6 n=7

786

n=8

C2 = number of bits in the longest substring consisting of 2 red bits separated by exactly one blue bit. Example : C2 C3 = largest number of consecutive and adjacent red bits, i.e, no intervening blue bits. Example : C3 C4 = largest number of consecutive and adjacent blue bits, i.e., no intervening red bits. Example : C4 C5 = number of bits in the longest substring consisting of consecutive red bits separated by exactly one blue bit. Example : C5 int (x)


(

)

T∆ 172 = C5 − 1

Table 12.

Gallery of time-1 characteristic functions χ1

N

of 18 equivalent classes of local rules exhibiting robust non-homogeneous period-1 patterns.

787

Table 12.

(Continued )

788

Table 12.

(Continued )

789

Table 12.

(Continued )

790

Table 12.

(Continued )

791

792


Each intersection of the characteristic function χ1N with the main diagonal is a period-1 point of N . Note that only a subset of all period-1 points of N are plotted in Table 12 due to lack of resolution.21 An examination of Table 12 and the bit-string laws in Tables 10 and 11 for allowable period-1 bit strings shows that each rule can support an infinite number of period-1 fixed points as I → ∞. Some of these fixed points may be isolated in the sense that there are no other nearby fixed points, as demonstrated by the four fixed points of 12 exhibited in Table 13. On the other hand, there are also fixed points which are dense in the sense that there are other fixed points located inside an arbitrarily small neighborhood of themselves, as demonstrated by the four fixed points of 12 exhibited in Table 14. A careful analysis of the time-1 characteristic functions χ1N in Table 2 and their strati-

fications in Table 5 of [Chua et al., 2005b] of the 69 period-1 rules listed in Tables 3 and 4 of [Chua et al., 2005a] shows that isolated period-1 fixed points and dense period-1 fixed points of N are located in disjoint subintervals of φ ∈ [0, 1]. These subintervals are identified in Table 15 for isolated period-1 points, and in Table 16 for dense period-1 points.

6.2. Laws governing period-2 bit strings There are 17 time-reversible period-2 rules. They are all invertible and bilateral.22 The laws governing the ten time-reversible and invertible period2 rules 1 , 5 , 19 , 23 , 33 , 37 , 50 , 51 , 108 , and 178 are given in Table 17A. The associated “table of transient regimes” is given in Table 17B. There are eight time-irreversible noninvertible period-2 rules. The laws governing the three independent period-2 rules 28 , 29 , and 156 are given in Table 18A. The associated “table of transient regimes” is given in Table 18B. In addition to the four globally equivalent time2 characteristic functions {χ228 , χ270 , χ2199 , χ2157 }, the remaining 12 independent time-2 characteristic functions χ21 , χ25 , χ219 , χ223 , χ229 , χ233 , χ237 , 21

χ250 , χ251 , χ2108 , χ2156 , and χ2178 are given in Table 19. The subintervals where isolated period-2 points may reside are shown in Table 20. The subintervals where dense period-2 points may reside are shown in Table 21.

6.3. Laws governing period-3 bit strings There is only one globally equivalent class of robust period-3 rules, namely, 62 , 118 , 131 , and 145 . The table of bit-string laws for rule 62 is already given in Table 1A along with the “table of transient regimes” of a robust period-3 attractor Λ3 ( N ) in Table 1B. The four time-3 characteristic functions χ362 , χ3118 , χ3131 , and χ3145 are given in Table 22. The intervals where isolated and dense period-3 points may reside are given in Table 23.

6.4. Laws governing Bernoulli στ -shift bit strings There are all together 112 rules whose attractors are characterized by Bernoulli στ -shift maps [Chua et al., 2005a]. Among them, 84 rules are invertible and hence time-reversible in view of the Corollary to Theorem 1.23 The remaining 28 rules are noninvertible, and hence time-irreversible. All 112 rules are nonbilateral, a characteristic expected from the “shifting” dynamics of all Bernoulli στ -shift attractors.

6.4.1. Shift left or shift right by one bit Among the 84 time-reversible Bernoulli στ -shift rules, there are 11 equivalence classes which have only one robust Bernoulli στ -shift attractors with β > 0, σ = 1 and τ = 1; namely, 2 , 10 , 24 , 34 , 42 , 46 , 130 , 138 , 152 , 162 , and 170 . The laws governing these 11 time-reversible Bernoulli στ -shift rules are presented in Table 24A. Also listed is rule 240 = T † ( 170 ). The associated “table of transient regimes” is given in Table 24B.

The decimal and binary coordinates of the 201 blue and red bars used for plotting the time-1 characteristic functions in Table 2 of [Chua et al., 2005b] are listed in Table A-1 as an Appendix. 22 These 17 time-reversible and invertible rules are listed in Table 7 of [Chua et al., 2005a]. 23 These 84 time-reversible and invertible rules are listed in Table 10 of [Chua et al., 2005a].

Table 13.

Infinitesimal perturbation of isolated period-1 fixed points of 12 .

793

Table 14.

Infinitesimal perturbation of dense period-1 fixed points of 12 .

794

795


Locations where isolated period-1 points of N may reside.

N

Interval

T †[ N ]

4

[ 0.5 , 2/3 ]

4

12

[ 0.5 , 2/3 ]

68

13

(0

, 0. 25 )

69

Interval

T [N ]

Interval

T *[ N ]

Interval

[ 0.5 , 2/3] 223 [ 1/3 , 0.5 ] 223 [ 1/3 , 0.5 ] 207 [ 1/3 , 0.5 ] 221

no

(0

, 0. 25)

79

[ 0.5 , 0.75 )

( 0.25 , 0.5 )

93

no

( 0.75 ,

1)

( 0.75 , 1 ) [0.25 , 0.75 ] 219 [0.25 , 0.75 ] 219 [0.25 , 0.75 ]

36

[ 0.25 , 0.75 ]

44

[ 0.25 , 0.75 ] 100 [0.25 , 0.375] 203 [0.375 , 0.75] 217 [0.625 , 0.75]

72

[ 0.5 ,

1)

72

[ 0.5 ,

76

[0.75 , 0.875)

76

[0.75 , 0.875) 205 (0.125 , 0.25] 205 (0.125 , 0.25]

36

237

[0.125 , 0.25]

[0.125 , 0.25] 77

1)

(0 ,

0.5 ]

0.5 ]

[0.125 , 0.25]

[0.125 , 0.25] 77

77

237 ( 0 ,

77

[0.75 , 0.875)

[0.75 , 0.875)

[0.75 , 0.875)

[0.75 , 0.875)

[ 0.25 , 0.5 )

[ 0 , 0.25 )

(0.125 , 0.25)

(0.125 , 0.25)

78

141

92 [ 0.75 , 1 )

[ 0.75 , 1 )

104 ( 0. 25 , 1 ) 104

197 ( 0. 5 , 0.75 )

( 0.75 , 1 )

( 0. 25 , 1) 233 ( 0 , 0.75 ) 233 ( 0 , 0.75 )

132 ( 0. 5 , 0.75 ) 132 ( 0. 5 , 0.75 ) 222 ( 0.25 , 0.5 ) 222 ( 0.25 , 0.5 ) 140 [ 0. 5 , 0.75 ) 196 [ 0.75 , 1 ] 206 ( 0.25 , 0.5 ] 220 [ 0 , 0.25 ) 164 [ 0.25 , 0.75 ) 164 [ 0.25 , 0.75 ) 218 ( 0.25 , 0.75 ) 218 ( 0.25 , 0.75 ) 172

( 0.25, 0.75 )

228

( 0.25 , 0.5 ) ( 0.75 ,

202

( 0.25 , 0.75 )

216

1)

( 0 , 0.25 ) ( 0.5 , 0.75 )

200 [ 0. 5 , 0.75 ] 200 [ 0. 5 , 0.75 ] 236 [ 0.25 , 0. 5 ] 236 [ 0.25 , 0. 5 ] 204

no

204

no

204

no

204

no

232 (0.375, 0.625) 232 (0.375, 0.625) 232 (0.375, 0.625) 232 (0.375, 0.625)

796


N

Interval

Locations where dense period-1 points of N may reside.

T †[ N ]

Interval

T [N ]

Interval

T *[ N ]

Interval

4

[0

, 1/3 ]

4

[0 ,

1/3 ]

223 [ 2/3 ,

1]

223 [ 2/3 ,

1]

12

[0

, 1/3 ]

68

[0 ,

2/3 ]

207 [ 2/3 ,

1]

221 [ 1/3 ,

1]

13

( 0.25 , 0. 5 )

69

( 0.25 , 0.75 )

36

[0

, 0.25 )

36

[0

, 0.25 )

[0

, 0.25 )

[0

, 0.25 )

44

100

79

( 0.5 , 0.75 )

219 ( 0.75 ,

1]

( 0.75 ,

1]

203

93

219 ( 0.75 , 217

( 0.5 , 0.75 ) 237

( 0.25 , 0.75 ) 1]

( 0.25 , 0. 5 ) ( 0.75 ,

1]

( 0.5 ,

1]

237 ( 0.5 ,

1]

205 ( 0.25 ,

1]

205 ( 0.25 ,

1]

72

[0

,

0.5 )

72

[0

,

76

[0

, 0.75 )

76

[0

, 0.75 )

77

( 0.25 , 0.75 )

77

( 0.25 , 0.75 )

78

( 0.5 , 0.75 )

92

( 0.25 , 0.75 ) 141 ( 0.25 , 0.5 ) 197 ( 0.25 , 0.75 )

0.5 )

77

( 0.25 , 0.75 )

77

( 0.25 , 0.75 )

104 [ 0

, 0.25 )

104 [ 0

, 0.25 )

233 ( 0.75 ,

1]

233 ( 0.75 ,

1]

132 [ 0

,

0.5 )

132 [ 0

,

0.5 )

222 ( 0.5 ,

1]

222 ( 0.5 ,

1]

140 [ 0 , 0.375 ) 196 [ 0 , 0.675 ) 206 ( 0.625 , 1 ] 220 ( 0.325 , 1 ] 164 [ 0

, 0.25 )

[0

, 0.25 )

172

164 [ 0 , 0.25 ) 218 ( 0.75 , 228

[0 , 0.1875 )

202

( 0.75 ,

1] 1]

218 ( 0.75 , 216

( 0. 5 , 0.625 ) [0

[0

0.5 )

,

200

0.5 )

,

200 ( 0.75 ,

204 [ 0 [0

1]

( 0.75 ,

1]

204 [ 0

, 0.25 )

[0

,

232 ( 0.75 ,

1]

236

1]

[ 0 , 0.25 ) ( 0.5 ,

204 [ 0

, 0.25 )

[0

( 0.75 ,

1]

236

1]

[ 0 , 0.25 ) ( 0.5 ,

1]

204 [ 0

, 0.25 )

[0

,

232

232

( 0.375 , 0. 5 ) ( 0.8125 , 1 ]

1]

,

1]

,

1] 1]

, 0.25 )

232 ( 0.75 ,

1]

( 0.75 ,

1]

Table 17A.

Firing Patterns

1 0

Table of bit-string laws for 17 time-reversible and invertible period-2 rules.

Law Governing Generic Attractor Bit Strings

Red bits separated from each other by three or more blue bits:

xi = 1 ⇒ xi −3 = 0 , xi −2 = 0 and xi −1 = 0



( )

( )

Attractor : Λ 1

T† 1

Time-1 Return map

⇓

ρ1[1] : φn −1

Example :

1

( )

φn

0

0

T 1

1

0

000011110001100001000010001100011

5

( )

797

2

19 0 1 4

000011110011001100011110011100111

6

( )

1

1

0

0

1

⇓

3

4

127

5

6 0

2

( )

T 5

0

1

1

⇓

2

95

( )

⇓

0

1

3

4

6

T* 5

1

(5)

0

0 2

5

2

95

0

1

3

4

6

( )

Attractor : Λ ( 19 ) At least 2 adjacent red bits separated from each other by two or more blue Time-1 Return map bits: xi = 1 ⇒ xi −2 = 0 , xi −1 = 0 , and xi +1 = 1, ρ1[19] : φn−1 φn

or xi −3 = 0 , xi −2 = 0 , and xi −1 = 1 Example :

5

( )

Example : 011011110011111101011110111101011

127

⇓

xi −3 = 0, xi −2 = 1, xi −1 = 1, and xi = 1 ⇒ xi +1 = 1 ρ1[5] : φn −1 φn 0 0 0 xi −3 = 1, xi −2 = 0, xi −1 = 0, and xi = 0 ⇒ xi +1 = 0

0

4

3

T† 5

Time-1 Return map

1

2

T 1

Attractor : Λ 5

0

⇓

*

(1) Any bit string except those containing one or more substrings made of 3 adjacent red bits and blue bits:

0

T† 19

0

⇓

1 4

19

( )

T 19

1

1

0

⇓

2

55

5

( )

T* 19

( 19 )

0

⇓

2

55

5

1 4

1 4

Table 17A.

Firing Patterns

23 0


2



Attractor : Λ( 23 ) At least 2 adjacent red bits separated Time-1 from each other by two or more blue Return map bits: xi = 1 ⇒ xi −2 = 0 , xi −1 = 0 , and xi +1 = 1, ρ1[23] : φn−1 φn

or xi −3 = 0 , xi −2 = 0 , and xi −1 = 1

1

(Continued )

0

0

( )

T† 23 ⇓

( )

T 23 ⇓

1

33 798

xi = 1 ⇒ xi −3 = 0 , xi −2 = 0 and xi −1 = 0

0 5

0 2 5

( )

T

000011000011111100000111000001111

0 2

( 33 )

33

1 4

0

5

( )

T 33

φn

⇓

0

4

⇓

Time-1 Return map

0

1

23 †

Attractor : Λ 33

0

123

1

*

5

( )

T 33

( 33 )

123

5

T† 37

Time-1 Return map

2

37

5

( )

1

1

1

3

4

6 0

1

3

4

6

0

1

⇓

3

4

91

6

( )

0

1

⇓

3

4

91

6

T* 37

( 37 )

0

0

⇓

T 37

φn 0

⇓

( )

( )

Attractor : Λ 37

xi = 1 ⇒ xi −4 = 0 , xi −3 = 0, xi − 2 = 0 , xi −1 = 0 ρ [37] : φ 1 n −1 , or xi = 1 ⇒ xi −5 = 0 , xi − 4 = 0 , xi −3 = 0, xi − 2 = 0 0 0 0 , xi −1 = 1

Example :

⇓

( 23 )

000011110001100001000010001100011

37

2

( )

Example :

Two or more red bits separated from each other by at least 4 adjacent blue bits:

0

T* 23

ρ1[33] : φn −1

4

23

1

000011110011001100011110011100111

Any number of adjacent red bits separated from each other by three or more blue bits:

2

1

23

Example :

4

0

Table 17A.

(Continued )

Firing Patterns


Attractor :

50

One or two adjacent red bits separated from each other by one or two blue bits:


ρ1[50] : φn−1

1 4 5


Λ( 50 )

799 0

xi ∈ {0, 1} , i = 1, 2,··· , I

1

⇓

4

50 T 50

1

0

5

0

5

( )

0

⇓

51

1 *

5

( )

0

⇓

51

1 4

T 51

( 51 )

1 4

T 51

φn

Example : 011011110011101100011110011100111

7

⇓

51

1 4

5

T† 51

Time-1 Return map

0

0

( )

( )

Attractor : Λ 51

7

⇓

1 4

5

T* 50

179

1 4

( )

1

( 50 )

ρ1[51] : φn −1

0

⇓

179

0

5

( )

φn

011011001101101010100110110100101

51

( )

T† 50

Time-1 Return map

Example :

Any bit string:


1 4

5

Table 17A.

Firing Patterns

108 2


One, two, or three adjacent red bits separated from each other by one or more blue bits, subject to following restrictions: xi −5 = 0, xi −4 = 0, xi −3 = 1, xi − 2 = 1 and xi −1 = 1 ⇒ xi = 0 and xi +1 = 0 , or xi −4 = 0, xi −3 = 0, xi −2 = 1 and xi −1 = 1 ⇒ xi = 0 and xi +1 = 0 , or xi −5 = 1, xi −4 = 1, xi −3 = 1, xi − 2 = 0 and xi −1 = 1 ⇒ xi = 0 and xi +1 = 0 , or xi −5 = 1, xi −4 = 0, xi −3 = 1, xi − 2 = 1 and xi −1 = 1 ⇒ xi = 0 and xi +1 = 0 , or xi −5 = 0, xi −4 = 0, xi −3 = 1, xi − 2 = 0 and xi −1 = 1 ⇒ xi = 0 and xi +1 = 0 , or

3 5 6

(Continued )


(

Attractor : Λ 108

Time-1 Return map

ρ1[108] : φn −1 1

800

0

(

1

0

1

0

0

1

4

1

5 7

xi = 1 ⇒ xi −1 = 0 and xi +1 = 0 , or

ρ1[178] : φn −1

011011010011010010011010010100101

0 3 6

)

⇓

7

0 3

201

6

7

1

1

Λ( 178 )

Attractor :

Example :

6

( 108 )

2 adjacent red bits followed by one single blue bit or single red bits followed by one or two blue bits:

xi −1 = 0 and xi +1 = 1 , or xi −1 = 1 and xi +1 = 0

1

5

⇓

(

1

3

201 T* 108

1

2

)

T 108

1

1

0

⇓

108

φn 1

)

T† 108

0010100010111000110011100011101

1

(

)

Example:

178


(

T† 178 ⇓

Time-1 Return map

)

4

178

(

φn

T 178

5

)

0

1

1

(

T* 178

( 178 )

⇓

178

7 1

⇓

4

178

0

1

5

)

7 1 4

5

7

Table 17B.

N = T∆

Table of transient regimes for 17 time-reversible and invertible period-2 rules.

1

Firing Patterns

0

5

Firing Patterns

0

2

19

Firing Patterns

0

1

( )


T∆ 1 = 1


T∆ 5 = 1

Transient Period T

T∆ 19 = 2

n=0 n=1 n=2 n=3 n=4

N = T∆

( )

n=0

801

n=1 n=2 n=3 n=4

N = n=0

T∆

n=1 n=2 n=3 n=4 n=5

4

∆

( )

Table 17B.

N =

23

Firing Patterns

0

1

33

Firing Patterns

0

5

2

(Continued )

4

( )

 C1 + 1   2 

( )

 C1    2


T∆ 23 = int 

Transient Period T

T∆ 33 = int 

n=0 n=1

T∆

n=2 n=3 n=4 n=5 n=6 n=7

802

N = n=0

T∆

n=1 n=2 n=3 n=4 n=5 n=6

C1 = number of bits in the longest substring consisting of one red bit separated by exactly one blue bit. Example : C1 int (x) integer portion of x (e.g., int (5/2) = 2).

∆

Table 17B.

N =

37

Firing Patterns

0

2

5

(Continued )


( )

T∆ 37 = Tdefect

n=0 n=1 n=2 n=3 n=4

T∆

n=5 n=6 n=7

803

n=8 n=9 n=10 n=11 n=12 n=13 n=14


Table 17B.

N =

50

Firing Patterns

1

4

5

51

Firing Patterns

0

1

4

(Continued )

( )

T∆ 50 = int 

Transient Period T

T∆ 51 = 1

n=0 n=1

T∆

n=2 n=3 n=4 n=5 n=7

804

n=8

N = T∆

5

∆

n=0 n=1 n=2 n=3 n=4

C3 = largest number of consecutive and adjacent red bits, i.e, no intervening blue bits. Example : C3

 C3 + 1   2 


( )

Table 17B.

(Continued )

(

)

108

Firing Patterns

2

3

5

6


N = 178

Firing Patterns

1

4

5

7

C3 (or C4 ) −1 Transient = T 17 8 int   ∆ Period T   2 ∆

N =

T∆ 108 = 2

n=0

T∆

n=1 n=2 n=3 n=4 n=5

805

n=0

T∆

n=1 n=2 n=3 n=4 n=5 n=6

C4 = largest number of consecutive and adjacent blue bits, i.e., no intervening red bits. Example : C4

(

)

Table 18A.

Firing Patterns

28

Table of bit-string laws for eight time-irreversible and noninvertible period-2 rules.

Law Governing Attractor Bit-Strings

At most 2 adjacent red bits separated from each other by one or two blue bits:

2

xi −1 = 1 and xi = 1 ⇒ xi +1 = 0 and xi + 2 = 1

3

xi −1 = 0 and xi = 0 ⇒ xi+1 = 1

4

Example :



( )

( )

Attractor : Λ 28

⇓

Time-1 Return map

ρ1[28] : φn −1

φn

6

( )

0

1

6

7

0

1

0

1

( )

( )

⇓

⇓

806

( )

T 29

1

0

( 29 ) Attractor : Λ 156

2

xi −1 = 1 and xi = 1 ⇒ xi+1 = 0 and xi + 2 = 1

ρ1[156] : φn −1

3

xi −1 = 0 and xi = 0 ⇒ xi+1 = 1 Example : 011011010010101101011010010100101

0

1

6

( )

0 2

⇓

T† 156 ⇓

)

2

(

T 156

6

) 2

⇓

1

(

T* 156

( 156 )

⇓

156

7 1

198

1

4

1

198

φn

3

29

(

)

Time-1 Return map

0

4

71 T* 29

Example :

0

3

2

⇓

At most 2 adjacent red bits separated from each other by one or two blue bits:

7

0 2

71

011011110011101100011110011100111

4

4 7

T† 29

φn

(

3

157

2

156

2

( )

Attractor : Λ 29

3

0

T* 28

( )

ρ1[29] : φn −1

4

2

199

Time-1 Return map

0

70 ⇓

28

29

2

T 28

011010010101101010100110110100101

Any bit string:

1

T† 28

)

2

6

7

3

4 7

Table 18B.

N =

Table of transient regimes for eight time-irreversible and noninvertible period-2 rules.

28

Firing Patterns

2

3

4

29

Firing Patterns

0

2

3

C  Transient T∆ 28 = 2 × int  3  − 1 Period T∆  2

( )

n=0 n=1

T∆

n=2 n=3 n=4 n=5 n=6

807

n=7 n=8

N = T∆

n=0 n=1 n=2 n=3 n=4

4


( )

T∆ 29 = 1

Table 18B.

N = 156

Firing Patterns

2

3

4

(Continued )

7

n=0 n=1 n=2 n=3

T∆

n=4 n=5 n=6

808

n=7 n=8 n=9 n=10

C3 = largest number of consecutive and adjacent red bits, i.e, no intervening blue bits. Example : C3 C4 = largest number of consecutive and adjacent blue bits, i.e, no intervening red bits. Example : C4 int (x)


(

)

Transient T 156 = C3 ( or C4 ) − 1 Period T∆ ∆

Table 19.

Gallery of time-2 characteristic functions χ2N of 13 equivalence classes of local rules exhibiting robust period-2 patterns. In addition, we have included χ2

N

for rules 70 , 199 and 157 , which are equivalent to χ228 , for future reference.

χ2

1

χ2

1

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

809

0 19

1

0.9

0

χ2

5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

1

χ2

1

23

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

19

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

5

φ

23

1

0.9

0

0

0

Table 19.

χ2

28

χ2

1

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

810

0 199

70

0.9

0

χ2

(Continued )

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

28

χ2

1

157

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

199

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

70

φ

157

1

0.9

0

0

0

Table 19.

χ2

29

χ2

1

1 0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

811

0 37

33

0.9

0

χ2

(Continued )

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

29

χ2

1

50

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

37

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

33

φ

50

1

0.9

0

0

0

Table 19.

χ2

51

χ2

1

108

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

812

0 156

1

0.9

0

χ2

(Continued )

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

0

51

χ2

1

178

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

156

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

108

φ

178

1

0.9

0

0

0


Locations where isolated period-2 points of N may reside.

N

Interval

T †[ N ]

1

[ 0.125 , 0.5 )

1

5

813

Interval

T [N ]

Interval

(0.0625 , 0.125)

(0.25 ,

(0.1875

(0.1875

(0.375 , 0.4375)

, 0.25)

0.3125)

95

5 (0.5625 , 0.625)

(0.75 ,

(0.6875

(0.6875

(0.875 , 0.9375)

, 0.75)

(0.25 ,

0.3125)

(0.375 , 0.4375) 95

(0.5625 , 0.625) , 0.75)

Interval

[ 0.125 , 0.5 ) 127 ( 0.5 , 0.875 ) 127 ( 0.5 , 0.875 )

(0.0625 , 0.125) , 0.25)

T *[ N ]

0.8125)

(0.75 ,

0.8125)

(0.875 , 0.9375)

19

[0.375 , 0.5]

19

[0.375 , 0.5]

55

[0.5 , 0.625]

55

[0.5 , 0.625]

23

(0.375

23

(0.375

23

(0.375

23

(0.375

28

, 0.625)

( 0 , 0.125 )

70

, 0.625)

(0.125 , 0.1875)

199

(0.375 , 0.4375)

(0.8125 , 0.875)

, 0.625)

(0.125 , 0.1875)

157

(0.5625 , 0.625) (0.8125 , 0.875)

( 0.875 , 1 )

29

( 0.5 , 0.625 )

71

(0.375 , 0.5)

29

(0.5 , 0.625)

33

( 0.125 , 0.5 )

33

( 0.125 , 0.5 ) 123 (0. 5 , 0.875) 123

(0. 5 , 0.875)

37

[ 0 , 0.5 ]

37

(0.125 , 0. 25)

(0.125 , 0. 25) 50

50 (0.75 , 0.875)

51 108

[ 0 , 0.5 ]

none (0.3125 , 0.4375)

71

, 0.625)

91 179

(0.75 , 0.875) 51 108

none (0.3125 , 0.4375)

(0.375 , 0.5)

[ 0.5 , 1 ] (0.125 , 0. 25)

91 179

(0.75 , 0.875) 51 201

none (0.0625 , 0.1875)

[ 0.5 , 1 ] (0.125 , 0. 25) (0.75 , 0.875)

51 201

none (0.0625 , 0.1875)

(0.8125 , 0.9375)

(0.8125 , 0.9375)

(0.5625 , 0.6875)

(0.5625 , 0.6875)

( 0 , 0.125 )

(0.125 , 0.25)

(0.125 , 0.25)

( 0 , 0.125 )

156 (0.5 , 0.625) 198 (0.375 , 0. 5) 198 (0.375 , 0. 5) 156 (0.5 , 0.625) (0.875 , 1)

(0.75 , 0.875) (0.125 , 0.25) 178

(0.75 , 0.875)

(0.125 , 0.25) 178

(0.125 , 0.25) 178

(0.75 , 0.875)

(0.75 , 0.875)

(0.875 , 1)

(0.125 , 0.25) 178

(0.75 , 0.875)

(0.75 , 0.875)

814


N

1

5

19 23

Interval [ 0 , 0.125 ) [ 0.5 , 1 )

Locations where dense period-2 points of N may reside.

T †[ N ]

1

[ 0 , 0.125 ) [ 0.5 , 1 )

T [N ]

127

Interval ( 0 , 0.5 ] [0.875 , 1 ]

T *[ N ]

127

[0.875 , 1 ]

[ 0 , 0. 25 )

[ 0 , 0. 25 )

(0.125 , 0.1875)

(0.125 , 0.1875)

(0.3125 , 0.375)

(0.3125 , 0.375)

(0.25 ,

0.5625)

5

(0.25 ,

0.5625)

95

(0.4375 ,

0.75)

95

(0.4375 ,

0.75)

(0.625 , 0.6875)

(0.625 , 0.6875)

(0.8125 ,0. 875)

(0.8125 ,0. 875)

( 0. 75

( 0. 75

( 0. 9375

( 0. 9375

, 1]

[ 0 , 0.25 ] [ 0. 5 , 1 ] [ 0 , 0.25 ] [ 0.75 , 1 ]

19 23

(0.625 , 0.75)

70

( 0 , 0. 5 ) ( 0.625 , 1 )

71

[ 0 , 0.125 ] 33

( 0 , 0.5 ]

[ 0 , 0.0625 )

, 1]

[ 0 , 0.25 ] [ 0. 5 , 1 ] [ 0 , 0.25 ] [ 0.75 , 1 ]

55 23

, 1]

[ 0 , 0. 5 ] [ 0.75 , 1 ] [ 0 , 0.25 ] [ 0.75 , 1 ]

55 23

(0.5

, 0.5625)

[ 0.75 , 1 ]

[ 0 , 0. 5 ] [ 0.75 , 1 ] [ 0 , 0.25 ] [ 0.75 , 1 ]

(0.25 , 0.375)

(0.125 , 0.25)

( 0.5 , 0.75 ) 199

(0. 5 , 0.875)

157 ( 0.25 , 0. 5 )

( 0 , 0.375 ) ( 0.5 ,

1)

(0.625 , 0.75) 71

[ 0.75 , 1 ] 33

, 1]

(0.25 , 0.375) (0.75 , 0.875)

29

Interval

[ 0 , 0.0625 )

( 0.125 , 0.5 ) 28

Interval

( 0 , 0.375 ) ( 0.5 ,

1)

29

[ 0.75 , 1 ]

( 0 , 0. 5 ) ( 0.625 , 1 ) [ 0.75 , 1 ]

123 (0.4375 , 0.5) 123 (0.4375 , 0.5) [ 0.75 , 1 ] [ 0.875 , 1 ] [ 0.875 , 1 ]

(0. 5

, 0.5625)

91

91

37

none

37

50

(0.25 , 0.75)

50

(0.25 , 0.75) 179 (0.25 , 0.75) 179 (0.25 , 0.75)

51

[0

51

[0

108 156

,

1]

[0 , 0.3125) (0.4375 , 0.8125) ( 0.125 , 0.5 )

108 198

(0.625 , 0.75) 178

(0.25 , 0. 75)

none ,

1]

[0 , 0.3125) (0.4375 , 0.8125) (0.25 , 0.375)

51 201 198

(0.5 , 0.875) 178

(0.25 , 0. 75)

none [0

,

1]

(0.1875 , 0.5625) ( 0.6875 , 1 ] (0.25 , 0.375)

51 201 156

(0. 5 , 0.875) 178

(0.25 , 0. 75)

none [0

,

1]

(0.1875 , 0.5625) ( 0.6875 , 1 ] ( 0.125 , 0.5 ) (0.625 , 0.75)

178

(0.25 , 0. 75)

Table 22.

χ3

62

Gallery of time-3 characteristic functions χ3N of four globally-equivalent local rules

χ3

1

118

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

815

0

χ3

131

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

χ3

145

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

131

exhibiting robust period-3 patterns.

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

φ

118

φ

145

1

0.9

0

¯

0

62

1

62 , 118 , 131 , 145

1

0.9

0

˘

0

816


Table 23. Locations where isolated or dense period-3 points may reside.

N

Intervals with

Intervals with

dense period-3

isolated period-3

points

points

plot in this column is calculated from the unfolding formula zn+1 = zn + exp[2πiφn ] where zn is a complex number and ∆

( 0.1875 62

( 0.6875 , 0. 875 ) ( 0.9375

118

131

, 0. 5 )

,

( 0.5 ,

1)

( 0.3125 , 0. 5625 )

( 0.125

, 0.25 )

( 0.6875 , 0. 875 )

( 0.875 ,

(0

, 0. 0625 )

( 0.0625 , 0.125 )

( 0.125 , 0. 3125 )

( 0.3125 , 0.25 )

1)

( 0.125 , 0.3125 )

(0

( 0.4375 , 0. 6875 )

( 0.75

,

0. 125 )

145 , 0.875 )

Observe from Table 24B that after the transient period T∆ ( N ) had expired, the dynamics on the attractor Λ( N ) of all 11 rules listed in Table 24A follows the same simple rule decreed in Theorem 1 (case 1) of [Chua et al., 2004] for β > 0, σ = 1, or σ = −1, and τ = 1; namely, σ = 1 : Shift left by one bit

(28)

σ = −1 : Shift right by one bit

(29)

In other words, each of the 44 Bernoulli στ -shift rules included in the left-most or right-most column of Table 24A follows the algorithm (28) if σ = 1, or (29) if σ = −1.

6.4.2. Unfolding Bernoulli orbits in complex plane Observe that Table 24A contains a new column under the heading “unfolded orbit in complex plane”, henceforth called orbit unfolding plot. Each

I k=0

(n)

and xk

(n)

2−(k+1) xk

(31)

is the kth component of the bit string (n)

( 0. 5 , 0. 8125 )

24

φn =

0.6875 )

( 0.875 , 0. 9375 )

(30)

x(n) = {x0

(n)

x1

(n)

x2

···

(n)

xI }

(32)

at the nth iteration under rule N . All orbit unfolding plots in Table 24A are calculated with an I = 1359 bit string and iterating over 1366 generations. It is instructive to interpret the orbit unfolding plots in Table 24A as the mathematical model of taking a walk in the complex plane via the nonlinear dynamics of N . Observe that the two unfolded plots of 170 and 240 in Table 24A are reminiscent of random walks in probability theory. Indeed, our unfolding formula (30) is nothing more than a modern version of Feller’s algorithm for illustrating an ideal coin toss as a random walk [Feller, 1950]. This interpretation is also consistent with the well-known result that the Bernoulli map φn+1 = 2φn mod 1 is a model of an ideal coin toss.

6.4.3. Shift left or shift right by one bit and followed by complementation There are two other rules which also exhibit a “random walk” orbit unfolding plot, namely, rules 15 and 85 . These two rules are globally-equivalent and time-reversible, with β < 0, σ = −1, and τ = 1 for 15 , and β < 0, σ = 1, and τ = 1 for 85 . The laws governing these two time-reversible Bernoulli στ -shift rules are given in Table 25A.24 The associated “table of transient regimes” is shown in Table 25B. Since β = −(1/2) < 0 in the time-1 map of 15 , and β = −2 < 0 in the time-1 map of 85 , the dynamics during each iteration of 15 and 85 involve two separate steps: (1) shift the previous bit string by one bit to the right for 15 , or to the left for 85 , and (2) complement (i.e. change the color of all bits) the bit string resulting from step 1. Observe that since T∆ ( N ) = 0 for both 15 and 85 , the return maps

Although rules 15 and 85 have only one Bernoulli στ -shift invariant orbit, they are not included in Table 24A because the slopes of their time-1 return maps are negative.

Table 24A. Characterization of all bit-string configurations which are allowed on the attractors of 12 equivalence classes of local rules exhibiting a single (α = 1) Bernoulli στ -shift attractor β > 0, |σ| = 1, and τ = 1. The last rule 240 is equivalent to 170 and is listed for pedagogical reasons.

Firing Patterns

Laws Governing Attractor Bit Strings

2

Only single red bits separated from each other by at least 2 blue bits :

xi = 1 ⇒ xi − 2 = 0 , xi −1 = 0,

1

Attractor Bit-String (N Generator

)

( )

( )

T† 2

Attractor : Λ 2

⇓

Time-1 Return map ρ1[2] : φn −1

xi +1 = 0 , and xi + 2 = 0

( )

φn

T 2

0

0

1

817 1

xi = 1 ⇒ xi −1 = 0 and xi +1 = 0 ,

3

or xi −1 = 0 and xi +1 = 1 ,

4

xi +1 = 0 , and xi + 2 = 0

⇓

247

5

( )

φn

4 6

1

1

0

⇓

2

175

5

( )

7

( ) 10

1

3 7 0

⇓

2

245

5

4 6

( )

T† 24

7 1

⇓

66

6

( )

0

1

6

7

φn

T 24 ⇓

2

0

231

5

1

( )

T* 24

Example : 1001000001000010010001000

1

6

T 10

0

7

4

T 10

*

ρ1[24] : φn −1

4

0 2

80

0

0

3

⇓


1

T† 10

Only single red bits separated Attractor : Λ ( 24 ) from each other by two or Time-1 more blue bits : Return map

xi = 1 ⇒ xi − 2 = 0 , xi −1 = 0,

191

5

0

( )

( )

Attractor : Λ 10

Example : 0010000001100100100011001

3

2

( )

or xi −1 = 1 and xi +1 = 0

24

⇓

T* 2

(2) 10

4

16

Example : 0010000001000100100000001

Only single red bits or adjacent pairs of two red bits separated from each other by at least 2 blue bits :


Unfolded Orbit In Complex Plane

( ) 24

0

⇓

2

189

5

3

4 7

Table 24A.

Firing Patterns


34

Only single red bits separated from each other by one or more blue bits :

1

(Continued )

Attractor Bit-String Generator ( N )

( )

( )

Attractor : Λ 34

T† 34 ⇓


42 818

1

xi = 1 ⇒ xi −1 = 0 and xi +1 = 0 ,

3

( )

φn

T 34

0

187

1

3 5

Example : 0011011001100011000000011

5

6

5

6

( )

T 42

φn

0

⇓

1

171 *

5

7

( )

( )

241 †

T

4

⇓

2

116

5

( )

1

1

4 6 0

( )

T* 46

7 0

⇓

209

1

3

139

( 46 )

7

( 46 )

⇓

0

6

5

T 46

φn

0

⇓

42

1

3

T 42

ρ1[46] : φn −1

7

4

112

Only 2 adjacent red bits Attractor : Λ( 46 ) separated from each other by Time-1 one or more blue bits : Return map

xi = 1 ⇒ xi −1 = 0 , xi +1 = 1 , and xi + 2 = 0 , or xi − 2 = 0 , xi −1 = 1 , and xi +1 = 0

0

⇓

Example : 0011011001000100110000001

2

7

T† 42

or xi −1 = 1 and xi +1 = 0

46

4

4

243

Time-1 Return map

1

3

( )

( )

Attractor : Λ 42

0

1

⇓

( 34 )

ρ1[ 42] : φn −1

0

5

( )

*

T 34

or xi −1 = 0 and xi +1 = 1 ,

5

5

⇓

Example : 0010010001010010000001001 Only single red bits or adjacent pairs of two red bits separated from each other by one or more blue bits :

4

48

xi = 1 ⇒ xi −1 = 0 and xi +1 = 0

5

1



4 6

7

Table 24A.

Firing Patterns

130


7

Attractor Bit String Generator ( N )

ρ1[130] : φn −1

xi +1 = 0 , and xi + 2 = 0

(

)

T† 130 ⇓

4

144

(

φn

T 130

0

0



Only single red bits separated Attractor : Λ ( 130 ) from each other by at least 2 Time-1 blue bits : Return map

xi = 1 ⇒ xi − 2 = 0 , xi −1 = 0,

1

(Continued )

1

7

)

1

⇓

2

190

5

(

138 819

3 7

n −1

1

152 3 4 7

(

T 138

1

( 138 )

Example : 1000100010010001001000100

)

174

5

(

ρ1[152] : φn −1

1

⇓

2

244

5

⇓

(

T 152

0

0

)

7 1

)

7 1

⇓

2

230

5

6

7

2

3

4

(

(

4 6

6

T* 152

152

7

)

194 φn

3

)

T† 152

Time-1 Return map

7 1

2

(

)

7

4 6

T 138

(

6

)

⇓

*

Attractor : Λ 152

4

⇓

xi = 1 ⇒ xi − 2 = 0 , xi −1 = 0 xi +1 = 0 ,and xi + 2 = 0

5

208

Example : 0010001111000100110000111 Only strings with 2 or more consecutive and adjacent blue bits separated from each other by only one red bit :

246 T† 138

n

0

0

1

2

(

Any number of consecutive Attractor : Λ ( 138 ) and adjacent red bits separated Time-1 from each other by two or Return map more blue bits : ρ [138] : φ φ

xi = 1 ⇒ xi − 2 = 0 , xi −1 = 0, and xi +1 = 0, or xi − 2 = 0 , xi −1 = 0, and xi +1 = 1

1

( 130 )

⇓

4 7

)

T* 130

Example : 0010000001000100100000001

3

⇓

188

) 5

7

Table 24A.

Firing Patterns

162


Attractor Bit String Generator ( N )

xi = 1 ⇒ xi −1 = 0 and xi +1 = 0

ρ1[162] : φn −1



(

Only single red bits separated Attractor : Λ ( 162 ) from each other by one or Time-1 Return map more blue bits :

1 5

(Continued )

)

T† 162 ⇓

4

176

(

φn

T 162

5

)

1

⇓

0

7

3

186

1

5

(

170

Any binary string.

( 162 ) (

Attractor : Λ 170

242

4

240

(

T 170

0

1

5

7

( 170 ) (

Attractor : Λ 240

1

240

Any binary string.

(

⇓

4

240

5

⇓

170

(

T 240

φn

7

)

T† 240


4

5

(

)

)

6

6

Example : 0010100011001111100110101

1

1

5

7

)

240 *

(

T 240

( 240 )

4 5

6

)

7 1

⇓

170

7

3

⇓

5

7

3

T 170

Example : 0010100011001111100110101

6

)

170 *

7

)

⇓

5

6

⇓

3

7

5

T† 170

φn

0

820

ρ1[170] : φn −1

1 4

(

)

Time-1 Return map

1

⇓

4 7

)

T* 162

Example : 0010000001000100100000001

7

3 5

7

Table 24B. Dynamic evolution from a generic initial bit string for rule N from Table 24A until it converges to an attractor at n = T∆ , as verified by a “left-shift”, or a “right-shift”, operation (by one pixel) shown in the last row at n = T∆ + 1.

N = T∆

2

Firing Patterns

1

10

Firing Patterns

1

24

Firing Patterns

34

Firing Patterns

( )


T∆ 2 = 1

3


T∆ 10 = 1

3

4


T∆ 24 = 2

1

5


T∆ 34 = 1

n=0 n=1 n=2

N = T∆

( )

n=0 n=1 n=2

821

N = T∆

( )

n=0 n=1 n=2 n=3

N = T∆

n=0 n=1 n=2

( )

Table 24B.

N = T∆

42

Firing Patterns

1

3

5

46

Firing Patterns

1

2

3

130

Firing Patterns

1

7

(Continued )


( )

T∆ 42 = 1

n=0 n=1 n=2

N =

5

( )


T∆ 46 = 2

n=0

T∆

n=1

822

n=2 n=3

N = n=0 n=1

T∆

n=2 n=3 n=4 n=5


(

)

 C3    2 

T∆ 130 = int 

Table 24B.

(

)

Firing Patterns

1

3

7


T∆ 138 = 1

152

Firing Patterns

3

4

7


T∆ 152 = C3 − 1

162

Firing Patterns

1

5

7


C  T∆ 162 = int  3   2 

N = 138 T∆

(Continued )

n=0 n=1 n=2

N =

(

)

n=0

T∆

n=1 n-=2

823

n=3 n=4

N = n=0 n=1

T∆

n=2 n=3 n=4 n=5

(

)

Table 24B.

N = 170

(Continued )

(

)

(

)

Firing Patterns

1

3

5

7

Transient T 170 = 0 Period T∆ ∆

Firing Patterns

4

5

6

7

Transient T 240 = 0 Period T∆ ∆

n=0 n=1

N =

240

824

n=0 n=1

C3 = largest number of consecutive and adjacent red bits, i.e, no intervening blue bits. Example : C3 int (x)


Table 25A.

Firing Patterns

Characterization of all bit-string configurations which are allowed on the invariant orbits of 15 and 85 .

Laws Governing Attractor Bit Strings Any binary string.




( )

Attractor : Λ 15

( )

T† 15 ⇓

15

2

85 0 1

Example : 0010100011001111100110101

2 3

ρ1[15] : φn −1

T 15 φn

2

1

3

15

0 Any binary string.

0

⇓

( )

T* 15

1

⇓

825


4 6

( )

Time-1 Return map

( 15 ) Firing Patterns

0


( )

Attractor : Λ 85

0

2

85

4 6



( )

T† 85 ⇓

85

2

0

1

3

15 0 2 4

Example : 0010100011001111100110101

( )


T 85 ⇓

φn

6

0 2

85

1

6

( )

T* 85

0

⇓

2

( 85 )

4

15

3

1

0

Table 25B.

N =

Dynamic evolution from a generic initial bit string for rules 15 and 85 from Table 25A.

( )

15

Firing Patterns

0

1

2

3


T∆ 15 = 0

85

Firing Patterns

0

2

4

6


T∆ 85 = 0

n=0 n=1 n=2 n=3 n=4 n=5 n=6 n=7

826

N = n=0 n=1 n=2 n=3 n=4 n=5 n=6 n=7

( )


shown in Table 25A are not attractors, but invariant orbits, as in the case of 170 and 240 . Observe the unfolded orbits of 15 and 85 in Table 25A are also reminiscent of random walks.

6.4.4. Shift left or shift right by one bit every two iterations There are eight other rules from Table 10 from [Chua et al., 2005a] which are not included in Table 24A because their time-1 return maps are not Bernoulli. However, their time-2 return maps are Bernoulli στ -shift maps with β > 0, σ = −1 and τ = 2. Since these eight rules are globally equivalent to rules 3 and 7 , only the laws governing these two rules are given in Table 26A. The corresponding “table of transient regime” is shown in Table 26B.

6.4.5. Time-reversible rules with two Bernoulli attractors The remaining 24 rules from Table 10 of [Chua et al., 2005a] have two distinct Bernoulli στ shift attractors. They are members of ten global equivalence classes; namely, 11 , 14 , 35 , 43 , 56 , 57 , 58 , 62 , 142 , and 184 . The laws governing each attractor of these ten rules are given in Table 27A. Observe that since each rule has two attractors, two rows are now devoted to each rule in this table. The associated “table of transient regimes” is shown in Table 27B. Observe that the time-1 return maps of some attractors exhibited in Table 27A have negative slope; namely, β = −(1/2) for attractors Λ1 ( 11 ), Λ2 ( 14 ), Λ1 ( 43 ), and Λ2 ( 142 ). An examination of Table 10 of [Chua et al., 2005a] shows that all together, there are ten Bernoulli στ -shift rules where one of their two attractors have β = −(1/2) or β = −2, and therefore have dynamics identical to those of rules 15 and 85 , respectively. These ten rules are 11 , 81 , 47 , 117 , 14 , 84 , 143 , 213 , 43 , 113 , 142 , and 212 .

6.4.6. Time-irreversible rules with two Bernoulli attractors Not all of the 112 Bernoulli στ -shift rules identified in [Chua et al., 2005a] are time-reversible. Indeed, there are 20 noninvertible time-irreversible 25

827

Bernoulli rules endowed with two robust Bernoulli στ -shift attractors. Each of these rules belongs to a global equivalence class with four distinct rules, a representative of each is listed in Table 28A; namely, rules 6 , 9 , 27 , 38 , and 134 . As in Sec. 6.4.5, each rule in Table 28A has two robust attractors endowed with two time-2 return maps (i.e. τ = 2), except for 9 , which has a time-2 return map and a time-3 return map. The laws governing the attractor bit strings for these attractors are listed in Table 28A, along with their “bit-string generators” and “orbit unfolding” plots. Since β > 0 for all attractors belonging to Table 28A, the dynamics involve only a leftshift if σ > 0, or a right-shift, if σ < 0, of |σ| bits every two iterations (except for the time-3 attractor Λ2 ( 9 ) which shifts left or right by two bits every three iterations). The “table of transient regime” associated with these attractors is exhibited in Table 28B.

6.4.7. Time-irreversible rules with three Bernoulli attractors There are eight time-irreversible noninvertible Bernoulli στ -shift rules endowed with three robust attractors; namely, 25 , 67 , 103 , 61 , 74 , 88 , 173 , and 229 .25 They are members of two global equivalence classes, where rules 25 , 74 , 88 , 173 , and 229 are exhibited in Table 29A. Each is endowed with three robust Bernoulli στ -shift attractors, each involving a different number of return times τ . The “table of transient regime” associated with these attractors is exhibited in Table 29B.

6.4.8. Deriving bit string laws for globally-equivalent rules is trivial Once the laws governing the attractor bit strings of any rule N are given, it is trivial to derive the laws governing the three other globally-equivalent rules T † ( N ), T ( N ), and T ∗ ( N ), in view of the Reflection Invariance Theorem and the complementation Invariance Theorem, respectively, from [Chua et al., 2004]. Applying these theorems to columns 2–4 of Tables 24A, 25A, 26A, 27A, 28A, and 29A, we can construct, painlessly, the corresponding results for the globally-equivalent rules listed in column 5.

The 20 time-irreversible rules are listed in Table 28A and the eight time-irreversible rules in Table 29A are given in Tables 11 and 12 of [Chua et al., 2005a], respectively.

Table 26A. and τ = 2.

Characterization of all bit-string configurations which are allowed on the sole attractor of Bernoulli στ -shift rule 3 and 7 with β > 0, σ = −1

Firing Patterns

3 0

Laws Governing Attractor Bit Strings Any number of consecutive and adjacent red bits separated from each other by at least 2 blue bits : .

1


( )

Attractor : Λ 3

φn


Orbit Unfolding Plot

( )

T† 3 φn

⇓

4

17 φn-1 Time-1 Return map

φn-2 Time-2 Return map

ρ1[3] : φn −1

ρ1[3] : φn − 2

φn

( )

T 3 ⇓

φn

2

63

0 1

7

Only strings with 2 or more red bits separated from each other by at least 3 blue bits : .

0 1 2

(3)

0


( )

φn

5

4

0

1 4

6

( )

T† 7 ⇓

φn

0 2

4

21 φn-1 Time-1 Return map

φn-2 Time-2 Return map

ρ1[7] : φn −1

ρ1[7] : φn −2

φn

φn

( )

T 7 ⇓

2

31

0

⇓

0

1

3

4

6

( )

T* 7

1 0

Example : 00011100001111000000011

3



Attractor : Λ 7

1

2

119

0

828


( )

⇓

0

5

T* 3

Example : 00100001100111110010010

Firing Patterns

0

0 2

1

(7)

87

1 4

6

Table 26B.

N =

Dynamic evolution from a generic initial bit string for rules 3 and 7 from Table 26A.

3

Firing Patterns

0

1

7

Firing Patterns

0

1

( )


T∆ 3 = Tdefect


T∆ 7 = Tdefect

n=0 n=1 n=2 n=3 n=4 n=5 n=6 n=7

829

N =

2

( )

n=0 n=1 n=2 n=3 n=4 n=5 n=6 n=7

Remark The transient period T defect depends on the length of the interval occupied by “defect” pixels in the initial bit string which move towards 1 each other until they are annihilated upon collision. In general, 1 < Tdefect ≤ ( I + 1) where the equality sign is attained only if the defects 2 included the boundary pixels at i = 0 and i = I.

Table 27A. Characterization of all bit-string configurations which are allowed in the two invertible attractors of the ten Bernoulli στ -shift rules 11 , 14 , 35 , 43 , 56 , 57 , 58 , 62 , 142 and 184 .

Firing Patterns

11

Laws Governing Attractor Bit Bit-Strings Strings Only strings with 2 or more red bits separated from each other by at least 2 blue bits :

Attractor Bit-String Attractor Bit String Generator Generator

( N ).


( )

T† 11

Time-1 Return map

Attractor : Λ 1 11

( )

Equivalent Equivalent Rules Rules and and Firing Firing Patterns Patterns ⇓

φn

0

81

4 6

3

Only strings with one or two red bits separated from each other by two blue bits :

0

Example : 00011000111000110000011

1

0

0

1

1 φ n-1

( ) 11

ρ1[11] :φ n - 1

( )

T 11 ⇓

φn

47

Time-1 Return map

Attractor : Λ 2 11

( ) φn

830

Firing Patterns


14

Only strings with 2 red bits separated from each other by at least 2 blue bits :

( 11 )

ρ1[11] :φ n - 1

117

φ n-1 φn

(N )

Time-1 Return map

( )

3 5

( )

Attractor Bit String Generator

Attractor : Λ1 14

1

T* 11 ⇓

Example : 10010010011001001001101

0 2

φn

0 2 4

.



( )

T† 14 ⇓

2

84

4 6

1 2

5

6

Example : 0001100011000011000011

( )

3

Only strings with one or two red bits separated from each other by one or two blue bits : Example : 00110100110010110010011

φ n-1

14

ρ1[14] :φ n - 1

( )

⇓

φn

Time-1 Return map

Attractor : Λ2 14

( )

T 14

φn

143 T

( )

ρ 1[14] : φ n - 1

φ n-1 φn

1 3 7

*

( 14 )

⇓

14

0 2

213

0 2 4 6

7

Table 27A.

Laws Governing Attractor Bit-Strings

35

Only strings with 1 or more red bits separated from each other by at least 2 blue bits :


( )φ

Attractor : Λ 1 35


0

φn

0

1

φn-1

43

Only strings with 2 or more red bits separated from each other by at least 2 blue bits :

( )

Only strings with 1 or 2 red bits separated from each other by 1 or 2 blue bits :

ρ1[35] : φn −1

φn

(N)

5

0

1

4

5

6

.



( )

T† 43 ⇓

0

113

1

4

5

( )

T 43

φn-1 φn

⇓

Time-1 Return map

43

ρ1[43] : φn −1

( )

Attractor : Λ 2 43

1 3 5

T* 43 ⇓

( 43 )

0

( )

φn

1

4

3

59

115

1 Example : 01010100110011001100101

1

6

( 43 )

0

5

φn-1

φn

Example : 00011000110000111110011

0

3

1

0

( )

Time-1 Return map

Attractor : Λ1 43

5

T* 35


0 0

1

φn

1

( )

4

( )

⇓

35

0

0

⇓

φn

0 831


ρ1[35] : φn −2

( )

( )

T 35

Time-1 Return map

Example : 01010100100101010010101 Firing Patterns

⇓

49

φn-2

Attractor : Λ 2 35

Equivalent Rules and Firing Patterns T† 35

φn

( 35 )

0

Only strings of single red bits separated from each other by 1 or 2 blue bits


1

Example : 00010000111100110000011

5

(N).

Time-2 Return map

n

0

Firing Patterns

(Continued )

ρ1[43] : φn −1

φn-1

φn

113

0 4 6

5

Table 27A.

56

Laws Governing Attractor Bit-Strings Bit Strings


Only strings with 1 red bit separated from each other by 1 or 2 blue bits :

( )

⇓

( )

0

98

5 6

φn-1

0

( 56 )

Only strings of 1 or 2 red bits separated from each other by 1 blue bit :

Equivalent EquivalentRules Rulesand and Firing Firing Patterns Patterns T† 56

1

Example : 00101010100100101010101

5


φn

3 4

(N).

Time-1 Return map

Attractor : Λ 1 56

0

Firing Patterns

(Continued )

ρ1[56] : φn −1

( )

T 56 ⇓

φn

227

Time-1 Return map

( )

Attractor : Λ 2 56

0

( )

1

Firing Patterns


57

Only strings with 1 red bit separated from each other by 1 or 2 blue bits :

( ) 56

( )

Example : 01101010110101011001101



( )

T† 57 ⇓

0

99

1 5

( )

T 57 φn

⇓

Time-1 Return map

99

ρ1[57] : φn −1

( )

5

T* 57

1 ρ1[57] : φn −1

1

6

⇓

( 57 )

0

( )

φn

1

5 7

.

φn-1

Attractor : Λ 2 57

0

3 4

6

( )

Only strings with 1 or 2 red bits separated from each other by 1 blue bit :

(N)

φn

57

0

5

0

4

0

φn

1

Example : 01010100100100101010101

185

φn-1

Time-1 Return map

Attractor : Λ 1 57

0 3

ρ1[56] : φn −1


0

832

Example : 01101101101011011010101

7

T* 56 ⇓

1

5 6

φn

0

1

φn-1

φn

57

0 3 4

5

Table 27A.

Firing Patterns

58



Only strings with 1 or 2 red bits separated from each other by 1 blue bit :

( )

1

0

5

Only strings of 2 or 3 red bits separated from each other by 1 or 2 blue bits

ρ 1[6] : φn - 1


1

φ n-1

4 5

( )φ

Time-1 Return map ρ 1[62] : φn - 1

Example : 00111001110011001110111 Only strings with 2 red bits separated from each other by 1 blue bit :

Example : 110110110110110110110

⇓

φn

92

1

( )

φ n-2

ρ 1[58] : φn - 2

177

(N)

4

5 7

.



( )

T† 62 ⇓

1

118

2 4

5

6

1

( )

T 62

φ n-2

( )

ρ 1[62] : φ n - 2

Attractor : Λ 2 62

( )

⇓

φn

Time-1 Return map φn

( 62 )

⇓

ρ 1[62] : φn - 1

φn

1

7

T

φ n-1

0

131 *

1

( 62 )

0

φn

φn

62

1

5

6

1

φ n-1

4

T* 58

Time-2 Return map

φn

1

⇓

n

1

( )

φn

( 58 )

Attractor : Λ1 62

0 0

3

1

0

2

ρ 1[58] : φ n - 1


0

62


3

4

T 58

Time-2 Return map

φn

0 Firing Patterns

2

114

φ n-1

φn

1 0

833

Example : 01110110110111011001100

( )

⇓

6

( )

Time-1 Return map


φn

( 58 ) Attractor : Λ 2 58


1

Example : 01010110110110110101011

4

( N ).

Time-1 Return map

Attractor : Λ1 58

1 3

(Continued )

145

0 4 7

Table 27A.

142



Only strings with 2 or more red bits separated from each other by 2 or more blue bits:

Attractor : Λ1 142

(

1

0

3 7


)

1

(

Attractor : Λ 2 142

(

)

⇓ 2

212

4 6

1 φ n-1

( 142 )



Time-1 Return map

Example : 00110000111100110011111

2

( N ).

φn

0

Firing Patterns

(Continued )

ρ 1[142] : φ n - 1

(

T 142

1 2

142

Time-1 Return map

)

)

⇓

φn

7

3 7

φn

(

T* 142

)

⇓

Firing Patterns


184

Only strings with single blue bit separated from each other by 1, 2, 3 or more red bits :

( 142 )

Attractor : Λ1 184

(

3

7

Example : 01001010100101000010100

4

φn

(N )

.

φn

6


(

T† 184

)

⇓

1

226

5 6

φ n-1


(

ρ 1[184] : φn - 1

φn

Time-1 Return map

)

φn

7


1

( 184 )

Only strings with single red bit separated from each other by 1, 2, 3, or more blue bits :

2

212

φ n-1

Time-1 Return map

)

Example : 1011010111010101011010

0

5

0

4

ρ 1[142] : φ n - 1


0

834

Example : 00110011011010101010101

(

T 184

)

⇓

1

226

5 6

(

7

T* 184

7

)

⇓

1

( 184 )

φ n-1 ρ 1[184] : φn - 1

φn

184

3 4

5 7

Table 27B.

N =

Dynamic evolution from a generic initial bit string for the ten rules listed in Table 27A.

11

Firing Patterns

0

1

3


11

Firing Patterns

0

1

3

Transient T Λ 2 11 = Tdefect Period T∆ ∆

(

)

n=0 n=1

T∆

n=2 n=3 n=4 n=5 n=6 n=7 n=8

835

n=9 n=10

N = n=0 n=1 n=2

T∆

n=3 n=4 n=5 n=6 n=7 n=8

(

)

Table 27B.

N

14

N

14

(Continued )

T∆

(

)

T∆ Λ1 14 = Tdefect

T∆

836

T∆

(

)

T∆ Λ 2 14 = Tdefect

T∆

Remark The transient period T defect depends on the length of the interval occupied by “defect” pixels in the initial bit string which move towards 1 each other until they are annihilated upon collision. In general, 1 < Tdefect ≤ ( I + 1) where the equality sign is attained only if the defects 2 included the boundary pixels at i = 0 and i = I.

Table 27B.

N =

35

Firing Patterns

0

1

5

35

Firing Patterns

0

1

5

(Continued )





(

)

(

)

n=0 n=1

T∆

n=2 n=3 n=4 n=5 n=6 n=7

837

N = n=0 n=1 n=2 n=3

T∆

n=4 n=5 n=6 n=7 n=8 n=9 n=10

Table 27B.

N =

43

Firing Patterns

0

1

(Continued )

(

)

(

)


5

3

n=0 n=1 n=2

T∆

n=3 n=4 n=5 n=6 n=7

838

N = n=0 n=1 n=2 n=3

T∆

n=4 n=5 n=6 n=7 n=8 n=9 n=10

43

Firing Patterns

0

1

3

5


Table 27B.

N =

56

Firing Patterns

56

Firing Patterns

3

4

(Continued )

(

)

(

)

5


5


n=0

T∆

n=1 n=2 n=3 n=4

N = 839

n=0 n=1 n=2 n=3

T∆

n=4 n=5 n=6 n=7 n=8 n=9

3

4

Table 27B.

N =

(Continued )

(

)

(

)

57

Firing Patterns

0

3

4

5


57

Firing Patterns

0

3

4

5


n=0 n=1

T∆

n=2 n=3 n=4 n=5 n=6

N = 840

n=0 n=1 n=2 n=3

T∆

n=4 n=5 n=6 n=7 n=8 n=9 n=10 n=11

Table 27B.

N =

(Continued )

58

Firing Patterns

1

3

4

5

58

Firing Patterns

1

3

4

5


(

)

(

)


n=0 n=1 n=2 n=3 n=4

T∆

n=5 n=6 n=7 n=8 n=9

841

n=10 n=11

N = n=0

T∆

n=1 n=2 n=3 n=4 n=5 n=6 n=7


Table 27B.

N =

(Continued )

(

)

62

Firing Patterns

1

2

3

4

5


62

Firing Patterns

1

2

3

4

5

Transient T Λ 2 62 = 0 Period T∆ ∆

n=0 n=1 n=2 n=3 n=4

T∆

n=5 n=6 n=7

842

n=8 n=9 n=10 n=11 n=12 n=13

N = n=0 n=1

(

)

Table 27B.

N =

(Continued )

142

Firing Patterns

1

2

3

7

142

Firing Patterns

1

2

3

7

(

)

(

)





n=0

T∆

n=1 n=2 n=3 n=4 n=5

N = n=0

843

n=1 n=2 n=3

T∆

n=4 n=5 n=6 n=7 n=8 n=9 n=10 n=11 n=12 n=13

Table 27B.

(Continued )

184

Firing Patterns

3

4

5

7

Transient T Λ 184 Period T∆ ∆ 1

) =T

N = 184

Firing Patterns

3

4

5

7

Transient T Λ 2 184 Period T∆ ∆

) =T

N =

(

defect

n=0 n=1

T∆

n=2 n=3 n=4 n=5 n=6

844

n=0 n=1

T∆

n=2 n=3 n=4 n=5 n=6 n=7 n=8 n=9

(

defect

Table 28A. Characterization of all bit-string configurations which are allowed in the two noninvertible attractors of the five Bernoulli στ -shift rules 6 , 9 , 27 , 38 , 134 .

ρ 1[6] : φn - 1

Attractor : Λ1 9

( )

Time-1 Return map ρ 1[ 9 ] : φ n - 1

φ n-1

1

ρ 1[6] : φn - 2

6

.


ρ 1[9] : φn - 2

φn

Time-3 Return map φ n-1

0 0 0 1

φn


( )

T† 9 ⇓

0

⇓

111

0

1

2

3 5

6

( )

⇓

φ n-3

(9)

( )

T 9

T* 9

0

1

7

65

φ n-2

φn

φn

1

(N)

4

6

Time-1 Return map ρ 1[ 9 ] : φ n - 1

215

1

2

1 1

( ) ( )

0

φn

φn

9

Attractor : Λ2 9

3

4

( )

Time-2 Return map

φn

1

7

φ n-2

φn

0 2

T* 6


0 0 0 0

0

1 Example : 111010000010100000111000 11

159

φn

(6)

Example : 01011000011100000011110 Only strings with 1 or 3 red bits separated from each other by at 1, 3, or 5 blue bits :

⇓

φn

⇓

0

3

1

4

( )

Time-2 Return map φ n-1

2

T 6

1

0 0

0

Only strings with 1, 2, or more red bits separated from each other by 1, 2, or more blue bits :

( )

⇓

20

ρ 1[6] : φn - 2

φn


φ n-2

( 6)

φn

0 0

845

9



φn

1 1

Time-1 Return map

Example : 0011001101000101000101 Firing Patterns

φ n-1

( )

ρ 1[6] : φn - 1

( N ).

Time-2 Return map

φn

Attractor : Λ2 6

1 1

φn

0

Only strings with 1 or 2 red bits separated from each other by 1, 2, or 3 blue bits :

Time-1 Return map

0

Example : 00001100110000010000110

( )

0 0 0

2

Attractor : Λ1 6

0

1

Only strings with 1 or 2 red bits separated from each other by 2, 3, or more blue bits :


0

6


0

Firing Patterns

ρ 1[9] : φn - 3

φn

125

0 2

3

4

5

6

Table 28A.

Firing Patterns

27

Laws Governing Attractor Bit-Strings Bit Strings Only strings with 1, 3 or more red bits separated from each other by one or more blue bits :

(Continued )

Attractor Bit-String Attractor Bit String Generator Generator Attractor : Λ 1 27

( )

Time-1 Return map ρ 1[27] : φ n - 1

φ n-1

1

1


φn

( )


0

83

φ n-2


( )

⇓

4

ρ 1[27] : φn - 2

φn

( )

T 27 ⇓

φn

39

Time-2 Return map

φn

φ n-1

0

φn

( )

Attractor : Λ 1 38

( )


Example : 10001100100001100110000

φn

φ n-1

1

Attractor : Λ 2 38

( )φ

φ n-1


5


( )

T† 38 ⇓

2 4

5

0

1

( )

φn

φn

⇓

155

3 4 7

( )

T* 38 ⇓

211

( 38 )

4

T 38

Time-2 Return map

φn

0

1

ρ 1[38] : φn - 2

n

.

52

0

0

1

φn

1


(N )

φ n-2

0 2

φn

Time-2 Return map

( 38 )

0

1 Example : 100110110100100110110

ρ 1[27] : φ n - 2

φn

5

Only strings with 1 or 2 red bits separated from each other by at 1 or 2 blue bits :

φ n-2


0

2

53 27

0

1

0

38

Only strings with 1 or 2 red bits separated from each other by 2 or more blue bits:

0


1

5

( )

0

846

Firing Patterns

1

1

1

2

T* 27 ⇓

Example : 110110010010110010010

1

6

3 4


1

Example : 0010000111010101111110

1


Time-2 Return map

φn

0

0

φn

( N ).

φ n-2

ρ 1[38] : φn - 2

φn

0

1

4

5 7

Table 28A.

Firing Patterns

134

Laws Governing Attractor Bit Bit-Strings Strings Only strings with 1 or 2 red bits separated from each other by 2, 3, or more blue bits :

(Continued )

Attractor Bit-String Attractor Bit String Generator Generator Attractor : Λ 1 134

(

)

φn

Time-2 Return map

Time-1 Return map ρ 1[134] : φ n - 1

( N .)

φ n-1

φn

φn


Equivalent Equivalent Rules Rules andand Firing Firing Patterns Patterns

(

T† 134

)

⇓

148

2 4 7

1


(


( 134 )

φ n-2

ρ 1[134] : φn - 2

φn

φn

Time-2 Return map φn

1

( 134 )

)

⇓

158

1 2

7

(

)

⇓

ρ 1[134] : φ n - 2

φ n-2

φn

3

4

T* 134

φ n-1

1

0

1 1

φn

)

(

T 134

1 1

0 0 0

0 0

Example : 00010101001101101010001

0

Only strings with 1 or 2 red bits separated from each other by 1, 2, or 3 blue bits :

0

7

Example : 00010000011001100100011

0

847

2

214

1 2 4 6

7

Table 28B.

N =

Dynamic evolution from a generic initial bit string for the five rules listed in Table 28A.

(

)

(

)

6

Firing Patterns

1

2



6

Firing Patterns

1

2



n=0

T∆

n=1 n=2 n=3 n=4 n=5

N = 848

n=0

T∆

n=1 n=2 n=3 n=4 n=5 n=6


Table 28B.

N =

9

Firing Patterns

0

3

0

3

(Continued )


(

)


n=0 n=1 n=2

T∆

n=3 n=4 n=5 n=6 n=7

849

N = n=0 n=1 n=2

T∆

n=3 n=4 n=5 n=6 n=7 n=8 n=9 n=10

9

Transient T Λ2 9 Period T∆ ∆

(

) =T

defect

Table 28B.

N =

(Continued )

27

Firing Patterns

0

1

3

4

Transient T Λ 27 Period T∆ ∆ 1

) =T

27

Firing Patterns

0

1

3

4

Transient T Λ 2 27 Period T∆ ∆

) =T

(

defect

n=0 n=1

T∆

n=2 n=3 n=4 n=5

850

n=6 n=7 n=8

N = n=0 n=1 n=2 n=3

(

defect

Table 28B.

N =

38

Firing Patterns

1

2

5

38

Firing Patterns

1

2

5

(Continued )

(

)

(

)





n=0

T∆

n=1 n=2 n=3 n=4

851

n=5

N = n=0 n=1 n=2 n=3

Table 28B.

134

Firing Patterns

1

2

7

N = 134

Firing Patterns

1

2

7

N =

(Continued )

(

)

(

)





n=0 n=1

T∆

n=2 n=3 n=4 n=5 n=6 n=7

852

n=8

n=0

T∆

n=1 n=2 n=3 n=4 n=5 n=6

Table 29A. Characterization of all bit-string configurations which are allowed in the three noninvertible attractors of the five Bernoulli στ -shift rules 25 , 74 , 88 , 173 , and 229 .

Firing Patterns

25 0 3 4

Laws Governing Attractor Bit Strings Strings with 1, 1, or more red bits separated from each other by 1, 3, or more blue bits :

Attactor Bit String Generator (N


)

Attractor : Λ1 25

( )


( )

T† 25

φn

⇓

φn

67 0

Example : 01110100011101000000000 01000011010000111111111 00111010001110100000000 10100001101000011111111 00011101000111010000000 Strings with 1 or 3 red bits separated from each other by 1, 2, or 4 blue bits :

φ n-1

ρ 1[25] : φ n - 1

φn

ρ 1[25] : φn - 2

0

1 0

1

1

φ n-2

φn 6

0

0

( 25 )

1

Attractor : Λ 2 25

( )

( )

T 25

φn

φn

⇓

853

103 φ n-1

ρ 1[25] : φn - 1

φn

φ n-3

ρ 1[25] : φn - 3

0

φn

0

1

2 5

1 6

0

0

1

0

0

0 0

Example : 1110100100101000011100001 1000010010000111010011101 0111001001110100001010000 0100100101000011100001110 0010010000111010011101000 Strings with 1 ,2 ,3 or 4 red bits separated from each other by 1, 2, 3, 4, or 5 blue bits :

1

1

1

( 25 )

Attractor : Λ 3 25

( )

( )

T* 25

φn

φn

⇓

61

0

1 1

0

1

ρ 1[25] : φn - 5

1

1

1 1 1 1

1 1 1

0

φn

0

φn

0 0 0 0

0

0 00 00 0

0 0 0 0 0 1

1 1

1 1

φ n-3

0 0

ρ 1[25] : φ n - 1

φ n-1

0

Example : 1100001111001110100001110 1011101000101000011101000 0010000110000111010000111 1001110101110100001110100 0101000001000011101000011

( 25 )

2

3

4

5

Table 29A.

Only strings with 1, 2, or 3 red bits separated from each other by 1, 2, or 3 blue bits :

)


( )

T† 74 ⇓

φn

88 φ n-1

3

φn

4 6

1


( )

( )

T 74

φn

φn

⇓

173 φ n-1

ρ 1[74] : φ n - 1

1

1

φn

ρ 1[74] :φ n - 2

φ n-2 φn

0 2

3 5

1

1

7

1

1

1

( 74 )

Attractor : Λ 3 74

( )

( )

T* 74

φn

φn

⇓

229

1

φn

ρ 1[74] : φn - 3

1 1 1 1 1 0

ρ 1[74] : φ n - 1

φ n-1

0

0

0 0

Example : 1101011011010011100011010 1100011011000110100111000 1100111011001110001101001 1101101011011010011100011 1101100011011000110100111

0 0

854

Only strings with 1, 2, or 3 red bits separated from each other by 1, 2, or more blue bits : Example : 1110000111000000000011100 1010001101000000000110100 1000011100000000001110000 1000110100000000011010001 0001110000000000111000011


ρ 1[74] : φ n - 1

0

6

Example : 100010010010000100000001 000100100100001000000010 001001001000010000000100 010010010000100000001000 100100100001000000010001 001001000010000000100010

00

3

Only single red bits separated Attractor : Λ 1( 74 ) from each other by two or more blue bits :

0

1

Attactor Bit String Generator (N

0

74


0

Firing Patterns

(Continued )

φ n-3

φn

0 2 5

1

1

( 74 )

6

7

Table 29A.

Example : 100010010000100000001000 010001001000010000000100 001000100100001000000010 000100010010000100000001 100010001001000010000000 010001000100100001000000

φ n-1

ρ 1[88] : φ n - 1

3

φn

6

1


( )

( )

T 88

φn

φn

⇓

229 φ n-1

ρ 1[88] : φ n - 1

1

1

1

φn

ρ 1[88] : φ n - 2

φ n-2 φn

0 2 5

1

1

6

1

1

7

( 88 )

Attractor : Λ 3 88

( )

( )

T* 88

φn

φn

⇓

229 0

1

φ n-1 φn

ρ 1[88] : φn - 3

1 1 1 1 1 0

ρ 1[88] : φn - 1

0

0 0

Only strings with 1, 2, or 3 red bits separated from each other by 1, 2, or 3 blue bits : Example : 1110010110110001101100011 1011000110111001101110011 0011100110101101101011011 0010110110001101100011011 1000110111001101110011011 1100110101101101011011010

⇓

74 1

0

855

Only strings with 1, 2, or 3 red bits separated from each other by 1, 2, or more blue bits : Example : 1000101100010110001011000 1100001110000111000011100 0110001011000101100010110 0111000011100001110000111 0101100010110001011000101

( )

φn

0

6

Equivalent Rules and Firing Patterns T† 88

0

4

( )

0

3


Attractor : Λ1 88

0

88

Only single red bits separated from each other by two or more blue bits :

Attactor Bit String Generator ( N )

0 0


0

Firing Patterns

(Continued )

φ n-3 φn

2

3 5

1

7

1

( 88 )

Table 29A.

Firing Patterns


173

Only single blue bits separated from each other by two or more red bits :

0 2 3 5 7

(Continued )

Attactor Bit String Generator ( N ) Attractor : Λ 1 173

(

)


(

)

T† 173 ⇓

φn

Example : 111011101101101111101111 110111011011011111011111 101110110110111110111111 011101101101111101111111 111011011011111011111110 110110110111110111111101

229 0

ρ 1[173] : φn - 1

φ n-1 φn

2 5

1

6

0

1

7

( 173 )


(

)

(

T 173 φn


(

⇓

ρ 1[173] : φ n - 2

1

φ n-2

φn

3

0 6

( 173 )

0

)

φn

)

74

0

856

Only strings with 1, 2, or 3 blue bits separated from each φ n other by 1, 2, or more red bits : Example : 000111111110010111001011 φ n-1 ρ 1[173] : φn - 1 φ n 010111111100011110001111 011111111001011100101110 0 0 0 1 011111110001111000111100 1 1 0 111111100101110010111001 0 Only strings with 1, 2, or 3 blue bits separated from each other by 1, 2, or 3 red bits : Example : 1010010010110001110010110 1110010011100101100011100 1100010011000111001011000 1001010010010110001110010 1001110010011100101100011 1001100010011000111001011


(

T* 173 φn

)

⇓

88

ρ 1[173] : φn - 1

φ n-1 φn

ρ 1[173] :φ n - 3

0 0 0 1 1 1 1 1 0 0 0 1 1

φ n-3

3

φn 4

0

0

( 173 )

6

Table 29A.

Firing Patterns


229

Only single blue bits separated from each other by two or more red bits :

0 2 5 6 7

Example : 011011110111111101110111 101101111011111110111011 110110111101111111011110 111011011110111111101111 011101101111011111110111 101110110111101111111011

(Continued )

Attactor Bit String Generator ( N ) Attractor : Λ 1 229

(


)


(

T† 229

)

⇓

φn

88

ρ 1[229] : φ n - 1

φ n-1

2

φn

3 5

1

7

0

1

( 229 )


(

)

857

Only strings with 1, 2, or 3 φn blue bits separated from each other by 1, 2, or more red bits : Example : 001110100111111111101001 φ n-1 ρ 1[229] : φn - 1 φ n 000111100011111111111000 010011101001111111111010 0 0 0 1 110001111000111111111110 1 110100111010011111111111 0 0 1

(

T 229 φn

)

⇓

173

ρ 1[229] : φn - 2

φ n-2

φn

3 4

0

6

0

( 229 )

Only strings with 1, 2, or 3 Attractor : Λ 3( 229 ) blue bits separated from each φn φn other by 1, 2, or 3 red bits : Example : 1001000110010001101001110 φ n-1 φ n-3 1001010010010100111000110 ρ 1[229] : φn - 1 φ n ρ 1[229] : φn - 3 φ n 1001110010011100011010011 0 0 0 1 1 1000110010001101001110001 1 0 0 0 1 0 0 1010010010100111000110100 1 1 1 ( 229 ) 1110010011100011010011100

(

T* 229

)

⇓

229 1 3

6

Table 29B.

N =

25

Firing Patterns

Dynamic evolution from a generic initial bit string for the two rules listed in Table 29A.

0

3

4


(

)


n=0 n=1 n=2 n=3

T∆

n=4 n=5 n=6 n=7 n=8 n=9

858

n=10 n=11


Table 29B.

N = n=0 n=1 n=2 n=3

T∆

n=4

859

n=5 n=6 n=7 n=8 n=9 n=10 n=11 n=12 n=13

25

Firing Patterns

0

3

4

(Continued )

(

)


Table 29B.

N = n=0 n=1

T∆

n=2 n=3 n=4 n=5

860

n=6 n=7 n=8 n=9 n=10 n=11 n=12 n=13 n=14

25

Firing Patterns

0

3

4

(Continued )


(

)


Table 29B.

N =

(Continued )

(

)

74

Firing Patterns

1

3

6


T∆ Λ1 74 = 0

74

Firing Patterns

1

3

6



n=0 n=1

N = n=0

861

n=1

T∆

n=2 n=3 n=4 n=5 n=6 n=7 n=8

(

)

Table 29B.

N = n=0 n=1 n=2 n=3 n=4 n=5

T∆

n=6 n=7

862

n=8 n=9 n=10 n=11 n=12 n=13 n=14 n=15 n=16 n=17 n=18

74

Firing Patterns

1

3

6

(Continued )

(

)


Table 29B.

N =

(Continued )

(

)

88

Firing Patterns

3

4

6


T∆ Λ1 88 = 0

88

Firing Patterns

3

4

6



88

Firing Patterns

n=0 n=1

N =

(

)

(

)

n=0

T∆

n=1 n=2 n=3 n=4 n=5

863

n=6

N = n=0

T∆

n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=8 n=9

3

4

6


Table 29B.

(Continued )

(

)

Firing Patterns

0

2

3

5

7


173

Firing Patterns

0

2

3

5

7

Transient T Λ 173 = T defect Period T∆ ∆ 2

N = 173

Firing Patterns

0

2

3

5

7

N = 173 n=0 n=1

N =

(

)

(

)

n=0 n=1

T∆

n=2 n=3 n=4 n=5

864

n=6 n=7

n=0

T∆

n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=8

Transient T Λ 173 = T defect Period T∆ ∆ 3

Table 29B.

N = 229

(Continued )

(

)

Firing Patterns

0

2

5

6

7


Firing Patterns

0

2

5

6

7

Transient Period T∆ T∆ ( Λ 2 229 ) = Tdefect

n=0 n=1

N = 865

n=0 n=1

T∆

n=2 n=3 n=4 n=5 n=6 n=7

229

Table 29B.

N = n=0 n=1 n=2 n=3

T∆

n=4 n=5

866

n=6 n=7 n=8 n=9 n=10 n=11 n=12 n=13 n=14

229

Firing Patterns

0

2

5

(Continued )

6

7

(

)



To demonstrate the triviality of this task, we have “translated” the results in Table 29A for rule 74 to the three other globally-equivalent rules; namely, 88 = T † ( 74 ), 174 = T ( 74 ), and 229 = T ∗ ( 74 ), respectively. They are shown in the last three pages of Table 29A. The “recipe” for “translating” each column of 74 in Table 29A into a corresponding column for rules 88 , 173 , and 229 can be summarized as follows:

Equivalent Attractor Bit-String Generating Algorithm 1. The “laws governing the attractor bit strings” (column 2), the “attractor bit-string generator ” (column 3), and the “orbit unfolding plot ” (column 4) for rule N and rule N † = T † ( N ) are identical. 2. To obtain the corresponding results for rules N = T ( N ) and N ∗ = T ∗ ( N ), simply change “0” to “1”, and vice versa, in columns 2 and 3, and “flip” (i.e. rotate by 180◦ ) the “orbit unfolding plot ” (column 4) about the abscissa (Im[zn ] = 0).

[Chua et al., 2005a] seems to confirm that indeed all points on these time-τ return maps are subsets of one of the four ideal invariant maps shown in Figs. 26 and 27. Unfortunately, our above rush to conclusion is wrong! We will show below that none of these timeτ maps actually lie on one of these ideal invariant maps. Our hasty conclusion was an illusion due to a lack of precision of the printer, as well as our retina! We will prove in Sec. 7.1 that the correct time-τ maps will converge to a subset of one of the ideal invariant maps of 170 , 240 , 85 , or 15 shown in Figs. 26 and 27, if, and only if, I → ∞, where I + 1 is the length of the bit string in Fig. 1. Since all of our examples from Parts I to VI are generated with a finite I, how can we be sure that they are in fact correct? Our goal in this section is to prove that these computer-generated examples are not only correct, but they can also be predicted by an exact formula which we will derive in closed form analytically. The above paradox can only be resolved in a rigorous mathematical way by exploiting the analytical closed form equation

7. Mathematical Foundation of Bernoulli στ -Shift Maps There are all together 112 rules where attractors and/or invariant orbits are characterized by a Bernoulli στ -shift map where β = ±2σ or β = ±2−σ , σ ∈ {1, 2, 3} and τ ∈ {1, 2, 3, 5}. Among them 84 are time-reversible (Table 2) and 28 are time-irreversible (Tables 28A and 29A). These 112 rules are collected in consecutive order in Table 30-1. Throughout the series of papers, the color red, blue, or green codes for complexity index κ = 1, 2, or 3, respectively. The three integer parameters {β, σ, τ } associated with these rules are listed in Tables 30-2 and 30-3. It is truly remarkable that almost half of the 256 CA rules can be so elegantly characterized, assuming that all of the “time-τ return maps” which we had obtained via exhaustive computer simulations lie exactly on the ideal invariant time-1 return maps of rules 170 and 240 in Fig. 26, or 85 and 15 in Fig. 27. A glance at Tables 5–8 in the preceding sections, as well as Tables 13–15 from 26

867

(33)

of the time-1 characteristic function of N , where the parameters {z2 , c2 , z1 , c1 , z0 , b1 , b2 , b3 } are given in Table 1 of [Chua et al., 2005b] for all rules, N = 0, 1, 2, . . ., 255.

7.1. Exact formula for time-1 Bernoulli maps for rules 170 , 240 , 85 , or 15 In this subsection we will derive an exact formula in closed form for the time-1 return map of the four invariant rules 170 , 240 , 85 , or 15 for a finite-I bit string.

7.1.1. Exact formula for Bernoulli right-copycat shift map 170 The time-1 characteristic function χ1170 of the

“right-copycat”26 rule 170 is given in Table 1 of

Rule 170 is dubbed a “right-copycat ” rule, while rule 240 is dubbed a “left-copycat ” rule in [Chua et al., 2005a]. These interpretations are quite useful in understanding the examples in Sec. 7.1.5.

868

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 30. There are 112 rules characterized by a Bernoulli στ -shift map. Observe that τ (N † ) = τ (N ) = τ (N ∗ ) = τ (N ); σ(N † ) = −σ(N ), σ(N ) = σ(N ), σ(N ∗ ) = −σ(N ); β(N † ) = [β(N )]−1 , β(N ) = β(N ), and β(N ∗ ) = [β(N )]−1 . Table 30-1.

112 Rules Characterized By A Bernoulli στ-shift Map 2

25

53

83

117

152

185

215

3

27

56

84

118

155

186

226

6

31

57

85

119

158

187

227

7

34

58

87

125

159

188

229

9

35

59

88

130

162

189

230

10

38

61

98

131

163

190

231

11

39

62

99

134

170

191

240

14

42

63

103

138

171

194

241

15

43

65

111

139

173

208

242

16

46

66

112

142

174

209

243

17

47

67

113

143

175

211

244

20

48

74

114

144

176

212

245

21

49

80

115

145

177

213

246

24

52

81

116

148

184

214

247

[Chua et al., 2005b]; namely, χ1170 (φn−1 ) =

I

2−(i+1)

(n−1)

(n−1)

xi+1

i=0

−

1 2

(34)

[•] is the step function defined in where (n−1) (n−1) Eq. (18). Substituting [xi+1 − (1/2)] = xi+1 , (n−1) xi+1

∈ {0, 1}, in Eq. (34) and setting τ = 1 in for Eq. (14), we obtain the time-1 return map φn = χ1170 (φn−1 ) =

I i=0



= 2

I+1 j=0

 = 2

I j=0

27

(n−1)

2−(i+1) xi+1

=

I+1 j=1

(n−1) 



(n−1)

= 2φn−1 + [2−(I+1) − 1]x0

(35)

where we have substituted the decimal representaI −(j+1) x(n−1) for x(n−1) and tion27 φn−1 = j j=0 2 (n−1)

the periodic boundary condition xI+1 (n−1) x0

(n−1) x0

(n−1)

= x0

,

∈ {0, 1}. Since = 0 ⇔ φn−1 < where 0.5, Eq. (35) can be recast as follow: Formula for time-1 map 170

(n−1)

2−j xj



2−(j+1) xj

(n−1)

+ 2−(I+1) xI+1 − x0

φn = 2φn−1 , = 2[φn−1 − µ(I)],

if 0 ≤ φn−1 < 0.5, if 0.5 ≤ φn−1 ≤ 2µ(I),

(n−1)

− x0

(36) where the Bernoulli map intercept

(n−1) 

2−(j+1) xj

µ(I) = 0.5 − 2−(I+2)

(n−1)

Note that φn−1 is the decimal value of the forward bit string x(n−1) = [x0 Eq. (1) at the (n − 1)th iteration via rule N .

(n−1)

, x1

(n−1)

, . . . , xI

(37)

] → φn−1 defined in

Chapter 6: From Time-Reversible Attractors to the Arrow of Time Table 30-2.

(Continued )

( β , σ ,τ ) N

( β , σ ,τ ) N

Left - Right Left - Right Global Transformation Complementation Complementation


T †[ N ] T [ N ] T ∗ [ N ]

N

β

σ

τ

16

191

247

53

2, ¼

1 , -2

2,2

39

83

27

2

17

63

119

56

½,2

-1 , 1

1,1

98

227

185

2 , -2

2,2

20

159

215

57

½,2

-1 , 1

1,1

99

99

57

½

-1

2

21

31

87

58

2, ½

1 , -1

1,2

114

163

177

9

¼,4

-2 , 2

2,3

65

111

125

59

½,2

-1 , 1

2,1

115

35

49

10

2

1

1

80

175

245

61

103

67

25

11

-½ , 2

-1 , 1

1,1

81

47

117

62

½,2

-1 , 1

2,1

118

131

145

14

2 , -½

1 , -1

1,1

84

143

213

63

½

-1

2

119

3

17

15

-½

-1

1

85

15

85

65

4, ¼

2 , -2

2,3

9

125

111

16

½

-1

1

2

247

191

66

2

1

1

24

189

231

17

2

1

2

3

119

63

67

2 , 1/8 , ¼ 1 , -3 , -2 2 , 3 , 5

25

61

103

20

¼,4

-2 , 2

2,2

6

215

159

74

2 , 4 , 1/8 1 , 2 , -3 1 , 2 , 3

88

173

229

21

2

1

2

7

87

31

80

½

-1

1

10

245

175

24

½

-1

1

66

231

189

81

-2 , ½

1 , -1

1,1

11

117

47

67

103

61

83

2,¼

1 , -2

2,2

27

53

39

N

β

σ

τ

2

2

1

1

3

½

-1

6

4,¼

7

25

869

½ , 8 , 4 -1 , 3 , 2 2 , 3 , 5

2 , 1/8 , ¼ 1 , -3 , -2 2 , 3 , 5

T † [ N ] T [ N ] T ∗[ N ]

27

½,4

-1 , 2

2,2

83

39

53

84

½ , -2

-1 , 1

1,1

14

213

143

31

½

-1

2

87

7

21

85

-2

1

1

15

85

15

34

2

1

1

48

187

243

87

2

1

2

31

21

7

35

½,2

-1 , 1

2,1

49

59

115

88

74

229

173

38

4,4

2,2

2,2

52

155

211

98

2,½

1 , -1

1,1

56

185

227

39

½,4

-1 , 2

2,2

53

27

83

99

2,½

1 , -1

1,1

57

57

99

42

2

1

1

112

171

241

103

61

25

67

43

-½ , 2

-1 , 1

1,1

113

43

113

111

¼,4

-2 , 2

2,3

125

9

65

46

2

1

1

116

139

209

112

½

-1

1

42

241

171

47

-½ , 2

-1 , 1

1,1

117

11

81

113

-2 , ½

1 , -1

1,1

43

113

43

48

½

-1

1

34

243

187

114

½,2

-1 , 1

1,2

58

177

163

49

2, ½

1 , -1

2,1

35

115

59

115

2,½

1 , -1

2,1

59

49

35

52

¼, ¼

-2 , -2

2,2

38

211

155

116

½

-1

1

46

209

139

½ , ¼ , 8 -1 , -2 , 3 1 , 2 , 3

½ , 8 , 4 -1 , 3 , 2 2 , 3 , 5

870


Table 30-3.

(Continued )

β ,σ ,τ

β ,σ ,τ


N

T † [ N ] T [ N ] T ∗[ N ]


N

T † [ N ] T [ N ] T ∗[ N ]

N

β

σ

τ

11

185

2,½

1 , -1

1,1

227

98

56

145

131

186

2

1

1

242

162

176

63

17

3

187

2

1

1

243

34

48

2,3

111

65

9

188

2

1

1

230

194

152

1

1

144

190

246

189

2

1

1

231

66

24

½,2

-1 , 1

2,1

145

62

118

190

2

1

1

246

130

144

134

4,¼

2 , -2

2,2

148

158

214

191

2

1

1

247

2

16

138

2

1

1

208

174

244

194

2

1

1

152

188

230

139

2

1

1

209

46

116

208

½

-1

1

138

244

174

142

2 , -½

1 , -1

1,1

212

142

212

209

½

-1

1

139

116

46

143

2 , -½

1 , -1

1,1

213

14

84

211

¼,¼

-2 , -2

2,2

155

52

38

144

½

-1

1

130

246

190

212

½ , -2

-1 , 1

1,1

142

212

142

145

2,½

1 , -1

2,1

131

118

62

213

½ , -2

-1 , 1

1,1

143

84

14

148

¼,4

-2 , 2

2,2

134

214

158

214

¼,4

-2 , 2

2,2

158

148

134

152

½

-1

1

194

230

188

215

¼,4

-2 , 2

2,2

159

20

6

155

4,4

2,2

2,2

211

38

52

226

2,½

1 , -1

1,1

184

184

226

158

4,¼

2 , -2

2,2

214

134

148

227

½,2

-1 , 1

1,1

185

56

98

159

4,¼

2 , -2

2,2

215

6

20

229 ½ , ¼ , 8 -1 , -2 , 3 1 , 2 , 3

173

88

74

162

2

1

1

176

186

242

230

½

-1

1

188

152

194

163

2,½

1 , -1

1,2

177

58

114

231

½

-1

1

189

24

66

170

2

1

1

240

170

240

240

½

-1

1

170

240

170

171

2

1

1

241

42

112

241

½

-1

1

171

112

42

173 2 , 4 , 1/8 1 , 2 , -3 1 , 2 , 3

229

74

88

242

½

-1

1

186

176

162

174

2

1

1

244

138

208

243

½

-1

1

187

48

34

175

2

1

1

245

10

80

244

½

-1

1

174

208

138

176

½

-1

1

162

242

186

245

½

-1

1

175

80

10

177

½,2

-1 , 1

1,2

163

114

58

246

½

-1

1

190

144

130

184

½,2

-1 , 1

1,1

226

226

184

247

½

-1

1

191

16

2

N

β

σ

τ

117

-2 , ½

1 , -1

1,1

47

81

118

2,½

1 , -1

2,1

62

119

2

1

2

125

4,¼

2 , -2

130

2

131


φn

871

0.5

1

I=2

170

I=3 I=4 I=5

0.8 (0.375, 0.75) (0. 75, 0.625)

I →∞

0.6

0.4 (0.625, 0.375)

µ (3) = 0.46875

0.2

µ (5) = 0.4921875 µ (4) = 0.484375

µ (2) = 0.4375

0 Fig. 30.

0.2

0.4

0.5

0.6

0.8

φn−1

1

Plot of the exact Bernoulli Formula [Eq. (36)] for rule 170 for I = 2, 3, 4, 5, and ∞.

is the intercept of the straight line φn = 2(φn−1 − µ(I)) with the horizontal axis φn−1 in the graph of Eq. (36) shown in Fig. 30. The value of the Bernoulli map intercept µ(I) is given in Table 31 as a function of I, where I + 1 is the length of the bit string (see Fig. 1). Observe that even though µ(I) differs from the limiting value of 0.5 in the ideal Bernoulli map shown in Fig. 26, it is indistinguishable in any mechanical printer, or the human eye, for I > 10. Since all time-1 return maps in Table 2 of [Chua et al., 2005a], as well as in this paper, were calculated with I > 60, it is not surprising that they are practically indistinguishable from the limiting Bernoulli map φn = 2φn−1

mod 1

(38)

Our illusion is indeed a victim of poor printer resolution. Observe that the Bernoulli στ -shift map

of 170 for I = 2 consists of only the three points shown in Fig. 30. It is the only orbit of 170 . Observe that it is also an “isle of Eden”. It follows from the graphs of the exact Bernoulli στ -shift maps in Fig. 30 that the mechanism responsible for the left-shift algorithm of 170 is not the classic “mod 1” Bernoulli-shift map defined by Eq. (38), and shown in Fig. 26, but rather it is the slope being equal to “2” in Eq. (36). This observation is of course obvious once we realize that multiplying the decimal equivalent of a binary bit string by the integer “2” is equivalent to shifting the string one bit to the left in binary arithmetic. The intercept µ(I) plays no role in this operation.

7.1.2. Exact formula for Bernoulli left-copycat shift map 240 The time-1 characteristic function χ1240 of the “left-

copycat” rule 240 is given in Table 1 of [Chua et al.,

872

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 31. Exact value of the Bernoulli map intercept µ(I) and ∆ the Bernoulli constant ν(I) = 2µ(I) as a function of I for bit strings {x0 , x1 , x2 , . . . , xI }.

I

µ (Ι ) = 0.5 – 2-(I+2) ν (Ι ) = 1 – 2-(I+1)

2

0.4375

0.875

3

0.46875

0.9375

4

0.484375

0.96875

5

0.4921875

0.984375

10

0.499755859375

0.99951171875

20

0.4999997615814209

0.9999995231628418

30

0.49999999976716936

0.99999999953433872

40

0.49999999999977263

0.99999999999954526

50

0.49999999999999978

0.99999999999999956

60

0.49999999999999999 †

†

0.99999999999999999 †

µ (60) = 0.49999999999999999783159565502899 ν (60) = 0.99999999999999999566319131005798

†

  I 1  −(j+1) (n−1)  = 2 xj 2

2005b]; namely, 1

χ 240 (φn−1 ) =

I

−(i+1)

2

i=0

(n−1)

(n−1) xi−1

1 − . 2

(n−1)

(39) (n−1)

(n−1)

φn = χ1240 (φn−1 )

=

I

(n−1)

2−(i+1) xi−1

i=0 I−1

j=−1

(n−1)

2−(j+2) xj

I−1 1 −(j+1) (n−1) = 2 xj 2 j=−1   I−1 1  −(j+1) (n−1)  1 (n−1) = 2 xj + x−1 2 2 j=0

1 (n−1) + x−1 2 1 1 (n−1) − 2−(I+2) xI = φn−1 + , 2 2 (n−1)

− 2−(I+2) xI

Substituting [xi−1 − (1/2)] = xi−1 , for xi−1 ∈ {0, 1}, in Eq. (39), we obtain

=

j=0

(40)

(n−1)

in view where we have substituted x−1 = xI of the periodic boundary condition shown in Fig. 1. (n−1) ∈ {0, 1}, Eq. (40) can be recast as Since xI follows: Formula for time-1 map of 240 1 φn = φn−1 , 2

(n−1)

if xI

=0

(lower branch),

1 (n−1) =1 = φn−1 + µ(I), if xI 2 (upper branch),

(41)


φn

873

0.5

1

240

I →∞ I=3

I=4 0.8

µ (4) = 0.484375

I=5

I=2

µ (5) = 0.4921875

(0. 625, 0.75)

0.6

(0.375, 0.625)

0.5

0.5 0.4

µ (2) = 0.4375 µ (3) = 0.46875

(0.75, 0.375)

0.2

0 Fig. 31.

0.4

0.2

7.1.3. Exact formula for Bernoulli shift map for 85 χ1

The time-1 characteristic function 85 of rule 85 is given in Table 1 of [Chua et al., 2005b]; namely, χ 85 (φn−1 ) = Substituting

I

0.6

0.8

φn−1

1

Plot of the exact Bernoulli Formula [Eq. (41)] for rule 240 for I = 2, 3, 4, 5, and ∞.

where µ(I) is given by the same Eq. (37) and represents the vertical intercept of φn = (1/2)(φn−1 ) + µ(I) with the vertical axis φn in the graph of Eq. (41) shown in Fig. 31. Observe that the graphs of Figs. 30 and 31 are related by a reflection symmetry with respect to the main diagonal, as expected since 240 = T † ( 170 ). It follows that the three-point “isle of Eden” shown in Fig. 30 must appear also in 240 for I = 2, as confirmed in Fig. 31.

1

0.5

−(i+1)

2

i=0 (n−1)

1 (n−1) − xi+1 . 2 (n−1)

(n−1)

xi+1

∈ {0, 1}, in Eq. (42), we obtain

φn = χ185 (φn−1 ) =

I

2−(i+1) −

i=0

I i=0

(n−1)

2−(i+1) xi+1

. (43)

Observe that Eq. (43) differs from the second line of Eq. (35) of 170 by a negative sign on the right, and a constant

ν(I) =

I

2−(i+1)

i=0

= 1 − 2−(I+1)

(44)

= 2µ(I) (42)

[(1/2) − xi+1 ] = 1 − xi+1 , for

which tends to 1 as I → ∞. Hence, we can obtain the corresponding formula for 85 from Eq. (36) as

874


7.1.5. Exact formula for Bernoulli shift maps with β = 2σ (left shift), or β = 2−σ (right shift), σ = 1 and τ = 1

follows: Formula for time-1 map of 85 φn = ν(I) − 2φn−1 , = ν(I) − 2[φn−1 − µ(I)],

if 0 ≤ φn−1 < 0.5, if 0.5 ≤ φn−1 ≤ 2µ(I). (45)

The graph of Eq. (45) is obtained by adding “ν(I)” to the “flipped” (rotating about the φn−1 axis by 180◦ ) graph of Fig. 30. Observe that this graph converges to the ideal Bernoulli shift map ρ 85 in Fig. 27 as I → ∞.

7.1.4. Exact formula for Bernoulli shift map for 15 The time-1 characteristic function χ115 of rule 15 is given in Table 1 of [Chua et al., 2005b]; namely, I 1 (n−1) 1 −(i+1) − xi−1 2 . (46) χ 15 (φn−1 ) = 2 i=0

(n−1)

(n−1)

xi−1

(n−1)

[(1/2) − xi−1 ] = 1 − xi−1 , for

Substituting

∈ {0, 1}, in Eq. (46) we obtain

φn = χ115 (φn−1 ) =

I

−(i+1)

2

−

i=0

I i=0

(n−1) 2−(i+1) xi−1

.

(47)

Formula for time-1 map of 15

from Table 1 of [Chua et al., 2005b] is given by I 1 −(i+1) (n−1) (n−1) (n−1) 1 + xi+1 − 2 −xi−1 − xi χ = 2 2 i=0

(49) if

(n−1) xI

=0

(lower branch), 1 (n−1) =1 = ν(I) − φn−1 + µ(I) , if xI 2 (upper branch),

[Chua et al., 2005b] for all rules listed in Table 24A must reduce to either Eq. (34) (for the left-shift rules 2 , 10 , 34 , 42 , 46 , 130 , 138 , and 162 ) or Eq. (39) (for the right-shift rules 24 and 152 ). Consequently, the above eight left-shift rules have exactly the same Bernoulli στ -shift map as Eq. (36) for 170 . Similarly, the right-shift rule 24 must have exactly the same Bernoulli στ -shift map as Eq. (41) of 240 . But since the “active” firing patterns of all rules in Table 24A are only a proper subset of either 170 or 240 , certain bit-string configurations on the attractors of 170 or 240 must be excluded, namely, those violating the laws governing attractor bit strings in column 2. It follows that the time-1 return maps of all of these rules must be a proper subset of those of 170 or 240 (column 3). The excluded bit strings manifest themselves as “gaps” in their time-1 return maps. Let us illustrate the above mechanisms with some examples. Example 1. Rule 2 . The characteristic function

Following the same procedure as above, we obtain the following:

1 φn = ν(I) − φn−1 , 2

The rules under this section are listed in Table 24A. Observe that the firing patterns of each rule (column 1) are either a subset of the firing patterns of 170 or 240 , else they are “transient firing patterns” in the sense that they cannot occur on an attractor and have been excluded by the “Laws governing attractor bit strings” (column 2). This means that the characteristic functions χ1N in Table 1 of

1 From Table 24A we find the only firing pattern : 001 of 2 allowed on the attractor is 001 (column 1). Substituting xi−1 = 0 and xi = 0 into Eq. (49) we find χ12 → χ1170 , and hence the bit strings on the

attractor of 2 are governed by the same exact formula as that of 170 ; namely, by Eq. (36).

(48) Example 2. Rule 10 . The characteristic function from Table 1 of [Chua et al., 2005b] is given by where ν(I) is given by Eq. (44). I The graph of Eq. (48) is obtained by adding 1 −(i+1) (n−1) (n−1) 1 −xi−1 + xi+1 − 2 χ 10 (φn−1 ) = ν(I) to the “flipped” graph of Fig. 31. Observe that 2 i=0 this graph converges to the ideal Bernoulli shift map (50) ρ 15 in Fig. 27 as I → ∞.


From Table 24A we find that both firing pat(n−1) 1 : 001 and 3 : 011 of 10 have x = 0. terns i−1 Consequently, Eq. (50) reduces again to Eq. (34) of 170 . Example 3. Rule 24 . The characteristic function

from Table 1 of [Chua et al., 2005b] is given by I (n−1) (n−1) 1 χ 24 (φn−1 ) = −1 + xi−1 − xi i=0

1 −(i+1) + 2 (51) 2 Observe from Table 24A that even though only the 4 : 100 of 24 is a subset of those of firing pattern 240 , the first pattern 011 is a “transient bit string” in view of the Laws in column 2 outlawing it from an attractor of 240 . For the second pattern 100, Eq. (51) reduces to Eq. (39) of 240 because (n−1) 1 (n−1) (n−1) − xi+1 + −1 + xi−1 − xi 2 (n−1) 1 (n−1) −1 + xi−1 + = xi−1 . (52) = 2 (n−1) − xi+1

Example 4. Rule 42 . The characteristic function

from Table 1 of [Chua et al., 2005b] is given by 1

χ 42 (φn−1 ) =

I i=0

+

(n−1)

−xi−1

(n−1) 2xi+1

(n−1)

− xi

1 −(i+1) 2 − 2

(53)

1 : 001, Rule 42 has three firing patterns; namely, 3 : 011, and 5 : 101 (Table 24A), all of them are subsets of those of 170 . 1 : 001, we have For the first firing pattern, 1 (n−1) (n−1) (n−1) + 2xi+1 − −xi−1 − xi 2 1 (n−1) 2xi+1 − = 2

(n−1)

= xi+1

(54)

3 : 011, we have For the second firing pattern, 1 (n−1) (n−1) (n−1) + 2xi+1 − −xi−1 − xi 2 1 (n−1) (n−1) + 2xi+1 − −xi = 2

=

(n−1) xi+1

(55)

875

5 : 101, we have For the third firing pattern, 1 (n−1) (n−1) (n−1) + 2xi+1 − −xi−1 − xi 2 1 (n−1) (n−1) −xi−1 + 2xi+1 − = 2

(n−1)

= xi+1

(56)

It follows from Eqs. (54)–(56) that Eq. (51) reduces to Eq. (34) of 170 .

7.1.6. Exact formula for Bernoulli shift maps for 184 The ten Bernoulli στ -shift rules listed in Table 27A have two qualitatively distinct robust attractors. Except for rules 35 , 58 , and 62 , whose attractors consist of a mixture of a time-1 return map and a time-2 return map, the other seven rules are endowed with two distinct time-reversible attractors. Each attractor is described by the leftshift Bernoulli-map of Eq. (36), or the right-shift Bernoulli-map of Eq. (41). We will derive this result for rule 184 as an illustration. The characteristic function of 184 from Table 1 of [Chua et al., 2005b] is given by I 3 (n−1) 1 1 − − + |(xi−1 χ 184 (φn−1 ) = 2 i=0 (n−1) (n−1) − 2xi + 3xi+1 − 3)| 2−(i+1) (57) Let us restrict Eq. (57) to each attractor. 1. Attractor Λ1 ( 184 ) It follows from the laws governing attractor Λ1 ( 184 ) in Table 27A that only two out of the four firing patterns are active on Λ1 ( 184 ); namely, (n−1) 3 : 011 and : 7 3 111. For : 011, since xi−1 = 0 (n−1)

= 1, the step function in Eq. (57) reduces and xi to 3 (n−1) 1 − − + |(0 − 2 + 3xi+1 − 3)| 2 3 (n−1) (n−1) 1 − − + |(3xi+1 − 5)| = xi+1 = 2 (58a) (n−1)

(n−1)

7 111, since xi−1 = 1 and xi and for : the step function in Eq. (57) reduces to

= 1,

876


3 (n−1) 1 − − + |(1 − 2 + 3xi+1 − 3)| 2 3 (n−1) (n−1) 1 − − + |(3xi+1 − 4)| = xi+1 = 2 (58b)

the initial string configuration is conserved, and is therefore an invariant characteristic, for all iterations n = 0, 1, 2, . . . , under 184 , i.e. for both “transient” and steady state (attractor or invariant orbit) regimes.

It follows that time-1 return map on attractor Λ1 ( 184 ) of 184 is given by the left-shift Bernoulli map in Eq. (36).

7.2. Exact formula for time-τ Bernoulli maps for rules 170 , 240 , 85 , or 15

2. Attractor Λ2 ( 184 ) It follows from the laws governing attractor Λ2 ( 184 ) in Table 27A that only two out of the four 4 firing patterns are active on Λ2 ( 184 ); namely, : (n−1) 5 101. For : 4 100, since x 100 and : = 0 and i (n−1) xi+1 = 0, the step function in Eq. (57) reduces to 3 (n−1) 1 − − + |(xi−1 − 2 · 0 + 3 · 0 − 3)| 2 3 (n−1) (n−1) 1 − − + |(xi−1 − 3)| = xi−1 = 2

The time-1 return maps of many rules exhibited in Table 2 of [Chua et al., 2005a] are not Bernoulli in the sense that the points on these maps do not lie on any one of the four ideal (I → ∞) time-1 return maps of 170 and 240 in Fig. 26, or 85 and 15 in Fig. 27. For example, the time-1 map of the attractor Λ4 ( 62 ) in Table 1A is certainly not a subset of any one of these four ideal Bernoulli στ shift maps. However, if we plot the time-2 return map ρ262 (φn−2 ) : φn−2 → φn , we find the points on this map is a subset of the ideal time-1 return map of 240 , shown in p. 1150 (Attractor 1) of [Chua et al., 2005a]. In particular, we find the composition [Chua, 1969] between two identical time-1 maps ρ162 (φn−1 ) : φn−1 → φn of attractor Λ4 ( 62 ) of 62 is a subset of the Bernoulli map shown in Fig. 31 for 240 for I = 2, 3, 4, 5, . . . , ∞.

(59a) (n−1)

(n−1)

5 101, since xi = 0 and xi+1 = 1, and for : the step function in Eq. (57) reduces to 3 (n−1) 1 − − + |(xi−1 − 2 · 0 + 3 · 1 − 3)| 2 3 (n−1) (n−1) 1 − − + |(xi−1 )| = xi−1 = 2 (59b)

It follows that time-1 return map on attractor Λ2 ( 184 ) of 184 is given by the right-shift Bernoulli map in Eq. (41). A careful analysis of the firing patterns of 184 shows that the basin of attraction for the left-shift attractor Λ1 ( 184 ) consists of all bit strings with a majority of red “1” bits. Conversely, the basin of attraction for the right-shift attractor Λ2 ( 184 ) of 184 consists of all bit strings with a majority of blue “0” bits. The special case of bit strings whose number of red bits is exactly equal to the number of blue bits can be easily proved to converge to a nonrobust time-reversible attractor Λ3 ( 184 ) consisting of alternating bits, thereby resulting in a period-2 checkerboard pattern, as shown in Fig. 32. In this 5 : 101 is active. case, only the firing pattern It is interesting to note, in passing, that the net sum of the red (“1”) and blue (“0”) bits in 28

7.2.1. Geometrical interpretation of time-τ maps 7.2.1.1. Time-2 map of 62 To illustrate geometrically [Chua, 1969] how the “composition” between two non-Bernoulli time-1 maps can give rise to a Bernoulli στ -shift map, Fig. 33 shows the loci of three points colored in violet, brown and cyan, respectively, which lie on two identical time-1 maps φn−1 → φn (upper-left graph) and φn−2 → φn−1 (lower-right graph) of attractor Λ4 ( 62 ) in Table 1A. If we project these three sample points onto the φn−2 versus φn plane (upper-right graph) via guide lines of corresponding colors, we will find that their corresponding projections fall exactly on a time-1 Bernoulli map of 240 .28 Observe that the horizontal axes of both upper and lower graphs are labeled φn−1 on the left, and φn−2 on the right. The variables of the vertical axes are both φn in the upper graphs, and φn−1 on the lower graphs. The symmetrical alignment of these four pairs of axes makes it quite easy to pick

This will be proved mathematically in Section 7.2.7 and 7.3 to be true for all I = 2, 3, 4, 5, . . . , ∞.


877

Row 69

184

Row 19

Row 0 Row 0

226

Row -19

184

226 Fig. 32.

Time reversal test for the “checkerboard” attractor of rule 184 .

Row 19

-19

Row 0

0

878


ρ 262 : φ n − 2

φn

φn

φn

1

1

0 .5

0 .5

0

φ n −1

0 .5

0

0 .5

1

0 .5

1

φn−2

φ n −1

φ n −1

1

1

0 .5

0

1

φ n −1

0

φn−2

Fig. 33. Geometrical illustration of graphical composition between two identical time-1 return maps of Λ4 ( 62 ) to obtain a corresponding time-2 return map.

any point on the time-1 map φn−1 → φn (upperleft graph), and verify that its corresponding projection lies on the time-2 map φn−2 → φn shown in the upper-right graph. The resulting graph shown in the upper-right corner is a time-2 Bernoulli στ shift map with β = 2σ , σ = −1, and τ = 2. The above result is truly remarkable because it shows that even though the time-1 map ρ162 (φn−1 ) : φn−1 → φn of the attractor Λ4 ( 62 ) in Table 1A cannot be defined by a simple formula, its time-2 map obeys the Bernoulli (“left-copycat”) right-shift rule, independent of I! Hence, given any initial configuration belonging to the basin of attraction of Λ4 ( 62 ), we can predict its evolution for all times n > T∆ + 1, where T∆ is the number of iterations during the transient regime. We need one more iteration beyond T∆ because the time-2 map ρ262 can be applied to only every second rows, and hence, we need to calculate a second row of bit strings, from χ162 , or from the truth table of 62 , to obtain the complete evolution for all times n > T∆ + 1. This is truly a global result since the evolution dynamics is obtained directly by shifting alternate rows by one bit to the right, regardless of the length (I + 1) of the bit string.

7.2.1.2. Time-2 map of 74 The above example of a Bernoulli time-2 map for 62 involves a right shift of only one bit. To show that some time-2 maps may involve shifting two bits, Fig. 34 shows the graphical composition between two identical time-1 maps of the attractor Λ2 ( 74 ) of 74 in Table 15-2 (p. 1158) of [Chua et al., 2005a]. The time-1 map ρ174 (Λ2 ) is extracted from Λ2 ( 74 ) in Table 29A, and is reproduced in the upper-left corner and the lower-right corner of Fig. 34. Again, observe that even though the time1 map ρ174 (Λ2 ) in the upper-left corner of Fig. 34 cannot be described by a simple formula, its time-2 map ρ274 (Λ2 ) is a subset of a Bernoulli στ -shift map with β = 2σ , σ = 2, and τ = 2. Observe that in this case, since σ = 2, the evolution for n > T∆ + 1 consists of shifting every second row after T∆ + 1 by two bits to the left.

7.2.1.3. Time-3 map of 74 To illustrate a non-Bernoulli time-1 map which can be described by a time-3 Bernoulli στ -shift map, Fig. 35 shows the composition between a


Fig. 34.

Graphical composition between two identical non-Bernoulli time-1 maps ρ1

Bernoulli στ -shift map ρ2

74

74

(Λ2 ).

ρ 374 ( Λ3 ) : φn−3

φn

φn

1

1

0 .5

0 .5

0

φn−2

0 .5

879

(Λ2 ) of 74 gives rise to a time-2

φn

0

0 .5

1

0 .5

1

φn−3

φn−2

φn−2

1

1

0 .5

0

Fig. 35.

1

0

φn−2

Graphical composition between a non-Bernoulli time-1 map ρ1

of attractor Λ3 ( 74 ) gives rise to a time-3 Bernoulli στ -shift map

ρ3 74

74

(Λ3 ).

φn−3

(Λ3 ) and a non-Bernoulli time-2 map ρ2

74

(Λ3 )

880


Fig. 36. Graphical composition between the non-Bernoulli time-1 map ρ1 (Λ1 ) (lower-right graph) and the non-Bernoulli 25 time-1 map ρ1 (Λ1 ) (upper-left graph) gives rise to the Bernoulli time-2 map ρ2 (Λ1 ) of attractor Λ1 with β = 2σ , σ = 1, 25 25 and τ = 2.

non-Bernoulli time-1 map29 ρ1 (Λ3 ) : φn−3 → φn−2 (lower-right corner) of attractor Λ3 of 74 in Table 15-2 [Chua et al., 2005a], and a non-Bernoulli time-2 map ρ274 (Λ3 ) : φn−2 → φn (upper-left corner) of the same attractor Λ3 ( 74 ). Observe that the resulting composition shown in the upper-right corner of Fig. 35 is a time-3 Bernoulli στ -shift map with β = 2σ , σ = −3, and τ = 3. In this case, since σ = −3 and τ = 3 the evolution dynamics on attractor Λ3 consists of shifting every third row by three bits to the right. The complete evolution for all n > T∆ + 1 requires that we calculate the next two rows T∆ + 2 and T∆ + 3 via χ174 , or via the truth table of 74 , and then shift rows T∆ + 1, T∆ + 2, and T∆ + 3 sequentially by three bits to the right (since σ < 0).

7.2.1.4. Time-5 map of 25 We end Sec. 7.2.1 with a geometrical interpretation of the three Bernoulli στ -shift attractors of 25 29

Observe that this map is identical to the time-1 map ρ1

74

exhibited in Table 15-1 of [Chua et al., 2005a], and in Table 29A. The time-1 map of attractor Λ1 ( 25 ) in Table 29A is not a Bernoulli map. However, its time-2 map ρ225 (Λ1 ) : φn−2 → φn is a Bernoulli στ -shift map with β = 2σ , σ = −1, and τ = 2 as depicted in Fig. 36. The dynamics on Λ1 ( 25 ) consists therefore of a shift of one bit to the right every second row, as in ρ262 (Λ4 ). The time-2 map of attractor Λ2 ( 25 ) in Table 29A is also not a Bernoulli map (see the upper-left graph of Fig. 37). However, the time3 map ρ325 (Λ2 ) : φn−3 → φn of Λ2 ( 25 ) in Table 29A is a Bernoulli στ -shift map with β = 2σ , σ = 3, and τ = 3, as depicted in the graphical composition between the time-1 map ρ125 (Λ2 ) : φn−3 → φn−2 (lower-right graph) and the time-2 map ρ225 (Λ2 ) : φn−2 → φn (upper-left graph) in Fig. 37. Since σ = 3, the dynamics on the second attractor Λ2 ( 25 ) consists of shifting every third row by three bits to the left. (Λ3 ) : φn−1 → φn .


ρ 325 ( Λ 2 ) : φn −3

φn

φn

1

1

0 .5

0 .5

0

0 .5

φn−2

881

φn

0

0 .5

1

0 .5

1

φn−3

φn−2

φn−2

1

1

0 .5

0

Fig. 37.

1

φn−2

0

Graphical composition between the non-Bernoulli time-1 map ρ1

25

φn−3

(Λ2 ) (lower-right graph) and the non-Bernoulli

time-2 map ρ2 (Λ2 ) (upper-left graph) gives rise to the Bernoulli time-3 map ρ3 (Λ2 ) of attractor Λ1 with β = 2σ , σ = 3, 25 25 and τ = 3.

The time-τ maps for the third attractor Λ3 ( 25 ) in Table 29A of 25 are not Bernoulli for τ = 1, 2, 3, and 4, as depicted in the lower-right graph for the time-1 map ρ125 (Λ3 ) : φn−4 → φn−3 in Fig. 38, in the lower-right graph for the time-2 map ρ225 (Λ3 ) : φn−5 → φn−3 in Fig. 39, in the upper-left graph for the time-3 map ρ325 (Λ3 ) : φn−3 → φn in Fig. 38, and in the upper-right graph for the time-4 map ρ425 (Λ3 ) : φn−4 → φn in Fig. 38.

However, the time-5 map ρ525 (Λ3 ) of attractor Λ3 of 25 in Table 29A is a Bernoulli στ shift map with β = 2σ , σ = 2, and τ = 5. We will demonstrate this by two independent graphical constructions. In Fig. 39, the time-5 Bernoulli map shown in the upper-right corner is obtained by a graphical composition between the time-2 map ρ225 (Λ3 ) : φn−5 → φn−3 (lower-right graph) and

the time-3 map ρ325 (Λ3 ) : φn−3 → φn (upperleft graph). In Fig. 40, the same time-5 Bernoulli map (upper-right graph) is obtained by a graphical composition between the time-1 map ρ125 (Λ3 ) : φn−5 → φn−4 (lower-right graph) and the time-4 map ρ425 (Λ3 ) : φn−4 → φn (upper-left graph)

copied from the upper right graph in Fig. 38. Since β = 2σ , σ = 2, and τ = 5, the global dynamics on Λ3 ( 25 ) consists of a left shift of two bits every fifth iterations. To obtain the complete dynamics for n ≥ T∆ + 1, it is necessary to calculate rows T∆ + 2, T∆ + 3, T∆ + 4, and T∆ + 5 via χ125 , or via the truth table of 25 .

7.2.2. Exact formula for time-2 Bernoulli left-shift map for rule 170 The time-τ Bernoulli στ -shift maps exhibited in Figs. 33–40 are generated by a computer with I > 60. Although all data points shown in the upper-right graphs in these figures appear to fall exactly on the ideal (I → ∞) compositions of Bernoulli maps, this observation is an illusion due to the poor printer resolution, as demonstrated earlier in Fig. 30 for the time-1 map ρ1170 : φn−1 → φn of 170 . Our goal in this section is to derive the exact formula for the time-2 map ρ2170 : φn−2 → φn of 170 analytically from the characteristic function χ1170 .

882


ρ 425 ( Λ3 ) : φn −4

φn

φn

1

1

0 .5

0 .5

0

φn−3

0 .5

φn

0

0 .5

1

0 .5

1

φn−4

φn−3

φn−3

1

1

0 .5

0

Fig. 38.

1

φn−3

0

Graphical composition between the non-Bernoulli time-1 map ρ1

time-3 map

ρ3 5

25

φn−4

(Λ3 ) (lower-right graph) and the non-Bernoulli

(Λ3 ) : φn−3 → φn (upper-left graph) gives rise to the non-Bernoulli time-4 map ρ4

25

right graph).

ρ 525 ( Λ3 ) : φn−5

φn

φn

1

1

0 .5

0 .5

0

φn−3

0 .5

0

(Λ3 ) : φn−4 → φn (upper-

φn

0 .5

1

0 .5

1

φn−5

φn−3

φn−3

1

1

0 .5

0

1

φn−3

0

φn−5

Fig. 39. Graphical composition between the non-Bernoulli time-2 map (lower-right graph) ρ2 (Λ3 ) : φn−5 → φn−3 and the 25 non-Bernoulli time-3 map (upper-left graph) ρ3 (Λ3 ) : φn−3 → φn gives rise to the Bernoulli time-5 map ρ5 (Λ3 ) : φn−5 → 25 25 φn with β = 2σ , σ = 3, and τ = 5.


ρ 525 ( Λ3 ) : φn−5

φn

φn

1

1

0 .5

0 .5

0

φn−4

0 .5

883

φn

0

0 .5

1

0 .5

1

φn−5

φn−4

φn−4

1

1

0 .5

0

1

0

φn−4

φn−5

Fig. 40. Graphical composition between the non-Bernoulli time-1 map (lower-right graph) ρ1 (Λ3 ) : φn−5 → φn−4 and 25 the non-Bernoulli time-4 map (upper-left graph) ρ4 (Λ3 ) : φn−4 → φn gives rise to the Bernoulli time-5 map ρ5 (Λ3 ) : 25 25 φn−5 → φn with β = 2σ , σ = 3, and τ = 5.

Starting from the time-1 characteristic function (n) (n−1) (•) from Eq. (34), we obtain xi = xi+1 for 170

= 22 ·

χ1

j=2

the time-1 map ρ1170 (•) and (n)

xi

(n−2)

= xi+2

x

=

(n−2) (n−2) (n−2) (n−2) [x0 , x1 , x2 , . . . , xI ]

φn =

i=0

=

I+2 j=2

(n) 2−(i+1) xi

=

i=0

(n−2)

2−(j−1) xj

I

,

=4·

I j=0

(61)

1 (n−2) (n−2) − 4 · x1 + 4 · 2−(I+3) xI+2 4

(n−2) 2−(i+1) xi+2

1 (n−2) = 4φn−2 − 4 · [1 − 2−(I+1) ]x0 2 1 (n−2) − 4 · [1 − 2−(I+1) ]x1 4 (n−2)

(62) (n−2)

∈ {0, 1} and x1 ∈ {0, 1}. where x0 (n−2) (n−2) and x1 have four distinct binary Since x0 combinations, we can recast the above time-2 map of 170 into the following explicit form

∆

where j = i + 2

φn = 4φn−2 ,

(n−2)

2−(j+1) xj

1 (n−2) (n−2) − 4 · x0 + 4 · 2−(I+2) xI+1 2

denotes the I + 1 bits of the bit string representing the right-hand side of Eq. (60). The decimal representation of Eq. (60) is given by I

(n−2)

2−(j+1) xj

(60)

for the time-2 map ρ2170 (•), where (n−2)

I+2

(n−2)

(n−2)

if x0

= 0 and x1

= 0,

= 4[φn−2 − 0.25ν(I)],

if

= 0 and

= 1,

= 4[φn−2 − 0.5ν(I)],

if

= 4[φn−2 − 0.75ν(I)],

if

(n−2) x0 (n−2) x0 (n−2) x0

= 1 and = 1 and

(n−2) x1 (n−2) x1 (n−2) x1

= 0, = 1,

(63)

884


where −(I+1)

ν(I) = 1 − 2

,

(64)

henceforth called the Bernoulli constant, depends on the length (I + 1) of the bit string. The above time-2 Bernoulli map ρ2170 : φn−2 → φn of 170 is plotted in Figs. 41(a) (for I = 2), 41(b) (for I = 3), 41(c) (for I = 4), and 41(d) (for I → ∞). The domain and range of the time-2 Bernoulli map depends of the length (I + 1) of the bit string and lies inside the light-yellow region. Observe that ρ2170 (•) is defined by 2I+1 points lying on four uniformly spaced parallel straight lines of slope m = 22 = 4. In the limit where I → ∞, ρ2170 converges to the ideal Bernoulli time-2 map φn = 22 · φn−2

mod 1

(65)

shown in Fig. 41(d). Observe that the time-2 Bernoulli map ρ2170 (•) converges rapidly to the ideal map (65) for I > 10. We cannot overemphasize that the time-2 return map for 170 shown in Fig. 41 for bit strings of length I + 1 = 3, 4, 5, and ∞ are derived analytically from the explicit formula of the function χ1170 in Table 1 of [Chua et al., 2005b]. This seemingly impossible task could not have been achieved without the explicit formula derived in [Chua et al., 2003]. Such analytical results stand in sharp distinction from those presented in [Wolfram, 2002], which are generated empirically via exhaustive, brute force computer simulations. We wish also to emphasize that Fig. 41 is a plot of the exact formula (63) of the time-2 return map of 170 for any initial state that falls on the basin of attraction of 170 . For I + 1 bits, there are 2I+1 distinct bit strings. The locations of all possible bit strings of length I + 1 are shown in Fig. 41. Observe that all of them fall on the graph of Eq. (63) for I + 1 = 3, 4, 5, and ∞. This means that 170 does not have a transient regime,30 thereby proving analytically that every orbit of rule 170 is an invariant orbit. Each orbit of 170 consists of a time sequence (of points consisting) of a proper subset of the 2I+1 points shown in Fig. 41. Indeed, observe that since the points located on the upper-right corner in Figs. 41(a)–41(c) lie on the main diagonal, it is a period-2 orbit of 170 , which is consistent with 30

the fact that φn−1 = 1 is a period-1 point of 170 for I → ∞. Since the time-2 map of 170 from any initial bit string is a subset of the points shown in Fig. 41, these points together actually represent the time-2 characteristic function χ2170 of 170 . We did not label the vertical axis χ2170 because all time-2 return maps of Bernoulli rules with β = 2σ , σ = 2, τ = 2 must also be a proper subset of the four lines (with slope m = 4) shown in Fig. 41. In such cases, Fig. 41 should be interpreted as a “template” showing where ρτN (•) lies, and not as the time-2 characteristic function χ2N of N .

7.2.3. Exact formula for time-2 Bernoulli right-shift map for rule 240 Our goal in this section is to derive the exact formula for the time-2 map ρ2240 : φn−2 → φn of 240 analytically from the characteristic function χ1240 . Starting from the time-1 characteristic function (n) (n−1) 1 χ 240 (•) from Eq. (39), we obtain xi = xi−1 for the time-1 map ρ1240 (•) and (n)

xi

(n−2)

= xi−2

(66) (n−2) xi−2

is defined for the time-2 map ρ2240 (•), where in Eq. (61). The decimal representation of Eq. (66) is given by φn =

I (n) 2−(i+1) xi i=0

I (n−2) = 2−(i+1) xi−2

=

i=0 I−2

(n−2)

2−(j+3) xj

∆

,

where j = i − 2

j=−2

= 2−2

I−2

(n−2)

2−(j+1) xj

j=−2

= 2−2

I (n−2) 2−(j+1) xj j=0

+ 2−2

−1

j=−2

(n−2)

2−(j+1) xj

(a)

Points belonging to the transient regime of any rule N must fall outside of the time-τ return plot ρτN (•) of N since return maps are defined only on attractors.


φn

φn ν(3) = 0.9375

ν(2) = 0.875

1

885

1

15

7

11 7

5

0.75

3

0.75

14

3

10 6

1

0.5

2

0.5

13

6

9 5

4

0.25

1

0.25

12

2

8 4

0

0 0

0.25

0.5

0.75

1

φn-2

0

0.25

0.5

0.75

φn-2

1

φn-2

(b) I = 3 : ν(3) = 0.9375

(a) I = 2 : ν(2) = 0.875

φn

1

0

0

φn

ν(4) = 0.96875

1

ν(οο) = 1

1

6

0.75

0.75

0.5

0.25

0

16

8

0

0

26

18

10

2

0.25

28

20

12

4

0.5

30

22

14

0.25

0.5

24

0.75

(c) I = 4 : ν(4) = 0.96875

1

φn-2

0 0

0.25

0.5

0.75

(d) I = οο: ν (οο) = 1

Fig. 41. The graph of the time-2 Bernoulli map ρ2 : φn−2 → φn for rule 170 for I = 2 in (a), I = 3 in (b), I = 4 in (c), 170 and I → ∞ in (d).

886

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science I

− 2−2

j=I−2+1

−1

=4

term (b), we have changed the index from “j” to

(n−2)

2−(j+1) xj

∆

“I + k + 1” by defining k = j − I − 1. For the last expression, we have changed the index “k” back to “j” for improved clarity. We can recast Eq. (67) in terms of the decimal variable φn−2 as follows:

(b)

I (n−2) 2−(j+1) xj j=0

−1

+4

−1

−4

= 4−1

φn = 4

(n−2) 2−(j+1) xI+j+1 j=−2

+4

j=−2 −1

=4

(n−2)

2−(j+1) xI+j+1 [1 − 2−(I+1) ] (n−2)

φn−2 + 4−1 {2−(−2+1) xI−2+1 (n−2)

(n−2)

+ 2−(−1+1) xI−1+1 }[1 − 2−(I+1) ]

2−(j+1) xj

−1

(n−2)

2−(j+1) xj

−1

−1

(n−2) 2−(k+I+1+1) xI+k+1 k=−2

j=0

I j=0

−1

I

+ 4−1

−1

−1

(68)

Simplifying Eq. (68), we obtain (n−2)

2−(j+1) xI+j+1

(n−2)

− 4−1 • 2−(I+1)

−1

(n−2)

2−(j+1) xI+j+1

(67)

(n−2)

where xI−1

j=−2

(n−2)

∈ {0, 1}, xI ∆

(n−2)

(n−2)

(where j = −1) and x−2 = xI−1 xI (where j = −2). Consequently, we can write (n−2) = xI+j+1 in (a). For the summation xj

}ν(I)

(69)

∈ {0, 1}, and

ν(I) = 1 − 2−(I+1)

Observe that in the above index manipulations in the summation term (a), we have (n−2) = used the periodic boundary conditions x−1 (n−2)

(n−2)

φn = 4−1 φn−2 + 4−1 {2xI−1 + xI

j=−2

(70)

is the same Bernoulli constant defined earlier in Eq. (64) as a function of the length (I +1) of the bit (n−2) (n−2) ) can assume four disstring. Since (xI−1 , xI tinct binary combinations of “0” and “1”, let us recast Eq. (69) into the following explicit formula for the time-2 Bernoulli στ -shift map for rule 240 , with β = 2σ , σ = 2, and τ = 2.

Formula for time-2 Bernoulli map ρ2240 : φn = =

1 φn−2 , 4

if xI−1 = 0 and xI

1 [φn−2 + ν(I)], 4

if xI−1 = 0 and xI

1 = [φn−2 + 2ν(I)], 4 =

1 [φn−2 + 3ν(I)], 4

The above time-2 Bernoulli map ρ2240

(n−2)

(n−2)

(n−2)

(n−2)

(n−2) xI−1

(n−2) xI

=0 =1 (71)

if

(n−2)

= 1 and

(n−2)

if xI−1 = 1 and xI :

φn−2 → φn of rule 240 is plotted in Figs. 42(a) (for I = 2), 42(b) (for I = 3), 42(c) (for I = 4) and 42(d) (for I → ∞). The domain and range of the time-2 Bernoulli map [Eq. (71)] depends on the length (I + 1) of the bit string in Fig. 1, as depicted in a light-yellow background color in

=0 =1

Fig. 42. Observe that ρ2240 (•) is defined by 4 ×

2I+1 points lying on four uniformly-spaced parallel straight lines of slope m = 2−2 = 1/4. Observe also that along each vertical line φn−2 = i × 2−(I+1) , i = 0, 1, . . . , 2(I+1) −1, there are four points 0 , 1 , 2 ..., , labeled by the same integer ,


φn

φn

ν(3) = 0.9375

ν(2) = 0.875

1

0.75

1

0

0.25

1

0

0 0

4

3

2 0.25

0.5

7

6

5

1

0.5

7

6

5

1

0.75

7

6

5

4

3

2

7

6

5

4

3

2

1

0

1

4

3

2

1

0

0.5

887

1

0.25

0.75

1

φn-2

1 0

5

3

5

3

5

3

9

7

9

7

9

7

11

11

11

13

13

13

15

15

15

15

0.25

0.5

0.75

1

φn-2

1

φn-2

(b) I = 3 : ν(3) = 0.9375

φn

ν(4) = 0.96875

30 22

1

3

9

7

13

0

(a) I = 2 : ν(2) = 0.875

φn

5

11

ν(οο) = 1

1

14 6 28 0.75

0.75

20 12 4 26

0.5

0.5

18 10 2 24

0.25

0.25

16 8 0

0 0

0.25

0.5

0.75

(c) I = 4 : ν(4) = 0.96875

1

φn-2

0 0

0.25

0.5

0.75

(d) I = οο: ν (οο) = 1


888

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science (n−1)

since they represent the same point, parameterized by one of the four straight lines to which it belongs.

7.2.4. Exact formulas for generalized Bernoulli maps Following the same analytical procedure as in Secs. 7.2.2 and 7.2.3, a single compact, and allinclusive formula can be derived for the time-τ Bernoulli map for rules 170 and 240 , respectively, for any return time τ = 1, 2, 3, . . ., and any length I + 1 of the bit string in Fig. 1.

= 2φn−1 − x0 ν(I) = 2φn−1 mod ν(I) which is Eq. (36), as expected. In the limit I → ∞, Eq. (76) reduces to: φn = 2φn−1

Substituting τ = 2 in Eq. (72), we obtain (n−2)

(n−2)

j=0

x0 (n−τ )

2−(j+1) xj

(77)

φn = 4φn−2 − 4 ν(I)[2−1 x0

rule 170 φn = 2τ φn−τ − 2τ ν(I)

mod 1

Example A-2. Time-2 Bernoulli Map for 170 .

7.2.4.1. Generalized Bernoulli map for τ −1

(76)

(72)

(n−2)

∈ {0, 1} and x1 = 4φn−2

(n−2)

+ 2−2 x1

],

∈ {0, 1}

mod ν(I)

(78)

which is Eq. (63), as expected. In the limit I → ∞, Eq. (78) reduces to:

where (n−τ )

∈ {0, 1}, τ ∈ {1, 2, 3, . . .}, xj j = 0, 1, 2, . . . , τ − 1,

φn = 4φn−2

mod 1

(79)

and ν(I) = 1 − 2−(I+1)

(73)

Substituting τ = 3 in Eq. (72), we obtain

The exact value of the Bernoulli constant ν(I) is listed in Table 31 as a function of I. By invoking the mod R function defined for any real number R,31 Eq. (72) can be recast into the following ultracompact formula: φn = 2τ φn−τ

mod ν(I)

(74)

Observe that in the limit I → ∞ (i.e. for an infinitely-long bit string), ν(I) → 1 in Eq. (73), and Eq. (74) reduces to the ideal τ th-iterated Bernoulli map: φn = 2τ φn−τ

mod 1

(75)


Substituting τ = 1 in Eq. (72), we obtain (n−1)

φn = 2φn−1 − 2ν(I) · 2−1 x0

31


,

(n−1)

x0

∈ {0, 1}

(n−3)

φn = 8φn−3 − 8ν(I)[2−1 x0 (n−3)

+ 2−3 x2

(n−3) x1

= 8φn−3

(n−3)

], x0

∈ {0, 1} and

(n−3)

+ 2−2 x1

∈ {0, 1},

(n−3) x2

∈ {0, 1}

mod ν(I) (80)

which is Eq. (74), as expected. The time-3 Bernoulli map ρ3170 : φn−3 → φn from Eq. (80) is plotted in Figs. 43(a) (for I = 2), 43(b) (for I = 3), and 43(c) (for I = 4). In the limit I → ∞, Eq. (80) reduces to the ideal time-3 Bernoulli map φn = 8φn−3

mod 1

The graph of Eq. (81) is shown in Fig. 43(d).

The “mod ν(I)” notation is defined as follows: φn = 2τφn−τ ,

= 2τφn−τ − ν(I),

if 2τφn−τ < ν(I)

if 2τφn−τ ≥ ν(I).

The mod ν(I) function in Eq. (74) can be calculated from standard mathematical softwares, such as Mathcad [2000].

(81)


7.2.4.2. Generalized Bernoulli map for rule 240 φn =

2−τ φn−τ

+

2−τ ν(I)

τ −1 j=0

(n−τ )

2j xI−j

(82)

where (n−τ )

τ ∈ {1, 2, 3, · · ·}, xI−j

∈ {0, 1},

j = 0, 1, 2, . . . , τ − 1,

(83)

and ν(I) = 1 − 2−(I+1) Example B-1. Time-1 Bernoulli Map for 240 .

Substituting τ = 1 in Eq. (82), we obtain 1 1 (n−1) φn = φn−1 + ν(I)xI 2 2 (n−1)

where xI expected.

(84)

∈ {0, 1}, which is Eq. (41), as

Example B-2. Time-2 Bernoulli Map for 240 .

Substituting τ = 2 in Eq. (82), we obtain 1 1 (n−2) (n−2) ] φn = φn−2 + ν(I)[2xI−1 + xI 4 4 (n−2)

(n−2)

where xI−1 ∈ {0, 1} and xI Eq. (69), as expected.

(85)

∈ {0, 1}, which is

Example B-3. Time-3 Bernoulli Map for 240 .

Substituting τ = 3 in Eq. (82), we obtain φn =

1 1 (n−3) φn−3 + ν(I)[22 xI−2 8 8 (n−3) + 2x I −1

(n−3)

(86)

(n−3) + xI ]

(n−3)

(n−3)

∈ where xI−2 ∈ {0, 1}, xI−1 ∈ {0, 1} and xI {0, 1}. The time-3 Bernoulli map ρ3240 : φn−3 → φn from Eq. (86) is plotted in Figs. 44(a) (for I = 2), 44(b) (for I = 3), 44(c) (for I = 4), and 44(d) (for I = ∞).

7.2.5. Analytical proof of period-3 “Isle of Eden” Λ2 ( 62 ) with I = 4 as subshift of time-1 Bernoulli στ -shift map (with σ = 1) φn = 2φn−1 mod ν(I) for rule 170 The period-3 “isle of Eden” Λ2 ( 62 ) depicted in Table 1A represents only a possible orbit of 62

889

whose “steady state” dynamics fall precisely on an attractor (or more precisely, an invariant orbit) Λ( 170 ) of the global invariant rule 170 described by the exact time-1 Bernoulli map φn = 2φn−1

mod ν(I)

(87)

derived analytically in Eq. (76). The three plots on the isle of Eden Λ2 ( 62 ) shown in the second part of Table 1A is obtained by a computer for a bit string of 63 bits. They appear to fall on the ideal Bernoulli map φn = 2φn−1 mod 1. However, our preceding rigorous analysis has revealed that this illusion is true only if I → ∞. Moreover, although a period-3 isle of Eden can exist for most bit lengths I + 1, those which fall precisely on the time-1 Bernoulli map defined by Eq. (87) can occur only if I + 1 is divisible by 3, in view of the left-shift law of the time-1 Bernoulli map governing Λ2 ( 62 ). For example, Fig. 5 shows one period-3 isle of Eden for I + 1 = 3 [Fig. 5(a)], five period-3 isles of Eden for I + 1 = 5 [Fig. 5(b)], one period-3 isle of Eden for I + 1 = 6 [Fig. 5(c)], seven period-3 isles of Eden for I + 1 = 7 [Fig. 5(d)], Eight period3 isles of Eden for I + 1 = 8 [Fig. 5(e)], and four period-3 isles of Eden for I + 1 = 9 [Fig. 5(f)]. Note that there is no period-3 isle of Eden for I + 1 = 4. Moreover, among all of these copious period-3 isles of Eden, they are described by a time-1 Bernoulli map only for I + 1 = 3, 6, and 9 in Fig. 5. To demonstrate the significance of the exact generalized Bernoulli map for rule 170 in Eq. (72), we will prove that the period-3 isles of Eden shown in Fig. 5(a) (for I + 1 = 3), Fig. 5(c) (for I + 1 = 6), and (i) of Fig. 5(f) (for I + 1 = 9) obey Eq. (87), and hence this particular period-3 isle of Eden, and only this one, is a subshift of the time-1 Bernoulli στ -shift map with β = 2σ , σ = 1, τ = 1 for I = 2, 5, and 8. Example C. Fig. 7: I = 2. Substituting ν(I) for

I = 2 from Table 31 into Eq. (87), we obtain φn = 2φn−1

mod 0.875

Substituting φ0 = 0.375 (for bit string Eq. (88), we obtain φ1 = 2(0.375) = 0.75

(88) 3

) into

mod 0.875 (89a)

which is bit string 6 in Fig. 7. Substituting φ1 = 0.75 (for bit string 6 ) into Eq. (88), we obtain φ2 = 2(0.75) mod 0.875 = 1.5 − 0.875 = 0.625

(89b)

890


φn

φn

ν(3) = 0.9375

ν(2) = 0.875

1

1

7

0.875

0.875

6

0.75

0

6 4

0.125

0

0

10 8

0.25

1

0.125

14 12

0.375

2

0.25

3 1

0.5

3

0.375

7 5

0.625

4

0.5

11 9

0.75

5

0.625

15 13

0.125 0.25 0.375 0.5 0.625 0.75 0.875

1

φn-3

2

0 0

φn

ν(4) = 0.96875

1 0.875

3

0.75 0.625 0.5 0.375

1

0.25 0.125

0

0 0

9

5

4

8

13

12

17

16

φn-3

1

φn-3

ν(οο) = 1

1 0.875

30

26

22

18

14

10

6

2

15

11

7

31

27

23

19

0.125 0.25 0.375 0.5 0.625 0.75 0.875

(b) I = 3 : ν(3) = 0.9375

(a) I = 2 : ν(2) = 0.875

φn

1

0

0.75 0.625

29

25

21

0.5 0.375

20

24

28

0.125 0.25 0.375 0.5 0.625 0.75 0.875

0.25 0.125

1

φn-3

0 0

0.125 0.25 0.375 0.5 0.625 0.75 0.875

(d) I = οο: ν (οο) = 1

(c) I = 4 : ν(4) = 0.96875


which is bit string 5 in Fig. 7. Substituting φ2 = 0.625 (for bit string 5 ) into Eq. (88), we obtain φ3 = 2(0.625) mod 0.875 = 1.25 − 0.875 = 0.375 which is bit string 3 in Fig. 7.

(89c)

Equations (89a)–(89c) proved that the closed three-bit string 3 → 6 → 5 → 3 is a period-3 isle of Eden which evolves according to the time-1 Bernoulli map (88) for I = 2. Example D. Fig. 10: I = 5. Substituting ν(I) for

I = 5 from Table 31 into Eq. (87), we obtain φn = 2φn−1

mod 0.984375

(90)


φn

φn

ν(3) = 0.9375

ν(2) = 0.875

1

0.625 0.5 0.375

6

7

5

6

7

4

5

6

7

6

7

0.5

4

5

6

7

0.375

3

4

5

6

2

5

7

3

4 4

5

6

2

3

3

4

1

2

3

4

0 0

1

2

0

1

0

1

0

3 3

2 2

1 1

5 5

4

3

2

1

0

0

7

2

0

0.125

6

1

0

0.25

1

0

0.875 0.75

15

9

13

7

11

5

11

15

3

5

9

13

7

15

9

13

1

3

7

11

5

13

15

7

11

1

5

9

3

15

9

13

3

7

11

5

15

9

13

1

3

7

11

5

13

15

7

11

1

5

9

3

15

9

13

3

7

11

5

1

0.75 0.625

1

0.25

7

0.125

0.125 0.25 0.375 0.5 0.625 0.75 0.875

φn

φn-3

1

1 0 0

φn

7 0.875 0.75 0.625 0.5 0.375 0.25

9

1 0.125

8

0

17 16

0.5

26

18

10

2

0.375

25

0.25

24

0.125

φn-3

0 0.125 0.25 0.375 0.5 0.625 0.75 0.875

1

The graph of the time-3 Bernoulli map ρ3

240

and I → ∞ in (d).

mod 0.984375 (91a)

which is bit string 54 in Fig. 10. Substituting φ1 = 0.84375 (for bit string 54 ) into Eq. (90), we obtain φ2 = 2(0.84375)

mod 0.984375

0

0.125 0.25 0.375 0.5 0.625 0.75 0.875

1

φn-3

: φn−3 → φn for rule 240 for I = 2 in (a), I = 3 in (b), I = 4 in (c),

Substituting φ0 = 0.421875 (for bit string 27 ) into Eq. (90), we obtain φ1 = 2(0.421875) = 0.84375

0

(d) I = οο: ν (οο) = 1

(c) I = 4 : ν(4) = 0.96875 Fig. 44.

ν

0.625

27

19

11

3

φn-3

0.75

28

20

12

4

1

0.875

29

21

13

5

ν(οο) = 1

1

30

22

14

6

31

23

15

0.125 0.25 0.375 0.5 0.625 0.75 0.875

(b) I = 3 : ν(3) = 0.9375

ν(4) = 0.96875

1

3

1

0.875

(a) I = 2 : ν(2) = 0.875

0

891

= 1.68750 − 0.984375 = 0.703125

(91b)

which is bit string 45 in Fig. 10. Substituting φ2 = 0.703125 (for bit string 45 ) into Eq. (90), we obtain φ3 = 2(0.703125) mod 0.984375 = 1.40625 − 0.984375 = 0.421875 which is bit string 27 in Fig. 10 .

(91c)

892


Equations (91a)–(91c) proved that the closed three-bit string 27 → 54 → 45 → 27 is a period-3 isle of Eden which evolves according to the time-1 Bernoulli map (90) for I = 5. Substituting ν(I) for I = 8 into the Bernoulli constant formula in Eq. (73), we obtain

Example E. Fig. 14(b): I = 8.

ν(8) = 1 − 2−(8+1) = 0.998046875 Substituting ν(8) into Eq. (87), we obtain φn = 2φn−1

mod 0.998046875

(92)

Substituting φ0 = 0.427734375 (for bit string 219 ) into Eq. (92), we obtain φ1 = 2(0.427734375) = 0.855468750

mod 0.998046875 (93a)

which is bit string 438 in Fig. 14(b). Substituting φ1 = 0.855468750(for bit string 438 ) into Eq. (92), we obtain φ2 = 2(0.855468750) mod 0.998046875 = 1.71093750 − 0.998046875 = 0.712890625 (93b) which is bit string 365 in Fig. 14(b). Substituting φ2 = 0.712890625 (for bit string 365 ) into Eq. (92), we obtain φ3 = 2(0.712890625) mod 0.998046875 = 1.42578125 − 0.998046875 = 0.427734375 (93c) which is bit string 219 in Fig. 14(b). Equations (93a)–(93c) proved that the closed three-bit string 219 → 438 → 365 → 219 is a period-3 isle of Eden which evolves according to the time-1 Bernoulli map (92) for I = 8. As a confirmation of the analytical results derived in Examples C, D and E, observe that the colored sequence of red and blue bits shown besides of the period-3 “isle of Eden” each bit string → → → in Figs. 7, 10 and string 14(b) shift to the left by exactly one bit (σ = 1) to , during each iteration from bit string (τ = 1), as predicted by Theorem 1 of [Chua et al., 2005a]. 32

It is interesting also to observe that under certain special conditions, a period-P Bernoulli attractor32 of a local rule N with finite I may fall precisely on more than one Bernoulli στ -shift maps. For example, the three period-3 “isles of Eden” analyzed in Examples C, D, and E also satisfy a time-2 Bernoulli στ -shift map with β = 2σ , σ = 1 and τ = 2, as can be verified in Figs. 7, 10 and 14(b), where each bit string shifts to the right by one bit (σ = 1) once every two iterations (τ = 2), i.e. string 365 → 438 → 219 → 365 .

7.2.6. Analytical proof of time-2 map of Λ2 ( 62 ) with I = 4 is a subshift of time-2 Bernoulli στ -shift map (with σ = −1) of Eq. (85) of rule 240 Our goal in this section is to prove analytically that the attractor Λ4 ( 62 ) depicted in the last part of Table 1A for rule 62 is a subshift of the time-2 Bernoulli στ -shift map (with β = 2σ , σ = −1, τ = 2) defined by Eq. (85) for rule 240 . To avoid clutter, let us choose the period-5 Bernoulli στ -shift attractor shown in Fig. 9, where β = 2σ , σ = −1, τ = 2. Observe that the 5 bit strings in the period-5 attractor string 7 → 28 → 19 → 14 → 25 → 7 in Fig. 9 evolves according to the following law: Laws governing Bernoulli Attractor Λ4 ( 62 )

Each bit string in Fig. 9 shifts to the right by one bit every second iterations. For example, bit string 19 is obtained by shifting bit string 7 one bit to the right, bit string 25 is obtained by shifting bit string 19 one bit to the right, bit string 28 is obtained by shifting bit string 25 one bit to the right, bit string 14 is obtained by shifting bit string 28 one bit to the right, and finally, bit string is obtained by shifting bit 7 string 14 one bit to the right.

We will now prove that the above laws governing the bit strings of Λ4 ( 62 ) can be obtained by changing the subscript (n−1) to (n−2) in Eq. (41),

For finite I, the iteration under any rule N must become periodic with some period P ≤ 2I+1 .


namely, φn =

=

893

(4) Choose

1 φn−2 , 2 1 φn−2 + 0.484375, 2

(n−2)

if x4

φ6 = 0.875

=0

1 φ8 = (0.875) 2

(lower branch) (n−2)

if x4

=1

= 0.4375

(upper branch) (94)

where we have substituted µ(4) = 0.484375 from Table 31. (1) Let us start with bit string 7 ; i.e. choose φ0 = 0.21875 1 φ2 = (0.21875) + 0.484375 2

(0)

(since x4 = 1)

= 0.59375

(95a)

which is bit string 19 . (2) Choose φ2 = 0.59375 1 φ4 = (0.59375) + 0.484375 2

(2)

(since x4 = 1)

= 0.296875 + 0.484375 = 0.78125

(95b)

which is bit string 25 . (3) Choose φ4 = 0.78125 1 φ6 = (0.78125) + 0.484375 2 = 0.390625 + 0.484375 = 0.875

(4)

(since x4 = 1) (95c)

which is bit string 28 .

Laws governing Bernoulli Attractor Λ4 ( 62 ) with I = 4

(6)

(since x4 = 0) (95d)

which is bit string 14 . (5) Choose φ8 = 0.4375 1 φ10 = (0.4375) 2

(8)

(since x4 = 0)

= 0.21875 which is bit string 7 .

Equations (95a)–(95e) proved that Λ4 ( 62 ) obeys a generalized Bernoulli στ -shift map with β = 2σ , σ = −1, τ = 2. This time-2 Bernoulli map is shown in Fig. 45, along with a Lameray diagram following the loci 7 → 19 → 25 → 28 → 14 → 7 in the φn−2 − φn plane.

7.2.7. Analytical proof of time-1 map of Λ4 ( 62 ) with I = 4 is a subshift of time-1 Bernoulli στ -shift map (with σ = 2) of Eq. (63) of rule 170 A more careful examination of Fig. 9 reveals a rather surprising happenstance; namely, the same attractor Λ4 ( 62 ) considered in the previous section has a “double identity” in the sense that its dynamics can also be predicted from a time-1 Bernoulli στ -shift map with β = 2σ , σ = 2, τ = 1, in accordance with the following laws:

Each bit string in Fig. 9 shifts to the left by two bit once every iteration. For example, bit string 19 is obtained by shifting bit string 28 two bits to the left, bit string 14 is obtained by shifting bit string 19 two bits to the left, bit string 25 is obtained by shifting bit string 14 two bits to the left, bit string 7 is obtained by shifting bit string two bits to the left, and bit string 28 is obtained by shifting bit string 7 two bits to the left. 25

(95e)

894


φn ν (4) = 0.96875 1 25

µ (4) = 0.484375 19 0.75 7 0.5

0.5 28

0.25 14 ν (4) = 0.96875

φn-2

0 0

0.25

0.5

0.75

1

I = 4 : ν(4) = 0.96875 Fig. 45.

Lameray diagram of Λ4 ( 62 ) on a time-2 generalized Bernoulli στ -shift map with β = 2σ , σ = −1, τ = 2.

We will now prove that the above laws governing the bit strings of Λ4 ( 64 ) can be obtained by changing the subscript (n − 2) to (n − 1) in Eq. (63), namely, (n−1)

φn = 4φn−1 ,

= 0 and x1

= 0,

= 4[φn−1 − 0.2421875],

if

= 0 and

= 1,

= 4[φn−1 − 0.484375],

if

= 4[φn−1 − 0.7265625],

if

where we have substituted ν(4) = 0.96875 from Table 31. (1) Let us start with bit string 7 ; i.e. choose φ0 = 0.21875 φ1 = 4(0.21875) = 0.875

(n−1)

if x0

(0)

(0)

(since x0 = 0, x1 = 0) (97a)

which is bit string 28 . (2) Choose φ1 = 0.875

(n−1) x0 (n−1) x0 (n−1) x0

= 1 and = 1 and

(n−1) x1 (n−1) x1 (n−1) x1

(96)

= 0, = 1,

φ2 = 4[0.875 − 0.7265625] (1)

(1)

(since x0 = 1, x1 = 1) = 0.59375 (97b) which is bit string 19 . (3) Choose φ2 = 0.59375 φ3 = 4[0.59375 − 0.484375] (2)

(2)

(since x0 = 1, x1 = 0) = 0.4375 (97c) which is bit string 25 .

895


(4) Choose φ3 = 0.4375 φ4 = 4[0.4375 − 0.2421875] (3)

(3)

(since x0 = 0, x1 = 1) = 0.78125 (97d) which is bit string 25 . (5) Choose

7.3. Λ4 ( 62 ) is a subshift of Λ( 240 )

φ4 = 0.78125 φ5 = 4[0.78125 − 0.7265625] (4)

(4)

(since x0 = 1, x1 = 1) = 0.21875

(97e)

which is bit string 7 . Equations (97a)–(97e) proved that Λ4 ( 62 ) for I = 4 obeys a generalized Bernoulli στ -shift map with β = 2σ , σ = 2, τ = 1. This time-1 Bernoulli map is shown in Fig. 46, along with a Lameray diagram following the loci 7 → 28 → 19 → 14 → → 7 in the φn−1 − φn plane. The basin of attraction of this period-5 attractor Λ4 ( 62 ) for I = 4 consists of the five bit strings { 1 , 2 , 4 , 8 , 16 }, as shown in Fig. 9. Observe that since none of these five bit strings lie on the time-1 Bernoulli map in Fig. 47, their dynamical evolutions do not follow the time-1 Bernoulli στ shift rule, with σ = 2, τ = 1, and must be calculated from the time-1 characteristic function χ1170 . From the Lameray diagram shown in Fig. 47, we see that each bit string converges to the attractor Λ4 ( 62 ) in only one iteration, and hence T∆ = 1. We remark that the double identity nature of Λ4 ( 62 ) is true only for I = 2 and I = 4, for I ≤ 8. We presented it here to demonstrate the exactness of the Generalized Bernoulli maps in Eq. (72), for rule 170 , and Eq. (82), for rule 240 . Although these equations were derived analytically for the two invariant rules 170 and 240 , the same equations apply in the more general context where 2τ in Eq. (73) is generalized to 2σ , and 2−τ in Eq. (82) is generalized to 2−σ , respectively. As demonstrated clearly in Secs. 7.2.6 and 7.2.7 the exponent “σ” need not be equal to τ for other Bernoulli rules, such as these listed in Tables 28A and 29A. 25

33

Finally, we stress that the two Generalized Bernoulli maps (with 2±τ generalized to 2±σ ) in Eqs. (72) and (82) are inverse functions33 of each other. Geometrically, this means that the graphs of these two maps are reflections of each other about the main diagonal, as is evident in Figs. 30–31, Fig. 41–42, and Figs. 43–44.

We have proved in Sec. 7.2.6 that the time-2 attractor Λ4 ( 62 ) of rule 62 falls precisely on an invariant orbit Λ( 240 ) of rule 240 when I + 1 = 5, as shown in Fig. 31. Our goal in this section is to prove analytically that this property holds for all finite I; namely, 1 φn = φn−2 , 2

(n−2)

if xI

1 = φn−2 + µ(I), 2

= 0 (lower branch) (n−2)

if xI

=1

(upper branch) where µ(I) is defined in Table 31. (98) Proof. We have already given a graphical verifica-

tion of Eq. (98) in Fig. 33 where the composition between two identical non-Bernoulli time-1 maps Λ4 ( 62 ) from the last part of Table 1A gives rise to the time-2 Bernoulli στ -shift map (with β = 2σ , σ = −1, τ = 2) in Fig. 33. To derive Eq. (98) analytically for all finite I, we will make use of the time-1 characteristic function χ162 given in Table 2 of [Chua et al., 2005b] to obtain the following explicit closed form relationship: 3 (n−1) (n−1) (n−1) n − |(2 − 2xi−1 − 2xi − xi+1 )| xi = 2 (99) It follows from Eq. (99) that 3 (n−1) (n−2) (n−2) (n−2) − |(2 − 2xi−1 − 2xi = − xi+1 )| xi 2 (100)

Although Figs. 31, 44 and 46 appear to be multivalued relations, they are in fact precisely-defined single-valued functions for finite I, as explained at length in Sec. 5.4 of [Chua et al., 2005a]. Indeed, they are defined uniquely by Eq. (82).

896


φn

ν(4) = 0.96875 0.2421875

0.7265625

0.484375

0.96875

1 7 14 0.75

28 0.5 19 0.25 25 0.7265625

0.484375

0.2421875

φn-1

0 0

0.25

0.5

0.75

1

I = 4 : ν(4) = 0.96875 Fig. 46.

Lameray diagram of Λ4 ( 62 ) on a time-1 generalized Bernoulli στ -shift map with β = 2σ , σ = 2, τ = 1.

φn

ν(4) = 0.96875 0.2421875

0.484375

0.7265625

0.96875

1 7

0.875

8 14

0.78125

16

0.75

1 0.59375

28

0.5

4

0.4375

19

2

0.25

0.21875

25

0 0

0.25

0.125 0.0625 0.25 0.03125

0.5

0.75

1

φn-1

0.5

I = 4 : ν(4) = 0.96875

1 , 2 , 4 Fig. 47. Transient regimes originating from the basin of attraction of Λ4 ( 62 ) consisting of bit strings {, 8 . 16 Note that each initial bit string converges to the time-1 attractor Λ4 ( 62 ) in only one iteration, in complete agreement , with the basin tree of Λ4 ( 62 ), depicted in Fig. 9 for I = 4.


Substituting Eq. (100) into Eq. (99) and rearranging terms, we obtain (n)

xi

3 (n−1) (n−1) − (2 − 2xi−1 − xi+1 ) 2 3 (n−2) (n−2) −2 − |(2 − 2xi−1 − xi+1 2 (n−2) − 2xi )|

=

(101)

Left-Copycat (right-shift) Law

(n)

xi

(n−2)

= xi−1

(102)

for all i = 0, 1, 2, . . . , I. (n) Substituting Eq. (102) for xi in the decimal representation Eq. (2), and simplifying, we obtain φn = =

=

I i=0 I

(n)

2−(i+1) xi

(n−2)

2−(i+1) xi−1

i=0 I−1

j=−1

(n−2)

2−(j+2) xj

,

∆

where j = i − 1

I−1 1 −(j+1) (n−2) 2 xj 2 j=−1   I−1 1 1 (n−2) (n−2)  =  2−(j+1) xj + x−1 2 2 j=0   I 1  −(j+1) (n−2)  = 2 xj 2

=

j=0

1 (n−2) (n−2) + x−1 − 2−(I+2) xI 2

1 1 (n−2) − 2−(I+2) xI = φn−2 + 2 2 from which we obtain Eq. (98).

combinations of these binary variables we have calculated all of them and organized the results in Table 32. Note that the five variables (n−2) (n−2) (n−2) (n−1) (n−1) {xi−1 , xi , xi+1 , xi−1 , xi+1 } in Eq. (101) are listed in columns 2–6, respectively. The value (n) of xi obtained by calculating Eq. (101) for each combination of the five binary variables is shown in ∆ (n) (n−2) Column 7. The difference ∆ = xi −xi−1 between (n)

We will prove below that for all bit-strings residing on the time-2 attractor Λ4 ( 62 ) of rule 62 , Eq. (101) reduces to the following formula

(103)

Proof of the Left-Copycat (Right-shift) Law. (n) The value of xi in Eq. (101) is determined by the (n−2) (n−2) (n−2) (n−1) , xi+1 , xi−1 , five binary variables {xi−1 , xi (n−1) xi+1 }, where each variable can assume either the value “0” or “1”. Since there are 25 = 32

897

(n−2)

column 7 for xi and column 2 for xi−1 of each row in Table 32 is shown in column 8. Observe from Table 32 that there are 17 rows (n) (n−2) where ∆ = 0, and hence xi = xi−1 ; namely, rows {0, 6, 7, 10, 11, 14, 15, 16, 17, 20, 21, 25, 26, 27, 29, 30, 31}. This implies that the three successive rows with color patterns shown in Figs. 48(a) or 48(b) for these rows must evolve according to the left-copycat (right-shift) law stipulated by Eq. (n) (102). In other words, the color of pixel xi is deter(n−2) mined by copying the color of pixel xi−1 of its left neighbor two iterations (n − 2) earlier. Hence, if (n−2) (n) xi−1 is red (resp. blue), then xi must also be red (resp. blue). The left-copycat rule is therefore identical to the movement of a knight in a game of chess, namely, from the “left top” to the “right bottom” position. Recall, however, that the left-copycat (rightshift) rule is a local rule and the global constraints prescribed by Eq. (98) may forbid the presence of some of the 17 possible three-row pattern combinations (those with ∆ = 0) in Table 32. A careful analysis of the laws governing bit strings for attractor Λ1 ( 62 ) in Table 27A reveals that only seven three-row pattern combinations satisfying the left-copycat (right-shift) law (out of 17) in Table 32 can occur in attractor Λ1 ( 62 ); namely, rows {7, 14, 17, 21, 25, 27, 30}. These seven rows are identified in Table 32 via a red color background. Among 32 combinations, 14 combinations, namely, rows {4, 6, 8, 9, 10, 12, 13, 15, 20, 22, 24, 26, 29, 31} are “impossible” combinations. These 14 rows are identified by a blue back(n−2) (n−2) (n−2) and xi+1 from these ground. When xi−1 , xi 14 rows are substituted into Eq. (101), they can(n−1) (n−1) not generate their corresponding xi−1 and xi+1 under any boundary condition, i.e., either “0” (n−2) (n−2) or “1” at xi−2 and xi+2 . For example, for N = 4, (n−2)

xi

(n−1) xi+1

(n−2)

= 0 and xi+1

= 1 cannot give rise to (n−2)

(n−2)

= 0 upon assigning xi+2 = 0 or xi+2 = 1, 2 3 010 and : 011 are firing since the patterns : patterns of rule 62 . All combinations involving

898


∆

(n)

Exact values of ∆ = xi

(n−2)

− xi−1

for 32 distinct combinations of the five binary variables

(n−2) (n−2) (n−2) (n−1) (n−1) (n) {xi−1 , xi , xi+1 , xi−1 , xi+1 }. The value of xi ∈ {0, 1} in column 5 is calculated (n−2) The value of xi−1 ∈ {0, 1} is listed in column 1. The identification number N in column 1 is

the decimal representation of the five-bit binary number.

from Eq. (101). calculated from

Chapter 6: From Time-Reversible Attractors to the Arrow of Time column i-1

column i

column i+1

xi(−n1− 2)

xi( n − 2)

xi(+n1− 2) xi(+n1−1)

xi(−n1−1) xi( n )

column i-1

column i

column i+1

iteration (n-2)

xi(−n1− 2)

xi( n − 2)

xi(+n1− 2)

iteration (n-2)

iteration (n-1)

xi(−n1−1)

xi(+n1−1)

iteration (n-1)

iteration (n)

xi( n )

(a) Fig. 48.

899

Assuming that the colors of the five pixels

iteration (n)

(b) (n−2) (n−2) (n−2) (n−1) (n−1) , xi+1 , xi−1 , xi+1 } {xi−1 , xi

are given by one of the 7 rows

(n)

(with red background) 7, 14, 17, 21, 25, 27 and 30 in Table 32, we find the calculated value of xi shown in column 7 is simply (n−2) (n) (n−2) a copy of the value of the upper left corner xi−1 . In other words, the colors of xi and xi−1 must match each other.

these 14 rows are corrupted in this sense and cannot be generated by the firing patterns of rule 62 . On the other hand, 11 combinations with the first columns colored in “orange”; namely, rows {0, 1, 2, 3, 5, 11, 16, 18, 19, 23, 28} cannot be generated from the laws governing the bit strings for attractor Λ1 ( 62 ) in Table 27A and therefore cannot exist on this attractor, even though they can be generated by rule 62 during the transient regime.

8. The Arrow of Time There are all together 28 Bernoulli στ -shift attractors which are time-irreversible. Among them 20 have two attractors (exhibited in Table 28A) and eight have three attractors (exhibited in Table 29A). The transient regimes of the attractors presented in Tables 28A and 29A are exhibited in Tables 28B and 29B, respectively. Our goal in this section is to provide a higher resolution picture of some of these attractors along with some time reversal tests needed for an in-depth analysis.

8.1. Rule 6 Rule 6 is endowed with two robust noninvertible Bernoulli στ -shift attractors with β = 2σ , σ = ±2, and τ = 2. These attractors are exhibited in Table 28A. Observe that although the time-1 maps of these attractors on the left are not Bernoulli, their associated time-2 maps are Bernoulli. The dynamics of the upper attractor consists of a left shift of two bits (σ = 2) every two iterations (τ = 2). The dynamics of the lower attractor consists of a right shift of two bits (σ = −2) every two iterations (τ = 2).

The “time reversal tests” for these two attractors are shown in Figs. 49(a) and 49(b), respectively. Observe that the presence of “white” pixels at the bottom of these figures confirms that these two attractors are time-irreversible.

8.2. Rule 9 Rule 9 is endowed with two robust noninvertible Bernoulli στ -shift attractors, one with β = 2σ , σ = −2, τ = 2, and the other with β = 2σ , σ = 2, τ = 3. These attractors are exhibited in Table 28A. Observe that the upper Bernoulli στ -shift attractor on the right pertains to a time-2 return map, whereas the lower attractor pertains to a time-3 return map. The dynamics of the upper attractor consists of a right shift of 2 bits (σ = −2) every two iterations (τ = 2). The dynamics of the lower attractor consists of a left shift of two bits (σ = 2) every three iterations (τ = 3). The “time reversal tests” for these two attractors are shown in Figs. 50(a) and 50(b), respectively. Observe that the presence of “white” pixels at the bottom of these figures confirms that these two attractors are time-irreversible.

8.3. Rule 25 Rule 25 is endowed with three robust noninvertible Bernoulli στ -shift attractors: Λ1 ( 25 ) : β = 2σ , σ = −1, τ = 2, Λ2 ( 25 ) : β = 2σ , σ = 3, τ = 3, and Λ3 ( 25 ) : β = 2σ , σ = 2, τ = 5. These attractors are exhibited in Table 29A. The dynamics of Λ1 ( 25 ) consists of a right shift of one bit (σ = −1) every two iterations (τ = 2). The dynamics of Λ2 ( 25 ) consists of a left shift of

900


Row 69

6

Row 19

Row 0 Row 0

20

Row -19

6

20

Row 19

-19

Row 0

0

(a) Fig. 49. (a) Time reversal test for the upper attractor of rule 6 in Table 28A. (b) Time reversal test for the lower attractor of rule 6 in Table 28A.


Row 69

6

Row 19

Row 0 Row 0

20

Row -19

6

20 (b) Fig. 49.

(Continued )

Row 19

-19

Row 0

0

901

902


Row 69

9

Row 19

Row 0 Row 0

65

Row -19

9

65

Row 19

-19

Row 0

0

(a) Fig. 50. (a) Time reversal test for the upper attractor of rule 9 in Table 28A. (b) Time reversal test for the lower attractor of rule 9 in Table 28A.


Row 69

9

Row 19

Row 0 Row 0

65

Row -19

9

65 (b) Fig. 50.

(Continued )

Row 19

-19

Row 0

0

903

904


Row 69

25

Row 19

Row 0 Row 0

67

Row -19

25

67

Row 19

-19

Row 0

0

(a) Fig. 51. (a) Time reversal test for the top attractor of rule 25 in Table 29A. (b) Time reversal test for the middle attractor of rule 25 in Table 29A. (c) Time-reversal test for the bottom attractor of rule 25 in Table 29A.


Row 69

25

Row 19

Row 0 Row 0

67

Row -19

25

67 (b) Fig. 51.

(Continued )

Row 19

-19

Row 0

0

905

906


Row 69

25

Row 19

Row 0 Row 0

67

Row -19

25

67 (c) Fig. 51.

(Continued )

Row 19

-19

Row 0

0


907

Row 69

74

Row 19

Row 0 Row 0

88

Row -19

74

88

Row 19

-19

Row 0

0

(a) Fig. 52. (a) Time reversal test for the top attractor of rule 74 in Table 29A. (b) Time reversal test for the middle attractor of rule 74 in Table 29A. (c) Time reversal test for the bottom attractor of rule 74 in Table 29A.

908


Row 69

74

Row 19

Row 0 Row 0

88

Row -19

74

88 (b) Fig. 52.

(Continued )

Row 19

-19

Row 0

0


Row 69

74

Row 19

Row 0 Row 0

88

Row -19

74

88 (c) Fig. 52.

(Continued )

Row 19

-19

Row 0

0

909

910


three bits (σ = 3) every three iterations (τ = 3). The dynamics of Λ3 ( 25 ) consists of a left shift of two bits (σ = 2) every five iterations (τ = 5). The “time reversal tests” for these three attractors are shown in Figs. 51(a)–51(c), respectively. Observe that the presence of “white” pixels at the bottom of these figures confirms that these three attractors are time-irreversible.

Just like rule 62 , rule 74 is endowed with both time-reversible and time-irreversible attractors.

9. Concluding Remarks 9.1. Attractors of 206 local rules The mathematical analysis presented in Sec. 7 provides the rigorous foundation for Theorem 1, as well as for all of the results from Sec. 5 on “Bernoulli στ -shift rules” in [Chua et al., 2005a]. Our in-depth mathematical analysis in Sec. 7 reveals that none of the attractors exhibited in the Bernoulli στ -shift maps (column 3) in Tables 12–15 in [Chua et al., 2005a] fall exactly on the parallel sloping lines (with slope m = ±2σ ) which were drawn for the ideal Bernoulli στ -shift maps where I → ∞. Although the discrepancies are negligible for I > 60 bits, they are fundamentally incorrect in the sense that none of the examples presented in Sec. 7.2.4 can be derived from the ideal Bernoulli maps, since any such attempts would result in grossly different coordinates in spite of the fact that the difference between the exact Bernoulli maps in Eq. (72) for rule 170 , and Eqs. (82) for rule 240 , is truly infinitesimal!34 Nevertheless, we will always obtain grossly incorrect answers if we use the ideal (mod 1) Bernoulli map to work out the examples in Sec. 7.2.4, no matter how large I is, as long as it is finite. Indeed, this extreme sensitivity phenomenon can be interpreted as a hallmark of spatial chaos. Let us summarize the two fundamental results from this paper in precise mathematical terms:

8.4. Rule 74 Rule 74 is endowed with three robust Bernoulli στ shift attractors: Λ1 ( 74 ) : β = 2σ , σ = 1, τ = 1, Λ2 ( 74 ): β = 2σ , σ = 2, τ = 2, and Λ3 ( 74 ): β = 2σ , σ = −3, τ = 3. The dynamics of Λ1 ( 74 ) consists of the most common case of a left shift of one bit (σ = 1) for each iteration (τ = 1). The dynamics of Λ2 ( 74 ) consists of a left shift of two bits (σ = 2) every two iterations (τ = 2). The dynamics of Λ3 ( 74 ) consists of a right shift of three bits (σ = −3) every three iterations (τ = 3). The “time reversal test” for these three attractors is shown in Figs. 52(a)–52(c), respectively. Observe that unlike the other attractors presented so far in this section, there are no “white” pixels at the bottom of Fig. 52(a). This implies that attractor Λ1 ( 74 ) is actually time-reversible. The other two attractors Λ2 ( 74 ) and Λ3 ( 74 ), however, are time-irreversible, as confirmed by the presence of “white” pixels at the bottom of Figs. 52(b) and 52(c).

Main Result 1: The Bernoulli Copycat Law The local dynamics of all 112 Bernoulli στ -shift maps listed in Table 30 evolves according to the compact local formula (for β = 2σ > 0) (n−τ )

xni = xi+σ

(104)

Depending on whether β = 2σ , or β = 2−σ , σ > 0, we can recast Eq. (104) as follow: Right-Copycat (Left-Shift) Law

(n−τ )

σ>0

(104a)

(n−τ )

σ 0) σ > 0 : φn = 2σ φn−τ ,

mod ν(I)

τ −1 1 1 (n−τ ) 2j xI−j σ < 0 : φn = |σ| φn−τ + |σ| ν(I) 2 2 j=0

(105) where ν(I) = 1 − 2−(I+1) ,

(106)

or by Eqs. (45) and (48) if β = −2−σ < 0. It is truly remarkable that the time-asymptotic (steady-state) dynamics of almost half (112 out of 256) of all local rules evolves according to one ultracompact and exact formula. Moreover, this formula implies the evolving spatial pattern is a travelling wave with a velocity equal to σ/τ . In addition, we have also completely characterized the steady-state dynamics of 69 period-1 rules and 25 period-2 rules. The four globally-equivalent local rules 62 , 118 , 131 and 145 are endowed with both a robust period-3 attractor, and a time-2 Bernoulli attractor with σ = −1. All together, our in-depth analysis has completely characterized the time asymptotic dynamics (i.e. attractors or invariant orbits) of 112 + 69 + 25 = 206 local rules (the four globally equivalent period-3 rules { 62 , 118 , 131 , 145 } are already included in our list of Bernoulli rules35 in Table 27A. There remains only 50 local rules (18 noninvertible, bilateral, and time-irrevertible rules and 32 noninvertible, nonbilateral, and time-irreversible rules) listed in Tables 17 and 18, respectively in [Chua et al., 2005a] which have not yet been given an in-depth investigation. Among these 50 local rules yet to be characterized in future research, only 10 bilateral rules { 18 , 22 , 54 , 73 , 90 , 105 , 122 , 126 , 146 , 150 } from Table 17, and 8 nonbilateral rules { 26 , 30 , 41 , 45 , 60 , 106 , 110 , 154 } from Table 18, warrant further research since the remaining 32 rules are globally equivalent to one of these 18 rules. 35

911

Although most of the examples and formulas on Bernoulli στ -shift maps presented in this paper are derived from the invariant rules 170 and 240 , or from the local rule 62 , the same analytical procedure can be applied to the other Bernoulli rules as well, including rules 25 and 74 . Indeed, the approach we developed in this paper represents a paradigm shift in research on Cellular Automata, which has hitherto been either empirical, as in [Wolfram, 2002], or highly abstract, as in automata theory. Our approach is both analytical and constructive, made possible by our discovery of an explicit unified formula for time-τ characteristic functions, which was derived from an associated nonlinear differential equation, or a nonlinear difference equation from [Chua et al., 2003]. Our discovery that the attractors of all Bernoulli rules are subshifts of corresponding Bernoulli στ shift maps of the four invariant rules 170 , 240 , 85 , and 15 allows us to analyze almost half of the 256 local rules from a unified perspective via the well-developed theory of nonlinear dynamics. Moreover, our approach provides an analytical bridge between cellular automata and nonlinear difference equations. Since the dynamics on an attractor of any one of the 112 local rules endowed with a Bernoulli στ -shift map consists of shifting the initial bit string either to the left (if σ > 0), or to the right (if σ < 0), by |σ| bits every τ iterations, it follows that as I → ∞, the iterates from any random bit string must visit the vicinity of any point on an attractor infinitely often. In other words, the orbits on any Bernoulli στ -shift attractor is essentially Ergodic [Cornfeld et al., 1982], as I → ∞.

9.2. Time reversality Another novel contribution of our analytical theory of cellular automata is the rigorous clarification of the concept of invertibility and time-reversibility in terms of attractors of local rules. The hitherto unsuccessful attempts by many researchers to view time-reversible cellular automata as an extension of the well-known time reversal theory from physics [Sachs, 1987] is doomed to fail in view of the ubiquitous presence of multiple attractors in cellular automata. Since the same local rule, such as rule 62 , is endowed with a time-reversible period-3 isle

The two Bernoulli attractors Λ1 ( 62 ) and Λ2 ( 62 ) of 62 in Table 27A are labeled differently from those in Table 1A. In particular, Λ1 ( 62 ) and Λ2 ( 62 ) in this table are labeled as Λ4 ( 62 ) and Λ2 ( 62 ), respectively, in Table 1A.

912


of Eden [e.g. Fig. 5(a)] and many time-irreversible period-3 isles of Eden (e.g. Fig. 5), as well as numerous period-3 attractors (e.g. Fig. 6) which are not time-reversible, any meaningful theory of timereversality for cellular automata must be couched in terms of attractors. Our theory allows us to classify each attractor of a local rule as either timereversible or time-irreversible. An easy time-reversal test 36 has been developed to carry out such tests efficiently on a computer. As a bonus, our timereversal test can also be used to identify the end of the transient regime from any initial bit string, for any local rules. There are 170 local rules, out of 256, which harbor time-reversible attractors. They include all 69 period-1 rules (29 are bilateral and invertible, 16 are nonbilateral and invertible and 24 are nonbilateral and noninvertible), 17 period-2 rules (all are invertible and bilateral), and 84 Bernoulli στ -shift rules (all are invertible and nonbilateral). A fundamental characteristic of a time-reversible attractor Λ( N ) of a local rule N is that the past of any orbit on Λ( N ) can be uniquely retrieved by iterat∆ ing its bilateral twin rule N † = T † (N ), in forward time. Conversely, the past of N † can be uniquely retrieved by iterating N in forward time. In other words, the notion of a “past” and a “future” time is entirely relative, as in Einstein’s theory of relatively [Davies, 1995]. Many counterintuitive phenomena on timeirreversality on cellular automata have emerged from our theory. For example, a periodic attractor need not be time-reversible! Indeed, there are eight period-2 rules which are time-irreversible. There are 28 Bernoulli στ -shift rules (which are periodic for finite I) which are also time-irreversible. There are many “isles of Eden” (e.g. Fig. 5) which are time irreversible. When viewed as a time-3 fixed point, all of the period-3 “isles of Eden” of rule 62 exhibited in Fig. 5 have neither a past nor a future. Time simply stood still for such invariant orbits, and has lost its meaning [Davies, 1995]! It is even more intriguing to observe that for each time-reversible attractor Λ( N ) of a local rule N , its associated bilateral twin attractor Λ† ( N ) T † (Λ( N )) is in fact a time machine [Hawking, 2005; Davies, 2003; Wells, 1895], the stuff of science fiction, which has attracted increasing 36

attention among many avant garde physicists and cosmologists since its existence does not violate the laws of physics [Thorne, 1994; Novikov, 1998]. In the universe of cellular automata, one can identify many concrete concepts and examples which mimic esoteric concepts and phenomena from quantum electrodynamics and relativity theory. For example, the left-right transform T † in cellular automata is analogous to the parity operator P in quantum field theory, the global complementation T in cellular automata is analogous to the particle–antiparticle reversal operator C in quantum field theory, and the left-right complementation T ∗ in cellular automata is analogous to the simultaneous left-right and particle–antiparticle operator (CP mirror) in quantum field theory. Finally, the forward and backward transformations Tn ( N ) and T−n ( N ) under rule N in cellular automata is analogous to the time-reversal operator T in quantum field theory. At a metaphorical level, one can associate “isles of Eden” from cellular automata with the “frozen time” phenomenon on black holes [Thorne, 1994; Novikov, 1998] from cosmology. One can even mimic the pair annihilation and production of matter and antimatter in particle physics by appropriate choice of initial states from time-reversible rules. For example, Appendix A-2(a) shows the collision of two double-stream of red pixels (mimicking an electron track) on the right with a symmetrical double-stream of blue pixels (mimicking a positron track) on the left by rule 184 , thereby annihilating each other, resulting in a checkerboard pattern (mimicking the emission of gamma radiation). By applying an excitation (simulated by pixels enclosed with the green rectangle) to an otherwise checkerboard pattern above it (mimicking the physical vacuum) by the associated bilateral twin rule 226 = T † ( 184 ), we find the spontaneous generation of a double-stream of red pixels on the right (mimicking an electron) and a symmetrical double-stream of blue pixels (mimicking a positron) on the left. Note the strong resemblance of the electron-positron annihilation and generation process depicted by the corresponding Feynman diagrams shown on the right of each space-time pattern. Observe that a minimum of two adjacent red (resp. blue) pixels are needed to emulate an

We often dubbed this test as the Litmus test: If some pixels at the bottom rows of the “time-reversal test pattern” turns white, then N is time-irreversible.


electron (resp. positron) in Appendix A-2 in order to distinguish it from the checkerboard background (emulating the vacuum condensate 0|e+ e− |0 from quantum physics). If we identify rules 184 and 226 as the same rule, via their global equivalence, then one can even think of the positron as an electron traveling backwards in time [Feynman, 1949]! We can also interpret the red stream from (a) and blue stream from (b) in Table A-2 as an electron-electron scattering. Another remarkable phenomenon exhibited by the bilateral twin rules 184 and 226 = T † ( 184 ) is that their respective transient regimes are timereversible, a property which we have so far restricted to evolutions on attractors. Indeed, the upper space-time diagram in Appendix A-2(a) ending at the middle horizontal line segment of the small green rectangle represents a transient regime of 184 . Similarly, the lower space-time diagram in Appendix A-2(b), beginning from the middle horizontal line segment of the small green rectangle also represents a transient regime of 226 . Observe that these two transient regimes satisfy the timereversal test, which was hitherto applicable only to space-time diagrams originating from attractors. In other words, the two transient regimes of 184 and 226 depicted in Appendix A-2 are time machines [Davies, 1995, 2003; Novikov, 1998] — the past of 184 is the future of 226 , and vice versa, over the corresponding duration of the transient regimes. It follows that the future and the past times cannot be defined absolutely, as in Einstein’s theory of relativity. As a further metaphor, we can imagine a contracting “toy” universe, as depicted in Appendix A-2(a), and an expanding “toy” universe, as depicted in Appendix A-2(b). Here, the singular point where the expansion begins is analogous to the “Big Bang” event from cosmology. There are two additional Bernoulli rules which exhibit the same space-time diagram as that of the 184 in Table A-2; namely, rules 56 and 57 . However, there is a subtle difference between the T ∗ T T † transformation (an element of the Vierergruppe from Fig. 17 of [Chua et al., 2004]) of rule 56 and that of rules 184 and 57 ; namely, T ∗ ( 56 ) = 185 = 56 but T ∗ ( 184 ) = 184 and T ∗ ( 57 ) = 57 . In other words, whereas both rules 184 and 57 possess the T T † = T † T = T ∗ symmetry, in 37

913

contrast, rule 57 violates the T symmetry. It follows that even though T † ( 184 ) = 184 , T † ( 57 ) = 57 , T ( 184 ) = 184 , and T ( 57 ) = 57 , the pair annihilation and pair production process of both rules 184 and 57 mimic the CP -symmetry property from classical quantum electrodynamics, where the parity operator P is analogous to T † ∈ , and the particle–antiparticle conjugation operator C is analogous to T ∈ [Chua et al., 2004]. In contrast, rule 56 mimics the “CP -violation” phenomenon observed from the subatomic particle Kaon [Davies, 1995]. The metaphors we have drawn from quantum field theory illustrate that in the universe of cellular automata, there are examples of simple, concrete, nonlinear dynamical systems which exhibit identical phenomena, thereby demonstrating that such phenomena are neither counter intuitive nor strange.

9.3. Paradigm shift via nonlinear dynamics Our proof of the “global equivalence theorem” in [Chua et al., 2004] based on Klein’s Vierergruppe is a major breakthrough becuase it allows us out of 256, among which Translating these results to other members of each equivalence class is a cinch, as demonstrated in Sec. 6.4.8. The truth table for each of the 256 local rules can be found in Table 5 of [Chua et al., 2003]. Rather than postulating these local rules as God given laws, we take the point of view that they are attractors of a family of 256 distinct nonlinear differential (resp. difference) equations [Chua et al., 2002]. The parameters defining the nonlinear differential equation for each of the 256 local rules are listed in Table 2 of [Chua et al., 2002] where ui−1 , ui , ui+1 ∈ {−1, 1}. The parameters defining the nonlinear difference equation for each of the 256 local rules are listed in Table 5 of [Chua et al., 2003], where ui−1 , ui , ui+1 ∈ {−1, 1}. For convenience of future research, we have collected these equations in Table A-3 using the conventional symbolic variables xi−1 , xi , xi+1 ∈ {0, 1}. We have also provided Tables A-4–A-8 for cross-referencing the locations where these rules have been cited.

We took this opportunity to delete row ε136 from Table 2 of [Chua et al., 2004] which has already appeared as ε122 , and relabelling the last 3 rows accordingly.

914


Since many local rules are endowed with multiple attractors exhibiting qualitatively different topological and dynamic structures, it does not make sense, from a nonlinear dynamics perspective, to pigeonhole the 256 local rules into four “taxonomic” classes, ` a la [Wolfram, 2002]. It is more illuminating to classify not local rules, but attractors, or invariant orbits, within each local rule. From a nonlinear dynamics perspective, each attractor associated with the 70 local rules we have characterized in this paper can be classified uniquely into a period-1, period-2, period-3, or Bernoulli στ -shift attractor. In particular, each Bernoulli attractor is parameterized by three integers (β, σ, τ ) where ∆

β = ±2σ . A Bernoulli στ -shift attractor with β > 0 simply implements a left shift (if σ > 0) or a right shift (if σ < 0) of |σ| bits every τ iterations. If β < 0, the same operation is followed by complementation (i.e. changing color of all bits) during each operation (i.e. every τ iterations). Among the 112 Bernoulli rules, 44 have β = −2σ < 0; they are members of 14 equivalence classes; namely, (see column 4 of Table 16 of [Chua et al., 2005a]) { 11 , 14 , 15 , 43 , 47 , 81 , 84 , 85 , 113 , 117 , 142 , 143 , 212 , 213 }. Since all of these 14 equivalence classes of Bernoulli rules with β < 0 have |σ| = 1 and τ = 1, their dynamics are described by the exact formula for rule 85 in Eq. (45) if σ = 1, or for rule 15 in Eq. (48) if σ = −1, and depicted in Fig. 27. It remains a mystery on why only time-1 Bernoulli maps have negative values for β. This paper culminates our research over the past three years into developing a rigorous nonlinear dynamics foundation [Chua, 1969; Shilnikov et al., 1998; Shilnikov et al., 2001] for a paradigm shift from an empirical-based approach a la [Wolfram, 2002], to an attractor-based analytical theory of cellular automata where all results are proved analytically and all formulas are exact and derived in closed form, for all one-dimensional Cellular Automata with a periodic boundary condition, as depicted in Fig. 1(a). It would be relatively easy to develop an analogous theory for a metric space made of one or two-sided infinite (I → ∞) bit strings by defining an appropriate metric as in [Devaney, 1992]. The induced topological space is totally-disconnected, perfect, and compact, and hence homeomorphic to the Cantor space [Kurka, 2003]. The local rules belonging to each of the 88 equivalence classes

given in Tables 2–4 of [Chua et al., 2004] can be proved to be topologically-conjugate of each other by exhibiting appropriate homemorphisms between pairs of local rules belonging to each equivalence class. Our Bernoulli στ -shift results can also be recast into a concise mathematical format and generalized significantly. However, in keeping with the self-contained and pedagogical nature of this tutorial-review paper, we have deliberately refrained from invoking various powerful tools from topological and symbolic dynamics [Blanchard et al., 2000; Kurka, 2003], which could have compressed these voluminous 6-part series into a single concise research paper. Our over-riding goal is to introduce cellular automata from the perspective of nonlinear dynamics for the lay readers unfamiliar with cellular automata. This paper also provides an intriguing bridge for research in neural science on pattern formation [Chua, 1998] and recognition [Chua & Roska, 2002], as well as in systems biology on gene regulation networks [Kitano, 2001]. There are common concepts in these areas which can be mapped onto attractors of cellular automata, and vice-versa. For example, each attractor Λ( N ) is hardwired to recognize only a small subset of local “firing” and “quenching” patterns [Chua et al., 2003]. The excluded patterns are henceforth called inactive local patterns. They too can include “firing” and “quenching” patterns. For example, for the period-3 “isle of Eden” Λ2 ( 62 ) in Tables 27A-3 and 27B-8, the active “fir] and 5 ( ), ing” patterns consists of 3 ( and the active “quenching” patterns consists of only ). The inactive “firing” patterns consist 6 ( ), 2 ( ), and 4 ( ), and the of 1 ( ) inactive “quenching” patterns consist of 0 ( ). Similarly, for the Bernoulli attracand 7 ( tor Λ1 ( 62 ) in Tables 27A-3 and 27B-8, the active ), 3 ( ), “firing” patterns consist of 1 ( ), and, and the active “quenching” and 4 ( ) and 7 ( ). patterns consist of 6 ( ) The inactive “firing” pattern consists of 2 ( and the inactive “quenching” pattern consists of 0 ( ). From a neural networks perspective, an active “firing” local pattern can be identified with an “excitatory” synapse and a “quenching” local pattern can be identified with an “inhibitory” synapse [Gerstner & Kistler, 2002; Chua & Roska, 2002].


From a gene regulatory systems perspective, an inactive “firing” local pattern can be identified with an unexpressed gene for an “excitatory” protein molecule, and an inactive “quenching “ local pattern can be identified with an unexpressed gene for an “inhibitory” protein molecule. Just as the dynamical mechanisms leading to a time-2 left-copycat (right-shift) attractor Λ1 ( 62 ) can be explained and predicted rigorously via the leftcopycat (right-shift) law presented in Sec. 7.3, and derived in Eq. (102), by invoking only the active “firing” and “quenching” patterns of Λ1 ( 62 ), so too can attractors associated with neural or gene regulatory networks be explained, and predicted, at least at a conceptual level. We close this paper with a resolution of the often-posed paradox on how the right-copycat rule 170 can be associated with a “coin toss” experiment, or “random walks”, as depicted in the “orbit unfolding plot” in Table 24A-4, when its characterderived izing Bernoulli map in Eq. (36) is deterministic and predictable! This paradox is due to the fallacious inference asserting that the Bernoulli “coin toss” interpretation implies that rule 170 can be used as a practical “random sequence generator”. It cannot. The Bernoulli map merely asserts that if one applies a random binary (0) (0) (0) (0) (0) sequence x0 = {x0 , x1 , x2 , . . . , xk−1 , xk , . . .} of “0”s and “1”s to a cellular automaton machine which implements rule 170 , the machine will spit out the same sequence as its output by shifting each pixel of x0 by one bit to its left — hence the name “left-shift law ”! However, although the output of rule 170 is indeed a random binary sequence, it is not generated by rule 170 . But rather it is given at the outset as its input. To the skeptic who had remained unimpressed by the triviality of rule 170 and who still wonders at what all the fuss is about rule 170 , it is important to realize that if one applies an initial state to any autonomous dynamical system, whether continuous or discrete, the solution will in general consist of a transient and a steady state component, where the steady state would normally converge to an equilibrium state, unless the system is locally active [Chua, 2005]. Indeed, all closed systems are destined to the same fate in view of the second law of thermodynamics. 38

915

Such systems tend to average out any randomness inherent in the initial state and must tend to a thermodynamic equilibrium, completely oblivious of its distant initial state. Such system has a fading memory [Boyd & Chua, 1985] and therefore cannot exhibit any complexity [Chua, 2005]. Viewed from this context, rule 170 , as well as all 112 Bernoulli στ -shift rules, are truly remarkable because they are endowed with an infinite memory in the sense that no bit on the initial state is ever averaged out in its dynamic evolution, no matter how far it reaches into the future. This is true for all 112 rules, including those with such a deceptively simple subshift as the local rule 2 in Table 5. This seemingly preposterous assertion follows from just one simple rule:38

Fig. 53. Scaled reproduction of a painting from page 53 (Fig. 27, 256 Farben, 1974/84) of [Kunstsammlung, 2005] (permission for reproduction granted by Studio Richter, March 31, 2006).

For τ > 0, the bit strings x1 , x2 , . . . , xτ −1 at n = 1, 2, . . . , τ − 1 must be calculated from the initial bit string x0 via the characteristic function χ1N given in Table 1 of [Chua et al., 2005b], or equivalently, from the truth table of local rule N . The value of {σ, τ } for each of 112 Bernoulli στ -shift rules is given in Table 19 of [Chua et al., 2005a] and in Tables 30-2 and 30-3. The 44 rules with β < 0 belong to the 14 equivalence classes listed in an earlier paragraph of this section.

916


With this deceptively simple yet profound recipe, one can, in principle, easily “cook” out random binary strings residing on any designated attractor of any one of the 112 Bernoulli στ -shift rules by following the rule’s Constitution (i.e. the laws governing the attractor bit strings) on the attractor, as stipulated in Tables 24A–29A for each attractor. We close this paper with a painting in Fig. 53, entitled “256 Farben”, by Gerhard Richter

[Kunstsammlung, 2005] from the Museum of Modern Art in New York city. Each of these 256 distinctly colored pixels epitomizes a unique “mini-universe” whose constituent attractors evolve according to definite bit-string laws. There are 256 parallel mini-universes, among them, 170 are endowed with at least one time-reversible attractor where time is relative in the sense that ”. “

Appendix The abscissa and its associated bit string (I + 1 = 63 bits) of each bar in the time-1 characteristic functions χ1N . There are 201 bars starting from bar no.

Table A-1.

n = 1 with decimal value φ(1) = 0 + 2−63 to no. n = 201 with decimal value φ(201) = 1.00 − 2−63 . The value of φ(n) in column 2 is rounded up to three decimal digits. The last column 62 is assumed to be “1” if n is “even”, and “0” if n is “odd ”. In the characteristic functions χ1N given in Table 2 of [Chua et al., 2005b], as well as in Tables 12, 19 and 22 (for χτN , τ = 1, 2, 3), all bars with an “even” n are coded in “red ”, and all with “odd ” n are coded in “blue”. The distance between each pair of adjacent red and blue bars is ∆φ = 0.005. n φ (n )

0

1

2

3

4

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9

1

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0.005

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0.01

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1

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 1

4

0.015

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6

0.025

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7

0.03

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8

0.035

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9

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0.07

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917

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0.085

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0.09

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0.095

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0

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

1

1

0

0

0

0

0

0

1

22

0.105

0

0

0

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

1

1

0

0

0

0

0

0

0

23

0.11

0

0

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

1

1

0

0

0

0

0

0

1

24

0.115

0

0

0

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

1

1

0

0

0

0

0

0

0

25

0.12

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

1

1

0

0

0

0

0

0

1

26

0.125

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

27

0.13

0

0

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

0

1

0

0

0

0

0

0

0

1

28

0.135

0

0

1

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

0

0

1

0

0

0

0

0

0

0

0

29

0.14

0

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

1

0

1

0

0

0

0

0

0

0

1

30

0.145

0

0

1

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

0

0

0

1

0

0

0

0

0

0

0

0

31

0.15

0

0

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

1

0

1

0

0

0

0

0

0

0

1

32

0.155

0

0

1

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

1

0

0

1

0

0

0

0

0

0

0

0

33

0.16

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

1

0

1

0

0

0

0

0

0

0

1

34

0.165

0

0

1

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

0

0

0

0

1

0

0

0

0

0

0

0

0

35

0.17

0

0

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

1

0

1

0

0

0

0

0

0

0

1

36

0.175

0

0

1

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

1

0

0

1

0

0

0

0

0

0

0

0

37

0.18

0

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

1

0

1

0

0

0

0

0

0

0

1

38

0.185

0

0

1

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

1

0

0

0

1

0

0

0

0

0

0

0

0

39

0.19

0

0

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

1

0

1

0

0

0

0

0

0

0

1

40

0.195

0

0

1

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

1

0

0

1

0

0

0

0

0

0

0

0

41

0.2

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

1

0

1

0

0

0

0

0

0

0

1

42

0.205

0

0

1

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

0

0

0

0

0

1

0

0

0

0

0

0

0

0

43

0.21

0

0

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

1

0

1

0

0

0

0

0

0

0

1

44

0.215

0

0

1

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

1

0

0

1

0

0

0

0

0

0

0

0

45

0.22

0

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

1

0

1

0

0

0

0

0

0

0

1

46

0.225

0

0

1

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

1

0

0

0

1

0

0

0

0

0

0

0

0

47

0.23

0

0

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

1

0

1

0

0

0

0

0

0

0

1

48

0.235

0

0

1

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

1

0

0

1

0

0

0

0

0

0

0

0

49

0.24

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

1

0

1

0

0

0

0

0

0

0

1

50

0.245

0

0

1

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

1

0

0

0

0

1

0

0

0

0

0

0

0

0

Table A-1.

(Continued )

918

n φ (n)

0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62

51

0.25

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

1

52

0.255

0

1

0

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

1

0

0

0

0

0

0

0

0

0

0

0

53

0.26

0

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

1

0

0

0

0

0

0

0

0

0

1

54

0.265

0

1

0

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

1

0

0

0

0

0

0

0

0

0

0

0

0

55

0.27

0

1

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

0

1

0

0

0

0

0

0

0

0

0

1

56

0.275

0

1

0

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

1

0

0

0

0

0

0

0

0

0

0

0

57

0.28

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

1

1

0

0

0

0

0

0

0

0

0

1

58

0.285

0

1

0

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

59

0.29

0

1

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

0

0

1

0

0

0

0

0

0

0

0

0

1

60

0.295

0

1

0

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

61

0.3

0

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

1

1

0

0

0

0

0

0

0

0

0

1

62

0.305

0

1

0

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

63

0.31

0

1

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

1

0

1

0

0

0

0

0

0

0

0

0

1

64

0.315

0

1

0

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

1

0

0

0

0

0

0

0

0

0

0

0

65

0.32

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

1

1

0

0

0

0

0

0

0

0

0

1

66

0.325

0

1

0

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

67

0.33

0

1

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

0

0

0

1

0

0

0

0

0

0

0

0

0

1

68

0.335

0

1

0

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

1

0

0

0

0

0

0

0

0

0

0

0

69

0.34

0

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

1

1

0

0

0

0

0

0

0

0

0

1

70

0.345

0

1

0

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

1

0

0

0

0

0

0

0

0

0

0

0

0

71

0.35

0

1

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

1

0

1

0

0

0

0

0

0

0

0

0

1

72

0.355

0

1

0

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

1

0

0

0

0

0

0

0

0

0

0

0

73

0.36

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

1

1

0

0

0

0

0

0

0

0

0

1

74

0.365

0

1

0

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

75

0.37

0

1

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

1

0

0

1

0

0

0

0

0

0

0

0

0

1

76

0.375

0

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

77

0.38

0

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

1

1

0

0

0

0

0

0

0

0

0

1

78

0.385

0

1

1

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

1

0

0

0

0

0

0

0

0

0

0

0

0

79

0.39

0

1

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

1

0

1

0

0

0

0

0

0

0

0

0

1

80

0.395

0

1

1

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

1

0

0

0

0

0

0

0

0

0

0

0

81

0.4

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

1

1

0

0

0

0

0

0

0

0

0

1

82

0.405

0

1

1

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

83

0.41

0

1

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

0

0

0

0

1

0

0

0

0

0

0

0

0

0

1

84

0.415

0

1

1

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

0

1

0

0

0

0

0

0

0

0

0

0

0

85

0.42

0

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

1

1

0

0

0

0

0

0

0

0

0

1

86

0.425

0

1

1

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

1

0

0

0

0

0

0

0

0

0

0

0

0

87

0.43

0

1

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

1

0

1

0

0

0

0

0

0

0

0

0

1

88

0.435

0

1

1

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

1

1

0

0

0

0

0

0

0

0

0

0

0

89

0.44

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

1

1

0

0

0

0

0

0

0

0

0

1

90

0.445

0

1

1

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

91

0.45

0

1

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

1

0

0

1

0

0

0

0

0

0

0

0

0

1

92

0.455

0

1

1

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

93

0.46

0

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

1

1

0

0

0

0

0

0

0

0

0

1

94

0.465

0

1

1

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

95

0.47

0

1

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

1

0

1

0

0

0

0

0

0

0

0

0

1

96

0.475

0

1

1

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

1

1

0

0

0

0

0

0

0

0

0

0

0

97

0.48

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

1

1

0

0

0

0

0

0

0

0

0

1

98

0.485

0

1

1

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

99

0.49

0

1

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

1

0

0

0

1

0

0

0

0

0

0

0

0

0

1

100

0.495

0

1

1

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

1

0

1

0

0

0

0

0

0

0

0

0

0

0

Table A-1. n φ (n)

(Continued )

0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62

101

0.5

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

1

0

0

0

0

0

0

0

0

0

102

0.505

1

0

0

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

1

0

0

0

0

0

0

0

0

0

0

0

0

103

0.51

1

0

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0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

1

1

0

0

0

0

0

0

0

0

0

0

0

145

0.72

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

1

1

1

0

0

0

0

0

0

0

0

0

1

146

0.725

1

0

1

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

147

0.73

1

0

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

1

0

0

0

0

1

0

0

0

0

0

0

0

0

0

1

148

0.735

1

0

1

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

1

0

1

0

0

0

0

0

0

0

0

0

0

0

149

0.74

1

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

1

0

0

1

1

0

0

0

0

0

0

0

0

0

1

150

0.745

1

0

1

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

1

1

0

0

0

0

0

0

0

0

0

0

0

0

Table A-1. n φ (n)

(Continued )

920

0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62

151

0.75

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

1

0

0

0

0

0

0

0

0

0

152

0.755

1

1

0

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

1

1

0

0

0

0

0

0

0

0

0

0

0

153

0.76

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

1

1

1

0

0

0

0

0

0

0

0

0

1

154

0.765

1

1

0

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

155

0.77

1

1

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

1

0

0

1

0

0

0

0

0

0

0

0

0

1

156

0.775

1

1

0

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

157

0.78

1

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

1

0

1

1

0

0

0

0

0

0

0

0

0

1

158

0.785

1

1

0

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

159

0.79

1

1

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

1

0

1

0

0

0

0

0

0

0

0

0

1

160

0.795

1

1

0

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

1

1

0

0

0

0

0

0

0

0

0

0

0

161

0.8

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

1

1

1

0

0

0

0

0

0

0

0

0

1

162

0.805

1

1

0

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

163

0.81

1

1

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

1

0

0

0

1

0

0

0

0

0

0

0

0

0

1

164

0.815

1

1

0

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

1

0

1

0

0

0

0

0

0

0

0

0

0

0

165

0.82

1

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

0

0

0

0

1

1

0

0

0

0

0

0

0

0

0

1

166

0.825

1

1

0

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

1

1

0

0

0

0

0

0

0

0

0

0

0

0

167

0.83

1

1

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

0

1

0

1

0

0

0

0

0

0

0

0

0

1

168

0.835

1

1

0

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

1

1

0

0

0

0

0

0

0

0

0

0

0

169

0.84

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

1

1

1

0

0

0

0

0

0

0

0

0

1

170

0.845

1

1

0

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

171

0.85

1

1

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

1

0

0

1

0

0

0

0

0

0

0

0

0

1

172

0.855

1

1

0

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

1

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

173

0.86

1

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

1

0

1

1

0

0

0

0

0

0

0

0

0

1

174

0.865

1

1

0

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

1

0

1

0

0

0

0

0

0

0

0

0

0

0

0

175

0.87

1

1

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

1

1

0

1

0

0

0

0

0

0

0

0

0

1

176

0.875

1

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

1

0

0

0

0

0

0

0

0

0

0

0

177

0.88

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

1

1

1

0

0

0

0

0

0

0

0

0

1

178

0.885

1

1

1

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

179

0.89

1

1

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

1

180

0.895

1

1

1

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

1

0

1

0

0

0

0

0

0

0

0

0

0

0

181

0.9

1

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

1

0

0

1

1

0

0

0

0

0

0

0

0

0

1

182

0.905

1

1

1

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

1

1

0

0

0

0

0

0

0

0

0

0

0

0

183

0.91

1

1

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

0

0

1

0

1

0

0

0

0

0

0

0

0

0

1

184

0.915

1

1

1

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

1

1

0

0

0

0

0

0

0

0

0

0

0

185

0.92

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

1

1

1

0

0

0

0

0

0

0

0

0

1

186

0.925

1

1

1

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

187

0.93

1

1

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

1

0

0

1

0

0

0

0

0

0

0

0

0

1

188

0.935

1

1

1

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

1

0

0

1

0

0

0

0

0

0

0

0

0

0

0

189

0.94

1

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

1

0

1

1

0

0

0

0

0

0

0

0

0

1

190

0.945

1

1

1

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

191

0.95

1

1

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

1

1

0

1

0

0

0

0

0

0

0

0

0

1

192

0.955

1

1

1

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

1

0

1

1

0

0

0

0

0

0

0

0

0

0

0

193

0.96

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

1

1

1

0

0

0

0

0

0

0

0

0

1

194

0.965

1

1

1

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

195

0.97

1

1

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

1

0

0

0

1

0

0

0

0

0

0

0

0

0

1

196

0.975

1

1

1

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

1

0

1

0

0

0

0

0

0

0

0

0

0

0

197

0.98

1

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

1

0

0

0

1

1

0

0

0

0

0

0

0

0

0

1

198

0.985

1

1

1

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

1

1

0

0

0

0

0

0

0

0

0

0

0

0

199

0.99

1

1

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

1

0

1

0

1

0

0

0

0

0

0

0

0

0

1

200

0.995

1

1

1

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

0

1

1

1

0

0

0

0

1

0

1

0

0

0

1

1

1

1

0

1

1

1

1

0

0

0

0

0

0

0

0

0

0

0

201

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

Chapter 6: From Time-Reversible Attractors to the Arrow of Time Table A-2. Space-time diagrams depicting evolutions of two globally-equivalent bilateral twin rules 184 and 226 . The red (resp. blue) pixel in the top right (resp. left) corner in (a) mimicks the initiation of a particle track and an anti-particle track along two intersecting diagonal directions, and annihilating each other upon collision. The red (resp. blue) perturbation pixel depicted in the bottom row of the green rectangle in (b) propagates along a red (resp. blue) diagonal track, mimicking the production of a particle–antiparticle pair. Since time increases in a downward direction in both space-time diagrams in (a) and (b), the blue arrow in both Feynman diagrams can be interpreted as an electron travelling backward in time. The checkerboard background made of alternating red and blue pixels mimicks the physical vacuum composed of a condensate of various particle pairs such as leptons–antileptons, quarks–antiquarks, etc.

Matter - Antimatter Pair Annihilation by Rule 184

Matter - Antimatter Production by Rule 226

921

922


n n Table A-3. Parameters defining the difference equation of rule N in terms of symbolic variables xn i−1 , xi , xi+1 ∈ {0, 1} with complexity index κ:

Chapter 6: From Time-Reversible Attractors to the Arrow of Time Table A-3.

(Continued )

923

924

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table A-3.

(Continued )

Chapter 6: From Time-Reversible Attractors to the Arrow of Time Table A-3.

(Continued )

925

Table A-4.

926

n

N

κ

1

0

1

2

4

1

3

8

1

4

12

1

5

13

1

6

32

1

7

36

2

8

40

2

9

44

2

10

64

1

11

68

1

12

69

1

13

72

2

14

76

1

15

77

1

16

78

3

17

79

1

18

92

3

19

93

1

20

94

2

21

96

2

22

100

2

23

104

2

24

128

1

25

132

2

26

133

2

27

136

1

28

140

1

29

141

3

30

160

1

31

164

2

32

168

1

33

172

3

34

192

1

35

196

1

TimeRever- Invertible sible

ε mκ ε11 ε 51 ε 81 ε111 ε121 ε171 ε112 ε142 ε162 ε 81 ε111 ε121 ε 232 1 ε 24 1 ε 25 ε 53 ε121 ε 53 ε121 ε 272 ε142 ε162 ε 282 1 ε 26 ε 352 ε 272 1 ε 27 1 ε 29 ε 53 1 ε 31 ε 412 1 ε 33 ε 83 1 ε 27 1 ε 29

T (N ) T (N ) T * (N ) N = N N = N N = N *

0

255 255

4

223 223

64

239 253

68

207 221

69

79

32

251 251

36

219 219

Cross index on 69 period-1 rules.

Figure Index Parts I, II, , VI

Table Index Parts I, II, , V

Table Index Part VI

Section Index Part VI

IV-12 IV-14 ; V-33 IV-12

I-9,15 ; II-11,13 ; III-2,6 ; IV-3,5 ; V-3,4,7,10 I-9,15; II-11,13; III-2,6 ; IV-3 ; V-9 I-9,13,15 ; II-11,13; III-2,5 ; IV-3,5 ; V-10 I-9,13,15 ; II-11,13 ; III-2,5 ; IV-4 ; V-9 I-9,13,15 ; II-11,13 ; III-2,5 ; IV-4 ; V-9 I-9,15 ; II-11,13 ; III-2,6 ; IV-5 ; V-9 I-10,16,17 ; II-11,13 ; III-3,6 ; V-10 I-10,13,16,17 ; II-11,13 ; III-3,5 ; IV-5 ; V-9 I-10,13,16,17 ; II-11,13 ; III-3,5 ; IV-4 I-9,13 ; II-11,13 ; III-2,5 ; IV-5 ; V-10 I-9,13 ; II-11,13 ; III-2,5 ; IV-4 ; V-7,9 I-9,13 ; II-11,13 ; III-2,5 ; IV-4 I-10,16,17 ; II-11,13 ; III-3,6 ; V-9 I-9,15 ; II-11,13 ; III-2,6 ; V-9 I-9,15 ; II-11,13 ; III-2,7 ; V-9 I-11,13,16 ; II-9,11,13 ; III-4,5,27 ; IV-4 ; V-10 I-9,13,15 ; II-11,13 ; III-2,5 ; IV-4 ; V-9 I-11,13,17 ; II-12,14 ; III-4,5,27 ; IV-4 ; V-10 I-9,13 ; II-12,14 ; III-2,5 ; IV-4 ; V-9 I-10,16,17 ; II-10,12,14 ; III-3,5,30 ; V-10 I-10,13,17 ; II-12,14 ; III-3,5 ; IV-5 ; V-9 I-10,13,17 ; II-11,13 ; III-3,5 ; IV-4 ; V-9 I-10,16,17 ; II-11,13 ; III-3,6 ; V-9 I-9 ; II-11,13 ; III-2,6 ; IV-5 ; V-9 I-10,17 ; II-11,13 ; III-3,6 ; V-9 I-10,17 ; II-11,13 ; III-3,6 ; V-10 I-9,13 ; II-11,13 ; III-2,5 ; IV-5 ; V-7,9 I-9,13 ; II-11,13 ; III-2,5 ; IV-4 ; V-9 I-11,13,17 ; II-11,13 ; III-4,5,27 ; IV-4 ; V-10 I-9 ; II-11,13 ; III-2,6 ; IV-5 ; V-9 I-10,17 ; II-11,13 ; III-3,6 ; V-10 I-9,13 ; II-11,13 ; III-2,5 ; IV-5 ; V-9 I-11,13,17 ; II-11,13 ; III-4,5,27 ; IV-4 ; V-9 I-9,13 ; II-11,13 ; III-2,5 ; IV-5 ; V-9 I-9,13 ; II-11,13 ; III-2,5 ; IV-4 ; V-9

2, 3, 9 2, 3, 10, 12, 15,16 2, 3, 9 2, 11,12,13,14,15,16 2, 11, 12, 15 , 16 2, 3, 9, 12 2, 3, 10, 15 , 16 2, 3, 9 2, 11, 12, 15 , 16 2, 3 2, 15 2, 15 2, 3, 10, 12, 15 , 16 2, 3, 10, 12, 15 , 16 2, 3, 10, 12, 15 ,16 2, 11, 12, 15 , 16 2, 15 , 16 2, 15 , 16 2, 15 , 16 2, 3, 10 2, 3 2, 12, 15 , 16 2, 3, 10, 12, 15 ,16 2, 3, 9 2, 3, 10, 12, 15 , 16 2, 3 2, 3, 9 2, 11, 12, 15 , 16 2, 15 , 16 2, 3, 9 2, 3, 10, 12, 15,16 2, 3, 9 2, 11, 12, 15 , 16 2, 3 2, 15 , 16

6.1 6.1 6.1 4.2, 6.1 4.2, 6.1 6.1 6.1 6.1 4.2, 6.1 6.1 4.2, 6.1 4.2, 6.1 6.1 6.1 6.1 4.2, 6.1 6.1 4.2, 6.1 6.1 6.1 6.1 4.2, 6.1 6.1 6.1 6.1 6.1 6.1 4.2, 6.1 6.1 6.1 6.1 6.1 4.2, 6.1 6.1 4.2, 6.1

93

96

235 249

100

203 217

8

253 239

12

221 207

13

93

72

237 237

76

205 205

IV-12 IV-12 IV-14 IV-12

79

77

77

92

141 197

77

93

13

78

197 141

IV-14 IV-14

69

79

69

94

133 133

13

40

249 235

44

217 203

104

233 233

128

254 254

132

222 222

133

94

192

238 252

196

206 220

IV-12

IV-3,12

94

197

78

160

250 250

164

218 218

224

234 248

228

202 216

136

252 238

140

220 206

IV-12

92

IV-12 IV-12 IV-14 IV-12

Table A-4.

927

n

N

κ

36

197

3

37

200

1

38

202

3

39

203

2

40

204

1

41

205

1

42

206

1

43

207

1

44

216

3

45

217

2

46

218

2

47

219

2

48

220

1

49

221

1

50

222

2

51

223

1

52

224

1

53

228

3

54

232

1

55

233

2

56

234

1

57

235

2

58

236

1

59

237

2

60

238

1

61

239

1

62

248

1

63

249

2

64

250

1

65

251

1

66

252

1

67

253

1

68

254

1

69

255

1

TimeRever- Invertible sible

ε mκ ε 53 1 ε 37 ε 83 ε162 1 ε 38 1 ε 24 1 ε 29 ε111 ε 83 ε162 ε 412 ε112 1 ε 29 ε111 ε 352 ε 51 1 ε 33 3 ε8 1 ε 39 ε 282 1 ε 33 2 ε14 1 ε 37 ε 232 1 ε 27 ε 81 1 ε 33 ε142 1 ε 31 ε171 1 ε 27 ε 81 1 ε 26 ε11

T (N ) T (N ) T * (N ) N = N N = N N = N *

141

92

200

236 236

216

172 228

78

217

44

204

204 204

205

76

220

140 196

76

12

202

228 172

203

100

218

164 164

219

36

206

68

196 140

44 36

207

68

222

132 132 4

12 4

168

248 234

172

216 202

232

232 232

233

104 104

248

168 224

249

40

236

200 200

96

237

72

252

136 192

253 234

8

64

224 168 96

250

160 160

40

251

32

238

192 136

239

64

254

128 128 0

IV-12 I-13,14 ; III-2 IV-14 IV-13 IV-13

72

235

255

IV-3,10

100

221

223

Figure Index Parts I, II, , VI

32 8 0

IV-13 IV-13 IV-13 IV-13 I-15,16 ; IV-13 IV-13 IV-13 IV-13 IV-13 IV-13

(Continued )

Table Index Parts I, II, , V

Table Index Part VI


I-11,13,17 ; II-11,13 ; III-4,5,27 ; IV-4 ; V-10 I-9 ; II-11,13 ; III-2,6 ; V-9 I-11,13,17 ; II-8,11,13 ; III-4,5,27 ; IV-4 ; V-9 I-10,13,17 ; II-11,13 ; III-3,5,30 ; IV-4 ; V-9 I-9 ; II-11,13 ; III-2,8 ; V-3,4,7,9 I-9 ; II-11,13 ; III-2,8 ; V-9 I-9,13 ; II-9,11,13 ; III-2,5 ; IV-4 ; V-9 I-9,13 ; II-11,13 ; III-2,5 ; IV-4 ; V-9 I-11,13,17 ; II-12,14 ; III-4,5,27 ; IV-4 ; V-9 I-10,13,17 ; II-12,14 ; III-3,5,30 ; IV-4 ; V-9 I-10,17 ; II-12,14 ; III-3,8,30 ; V-10 I-10,17 ; II-12,14 ; III-3,8 ; V-10 I-9,13 ; II-12,14 ; III-2,5 ; IV-4 ; V-9 I-9,13 ; II-12,14 ; III-2,5 ; IV-4 ; V-7,9 I-10,17 ; II-10,12,14 ; III-3,8 ; V-9 I-9 ; II-12,14 ; III-2,8 ; V-9 I-9,13 ; II-11,13 ; III-2,5 ; IV-5 ; V-9 I-11,13,17 ; II-11,13 ; III-4,5,27 ; IV-4 ; V-9 I-7,9,17 ; II-11,13 ; III-2,8 ; V-9 I-10,17 ; II-11,13 ; III-3,8 ; V-9 I-9,13 ; II-8,11,13 ; III-2,5 ; IV-6 ; V-9 I-10,13,17 ; II-11,13 ; III-3,5 ; IV-6 ; V-9 I-9 ; II-11,13 ; III-2,8 ; V-9 I-10,17 ; II-11,13 ; III-3,8 ; V-9 I-9,13 ; II-9,11,13 ; III-2,5 ; IV-6 ; V-7,9 I-9,13 ; II-12,14 ; III-2,5 ; IV-6 ; V-10 I-9,13 ; II-12,14 ; III-2,5 ; IV-6 ; V-9 I-10,13,17 ; II-12,14 ; III-3,5 ; IV-6 ; V-9 I-8,9 ; II-12,14 ; III-2,8 ; IV-6 ; V-9 I-9 ; II-12,14 ; III-2,8 ; IV-6 ; V-9 I-9,13 ; II-12,14 ; III-2,5 ; IV-6 ; V-9 I-9,13 ; II-12,14 ; III-2,5 ; IV-6 ; V-10 I-9 ; II-10,12,14 ; III-2,8 ; IV-6 ; V-9 I-9 ; II-12,14 ; III-2,8 ; IV-6 ; V-3,4,7,10

2, 15 , 16 2, 3, 10, 12, 15,16 2, 15 , 16 2, 12, 15 , 16 2, 3, 10, 12, 15,16 2, 3, 15 , 16 2, 15 , 16 2, 15 , 16 2, 15 , 16 2, 12, 15 , 16 2, 3, 15 , 16 2, 3, 15 , 16 2, 15 , 16 2, 15 , 16 2, 3, 15 , 16 2, 3, 15 , 16 2, 3 2, 3, 15 , 16 2, 10, 12, 15, 16 2, 3, 15 , 16 2, 3 2, 3 2, 3, 15 , 16 2, 3, 15 , 16 2, 3 2, 3 2, 3 2, 3 2, 3 2, 3 2, 3 2, 3 2, 3 2, 3

6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 4.2, 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1 6.1

Table A-5. TimeRever- Invertible sible

928

n

N

κ

1

1

1

Yes

Yes

2

5

1

Yes

Yes

3

19

1

Yes

Yes

4

23

1

Yes

Yes

5

28

2

No

No

6

29

3

No

No

7

33

2

Yes

Yes

8

37

2

Yes

Yes

9

50

1

Yes

Yes

10

51

1

Yes

Yes

11

55

1

Yes

Yes

12

70

2

No

No

13

71

3

No

No

14

91

2

Yes

Yes

15

95

1

Yes

Yes

16

108

2

Yes

Yes

17

123

2

Yes

Yes

18

127

1

Yes

Yes

19

156

2

No

No

20

157

2

No

No

21

178

1

Yes

Yes

22

179

1

Yes

Yes

23

198

2

No

No

24

199

2

No

No

25

201

2

Yes

Yes

ε mκ ε 21 ε 61 ε151 ε161 ε 82 ε 23 ε102 ε122 1 ε 22 1 ε 23 ε151 ε 82 ε 23 ε122 ε 61 ε 302 ε102 ε 21 ε 402 ε 82 1 ε 35 1 ε 22 ε 402 ε 82 ε 302

T † (N ) T (N ) T * (N ) N = N † N = N N = N *

1

127 127

Yes

No

No

5

95

95

Yes

No

No

19

55

55

Yes

No

No

23

23

23

Yes

Yes

Yes

70

199 157

No

No

No

71

71

29

No

No

Yes

33

123 123

Yes

No

No

37

91

91

Yes

No

No

50

179 179

Yes

No

No

51

51

51

Yes

Yes

Yes

55

19

19

Yes

No

No

28

157 199

No

No

No

29

29

71

No

No

Yes

91

37

37

Yes

No

No

95

5

5

Yes

No

No

108 201 201

Yes

No

No

123

33

33

Yes

No

No

127

1

1

Yes

No

No

198 198 156

No

No

Yes

199

28

No

No

No

178 178 178

Yes

No

Yes

179

50

Yes

No

No

156 156 198

No

No

Yes

157

70

No

No

No

201 108 108

Yes

No

No

70 50 28

Figure Index Parts I, II, …, VI

IV-15 ; VI-19 IV-15 ; VI-20 IV-15

IV-6,10,15

I-21 ; IV-15

VI-21

IV-15

Table A-6.

n

N κ

Time-Rever -sible 1

1

62

2

3

4

ε mκ

Invertible 1

2

3

Cross index on 25 period-2 rules.

T † (N ) T (N ) T * (N ) N = N † N = N N = N *

Table Index Parts I, II, …, V

Table Index Part VI


I-9,5 ; II-11,13 ; III-2,6,8 ; IV-7 ; V-10 I-9,5 ; II-11,13 ; III-2,6,8 ; IV-7 ; V-10 I-9,5 ; II-12,14 ; III-2,6,8 ; IV-7 ; V-9 I-9,5 ; II-12,14 ; III-2,7,8,9,10,11,12 ; IV-7 ; V-9 I-10,13,16,17 ; II-12,14 ; III-3,5 ; IV-8 ; V-10 I-10,11,13,16,17 ; II-12,14 ; III-4,6,10,27 ; IV-8 ; V-9 I-10,16,17 ; II-11,13 ; III-3,6,8 ; IV-7 ; V-9 I-10,16,17 ; II-11,13 ; III-3,6,8 ; IV-7 ; V-10 I-9,5 ; II-12,14 ; III-2,6,8 ; IV-7 ; V-9 I-9,5 ; II-12,14 ; III-2,7,8,9,10,11,12 ; IV-7 ; V-3,4,7,9 I-9,5 ; II-12,14 ; III-2,6,8 ; IV-7 ; V-9 I-10,13,17 ; II-9,11,13 ; III-3,5 ; IV-8 ; V-10 I-10,11,13,17 ; II-11,13 ; III-4,6,10,27 ; IV-8 ; V-9 I-10,16,17 ; II-12,14 ; III-3,6,8,30 ; IV-7 ; V-10 I-9,15 ; II-12,14 ; III-2,6,8 ; IV-7 ; V-10 I-10,16,17 ; II-11,13 ; III-3,6,8 ; IV-7 ; V-9 I-10,17 ; II-12,14 ; III-3,6,8 ; IV-7 ; V-9 I-9,15 ; II-12,14 ; III-2,6,8 ; IV-7 ; V-10 I-10,13,17 ; II-12,14 ; III-3,6,10 ; IV-8 ; V-9 I-10,13,17 ; II-12,14 ; III-3,5,30 ; IV-8 ; V-10 I-9 ; II-12,14 ; III-2,7,8,9,10,11,12 ; IV-7 ; V-9 I-9 ; II-12,14 ; III-2,6,8 ; IV-7 ; V-9 I-10,13,17 ; II-9,11,13 ; III-3,6,10 ; IV-8 ; V-9 I-10,13,17 ; II-11,13 ; III-3,5,30 ; IV-8 ; V-10 I-10,17 ; II-11,13 ; III-3,6,8 ; IV-7 ; V-9

2,3,17,19,20,21 2,3,17,19,20,21 2,3,17,19,20,21 2,3,17,19,20,21 18,19,20,21 18,19,20,21 2,3,17,19,20,21 2,3,17,19,20,21 2,3,17,19,20,21 2,3,17,19,20,21 2,3,20,21 19,20,21 20,21 2,3,20,21 2,3,20,21 2,3,17,19,20,21 2,3,20,21 2,3,20,21 18,19,20,21 19,20,21 2,3,17,19,20,21 2,3,20,21 20,21 19,20,21 2,3,20,21

6.2 6.2 6.2 6.2 4.3, 6.2 4.3, 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 4.3, 6.2 6.2 6.2 6.2 6.2 6.2 6.2

Cross index on four period-3 rules. Figure Index Parts I, II, …, VI


Table Index Part VI


4

2 Y Y N Y Y Y N Y

2

118 2 Y Y N Y Y Y N Y

3

131 2 Y Y N Y Y Y N Y

4

145 2 Y Y N Y Y Y N Y

ε 222

118

131 145

N

N

N

IV-7,8,10,11,16 ; VI- 2–18, 29(h),33,45-47

I-10,13,16,17 ; II-10,12,14 ; III-3,5,30 ; IV-9,10,13,16 ; V-9

1,2,3,22,23,27

2,3,4,6.3,7.2

ε ε ε

62

145 131

N

N

N

IV-11,16 ; VI- 15-18

I-10,13,17 ; II-12,14 ; III-3,5,30 ; IV-9,10,13,16 ; V-9

1,2,3,22,23,27

2,3,4,6.3,7.2

145

62

118

N

N

N

IV-16

I-10,13,17 ; II-11,13 ; III-3,5 ; IV-9,10,13,16 ; V-9

1,2,3,22,23,27

2,3,4,6.3,7.2

131

118

62

N

N

N

IV-16

I-10,13,17 ; II-12,14 ; III-3,5 ; IV-9,10,13,16 ; V-9

1,2,3,22,23,27

2,3,4,6.3,7.2

2 22 2 22 2 22

Table A-7. n

N

κ

TimeInverRevertible sible

1

2

1

Yes Yes

ε 31

16

191

247

No

No

Yes Yes

ε

17

63

119

No

20

159

215

21

31

65

2

3

1

3

6

2

No

4

7

1

Yes Yes

5

9

2

No

6

10

1

Yes Yes

No

No

ε ε 71 2 1

ε ε 91

2 2

7

11

1

Yes Yes

8

14

1

Yes Yes

ε ε131

Yes Yes

ε

9

15

1

1 10

1 15

10 16

1

Yes Yes

11 17

1

Yes Yes

ε ε

No

ε

12 20

2

No

929

13 21

1

Yes Yes

14 24

2

Yes Yes

1 3 1 4 2 1

ε ε 52 1 7

15 25

2

No

No

16 27

3

No

No

17 31

1

Yes Yes

ε ε13 ε 71

Yes Yes

ε

Yes Yes

ε

18 34 19 35

1 1

20 38

2

No

No

21 39

3

No

No

22 42

1

Yes Yes

23 43

1

Yes Yes

24 46

3

Yes Yes

2 6

1 18 1 19

ε132 ε13 ε 1 ε 21 ε 33 1 20

25 47

1

Yes Yes

26 48

1

Yes Yes

ε ε181

Yes Yes No

27 49 28 52

1 2

No



Table Index Part VI


No

V-7,8,17,32 ; VI-23

I-9,13,15 ; II-8,11,13 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1,6.4,7.1

No

No

IV-18 ; V-9,10,18 ; VI-28

I-9,13,15 ; II-11,13 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,6,26,30

5.3,6.4

No

No

No

IV-18 ; V-9 ; VI-48

I-10,14,16,17 ; II-9,11,13 ; III-3,5 ; IV-11,14,16 ; V-9

28,30

6.4,8

87

No

No

No

V-9 ; VI-28

I-9,13,15 ; II-11,13 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,6,26,30

5.3,6.4

111

125

No

No

No

IV-18,19 ; VI-49

I-10,14,16,17 ; II-11,13 ; III-3,5 ; IV-11,14,16 ; V-10

28,30

6.4,8

80

175

245

No

No

No

V-19 ; VI-23

I-9,13,15 ; II-8,11,13 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1,6.4,7.1

81

47

117

No

No

No

IV-17 ; V-20 ; VI-29

I-9,13,15 ; II-11,13 ; III-2,5 ; IV-10,13,16 ; V-10

2,3,7,27,30

5.4,6.4

84

143

213

No

No

No

IV-21,22,23 ; V-9 ; VI-29

I-9,13,15 ; II-9,11,13 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,7,27,30

5.4,6.4

85

15

85

No

Yes

No

V-9 ; VI-25,27

I-9,13,15 ; II-11,13 ; III-2,5,9 ; IV-10,13,16 ; V-3,4,9

2,3,5,25,30

5.1,5.2,6.4,7

2

247

191

No

No

No

V-31

I-9,13 ; II-11,13 ; III-2,6 ; IV-10,13,16 ; V-9

2,3,5,30

5.1

3

119

63

No

No

No

V-21

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-7,9

2,3,6,30

5.3

6

215

159

No

No

No

I-17 ; V-9

I-10,14,17 ; II-12,14 ; III-3,5 ; IV-11,14,16 ; V-9

30

7

87

31

No

No

No

V-9

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,6,30

5.3

66

231

189

No

No

No

VI-23

I-10,14,16,17 ; II-12,14 ; III-3,5 ; IV-10,13,16 ; V-10

2,3,5,30

5.1,6.4,7.1

67

103

61

No

No

No

IV-19 ; VI-36,37,38,39,40,50

I-10,14,16,17 ; II-12,14 ; III-3,5 ; IV-12,15,16 ; V-10

29,30

6.4,7.2,8

83

39

53

No

No

No

I-10,11,14,16,17 ; II-12,14 ; III-4,5 ; IV-11,14,16 ; V-9

28,30

6.4,8

87

7

21

No

No

No

V-9

I-9,13,15 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,6,30

5.3

48

187

243

No

No

No

VI-23

I-9,13,15 ; II-8,11,13 ; III-2,5 ; IV-10,13,16 ; V-7,9

2,3,5,30

5.1,6.4,7.1

49

59

115

No

No

No

VI-29

I-9,13,15 ; II-11,13 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,8,27,30

5.4,6.4,7.1

52

155

211

No

No

No

I-10,14,16,17 ; II-9,11,13 ; III-3,5 ; IV-11,14,16 ; V-10

28,30

6.4,8

53

27

83

No

No

No

I-10,11,14,16,17 ; II-11,13 ; III-4,5 ; IV-11,14,16 ; V-9

30

6.4

112

171

241

No

No

No

VI-23

I-9,13,15 ; II-8,11,13 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,5,27,30

5.1,6.4,7.1

113

43

113

No

Yes

No

VI-29

I-9,13,15 ; II-11,13 ; III-2,6,9 ; IV-10,13,16 ; V-9

2,3,7,30

5.4,6.4

116

139

209

No

No

No

V-11,12 ; VI-23

I-10,11,14,16,17 ; II-9,11,13 ; III-4,5 ; IV-10,13,16 ; V-10

2,3,5,30

5.1,6.4,7.1

117

11

81

No

No

No

I-9,13,15 ; II-11,13 ; III-2,5 ; IV-10,13,16 ; V-10

2,3,7,30

5.4,6.4

ε mκ T † ( N ) T ( N ) T * ( N ) N = N † N = N N = N *

1 4

1 10

Cross index on 112 Bernoulli στ -shift rules.

34

243

187

No

No

No

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1

ε

1 19

35

115

59

No

No

No

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,8,30

5.4

ε

2 13

38

211

155

No

No

No

I-10,14,17 ; II-12,14 ; III-3,5 ; IV-11,14,16 ; V-10

30

6.4

Table A-7. N

κ

29 53

3

n

30 56

2


ε mκ T † ( N ) T ( N ) T * ( N ) N = N † N = N N = N *

Table Index Part VI


I-10,11,14,17 ; II-12,14 ; III-4,5 ; IV-11,14,16 ; V-9

30

6.4

39

83

27

No

No

No

Yes Yes

ε

98

227 185

No

No

No

VI-29

I-10,14,16,17 ; II-12,14 ; III-3,5 ; IV-10,13,16 ; V-9

2,3,7,27,30

5.4,6.4

ε ε 43

99

99

57

No

No

Yes

VI-29

I-10,14,16,17 ; II-12,14 ; III-3,6,10 ; IV-10,13,16 ; V-9

2,3,7,27,30

5.4,6.4

114

163 177

No

No

No

VI-29

I-10,11,14,16,17 ; II-12,14 ; III-4,5 ; IV-10,13,16 ; V-9

2,3,8,27,30

5.4,6.4,7.1

ε ε 62

115

35

49

No

No

No

I-9,13,15 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,8,30

5.4

103

67

25

No

No

No

I-10,14,16,17 ; II-12,14 ; III-3,5,30 ; IV-12,15,16 ; V-10

30

118

131 145

No

No

No

17

No

No

No

2 2

9

125 111

No

No

2 5 2 6

24

189 231

No

25

61

103

No

2

Yes Yes

32 58

3

Yes Yes

33 59

1

Yes Yes

34 61

2

No

No

2 19 2 20

1 19

35 62

2

Yes Yes

36 63

1

Yes Yes

ε ε 41

No

No

ε

2


ε13

No

31 57

37 65


(Continued )

2 22

119

3

IV-7,8,10,11 ; VI-7-4,18,29,33 I-10,14,16,17 ; II-10,12,14 ; III-3,5,30 ; IV-10,13,16 ; V-9

930

2,3,8,27,30

2.4,3,4.4,5.4,6.4,7

I-9,13,15 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,6,30

5.3

No

I-10,14,17 ; II-11,13 ; III-3,5 ; IV-11,14,16 ; V-10

30

6.4

No

No

I-10,14,17 ; II-8,11,13 ; III-3,5 ; IV-10,13,16 ; V-10

2,3,5,30

5.1

No

No

No

I-10,14,17 ; II-11,13 ; III-3,5 ; IV-12,15,16 ; V-10

30

6.4

I-10,14,16,17 ; II-8,11,13 ; III-3,5 ; IV-12,15,16 ; V-9

29,30

6.4,7.2,8

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1,6.4

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-10

2,3,7,30

5.4,6.4

38 66

2

Yes Yes

39 67

2

No

No

ε ε

40 74

2

No

No

ε 252

88

173 229

No

No

No

ε ε101

10

245 175

No

No

No

11

117

47

No

No

No

27

53

39

No

No

No

I-10,11,14,17 ; II-12,14 ; III-4,5 ; IV-11,14,16 ; V-9

30

6.4

14

213 143

No

No

No

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,7,30

5.4,6.4

15

85

15

No

Yes

No

I-9,13 ; II-12,14 ; III-2,6,9 ; IV-10,13,16 ; V-3,4,7,9

2,3,5,25,30

5.1,5.2,6.4,7

7

No

No

No

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,6,30

5.3

41 80

1

Yes Yes

42 81

1

Yes Yes

1 9

IV-17,18,19 ; VI-34,35,51

IV-21

43 83

3

No

44 84

1

Yes Yes

45 85

1

Yes Yes

ε ε131 ε141

Yes Yes

ε

1 7

31

21

No

No

ε

2 25

74

229 173

No

No

No

I-10,14,17 ; II-12,14 ; III-3,5 ; IV-12,15,16 ; V-9

30

6.4,8

Yes Yes

ε

2 19

56

185 227

No

No

No

I-10,14,17 ; II-8,12,14 ; III-3,5 ; IV-10,13,16 ; V-9

2,3,7,30

5.4

Yes Yes

ε

2 20

57

57

99

No

No

Yes

I-10,14,17 ; II-11,13 ; III-3,6,10 ; IV-10,13,16 ; V-9

2,3,7,30

5.4

50 103 2

No

No

ε

61

25

67

No

No

No

I-10,14,17 ; II-11,13 ; III-3,5,30 ; IV-12,15,16 ; V-10

30

6.4

51 111 2

No

No

125

9

65

No

No

No

I-10,14,16,17 ; II-11,13 ; III-3,5 ; IV-11,14,16 ; V-10

30

6.4

52 112 1

Yes Yes

ε 22 1 ε 20

42

241 171

No

No

No

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1

53 113 1

Yes Yes

43

113

43

No

Yes

No

I-9,13 ; II-12,14 ; III-2,6,9 ; IV-10,13,16 ; V-9

2,3,7,30

5.4,6.4

54 114 1

Yes Yes

1 ε 21 ε 43

58

177 163

No

No

No

I-10,11,14,17 ; II-12,14 ; III-4,5 ; IV-10,13,16 ; V-9

2,3,8,30

5.4

55 115 1

Yes Yes

59

49

35

No

No

No

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,8,30

5.4

56 116 3

Yes Yes

ε191 ε 33

46

209 139

No

No

No

I-10,11,14,17 ; II-12,14 ; III-4,5 ; IV-10,13,16 ; V-10

2,3,5,30

5.1

46 87 47 88 48 98 49 99

1 2 2 2

No

3 1

2 6

IV-17 ; VI-24,27

IV-9,17

Table A-7. κ


ε mκ

57 117 1

Yes Yes

ε101

47

81

11

No

No

No

62

145 131

No

No

No

63

17

3

No

No

111

65

9

No

144

190 246

145

62

n

N

Yes Yes

ε

59 119 1

Yes Yes

60 125 2

No

ε ε 22

58 118 2

No

61 130 2

Yes Yes

62 131 2

Yes Yes

2 22 1 4

ε ε 222

2 34

63 134 2

No

64 138 1

Yes Yes

ε 1 ε 28

Yes Yes

ε

65 139 3

No

2 36

3 3

(Continued )


Table Index Part VI


I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,7,30

5.4,6.4

I-10,14,17 ; II-10,12,14 ; III-3,5,30 ; IV-10,13,16 ; V-9

2,3,8,30

3,4.4,5.4

No

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-7,9

2,3,6,30

5.3

No

No

I-10,14,17 ; II-12,14 ; III-3,5 ; IV-11,14,16 ; V-10

30

6.4

No

No

No

I-10,14,17 ; II-8,11,13 ; III-3,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1,6.4,7.1

118

No

No

No

I-10,14,17 ; II-11,13 ; III-3,5 ; IV-10,13,16 ; V-9

2,3,8.30

4.4,5.4

148

158 214

No

No

No

I-10,14,17 ; II-9,11,13 ; III-3,5 ; IV-11,14,16 ; V-9

28,30

6.4,8

208

174 244

No

No

No

I-9,13 ; II-8,11,13 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1,6.4,7.1

209

46

116

No

No

No

I-10,11,14,17 ; II-11,13 ; III-4,5 ; IV-10,13,16 ; V-10

2,3,5,30

5.1

212

142 212

No

Yes

No

I-9,13 ; II-9,11,13 ; III-2,6,9 ; IV-10,13,16 ; V-9

2,3,7,27,30

5.4,6.4

213

14

84

No

No

No

I-9,13 ; II-11,13 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,7,30

5.4,6.4

130

246 190

No

No

No

I-10,14,17 ; II-12,14 ; III-3,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1

I-10,14,17 ; II-12,14 ; III-3,5 ; IV-10,13,16 ; V-9

2,3,8,30

4.4,5.4

I-10,14,17 ; II-12,14 ; III-3,5 ; IV-11,14,16 ; V-9

30

6.4

I-10,14,17 ; II-12,14 ; III-3,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1,6.4

IV-11

VI-23

VI-23

66 142 1

Yes Yes

67 143 1

Yes Yes

ε ε

Yes Yes

ε

69 145 2

Yes Yes

131

118

62

No

No

No

70 148 2

No

ε ε 362

134

214 158

No

No

No

194

230 188

No

No

No

211

38

52

No

No

No

I-10,14,17 ; II-12,14 ; III-3,5,30 ; IV-11,14,16 ; V-10

30

6.4

214

134 148

No

No

No

I-10,14,17 ; II-10,12,14 ; III-3,5 ; IV-11,14,16 ; V-9

30

6.4

20

No

No

No

I-10,14,17 ; II-12,14 ; III-3,5 ; IV-11,14,16 ; V-9

30

I-9,13 ; II-8,11,13 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1,6.4,7.1

I-10,11,14,17 ; II-11,13 ; III-4,5 ; IV-10,13,16 ; V-9

2,3,8,30

5.4

I-9,13 ; II-8,11,13 ; III-2,6,9 ; IV-10,13,16 ; V-3,4,7,9

2,3,5,24,30

5.1,5.2,6.4,7

68 144 2

931


T † (N ) T (N ) T * (N ) N = N † N = N N = N *

No

1 30 1 13 2 34

2 22

71 152 2

Yes Yes

72 155 2

No

No

73 158 2

No

No

ε ε132 ε 362

No

Yes Yes

74 159 2 75 162 1

No

2 38

215

ε

1 32

176

186 242

No

No

No

177

58

114

No

No

No

240

170 240

No

Yes

No

241

42

112

No

No

No

I-9,13 ; II-11,13 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1

ε 1 ε 28

229

74

88

No

No

No

I-10,14,17 ; II-11,13 ; III-3,5,30 ; IV-12,15,16 ; V-9

30

6.4,8

244

138 208

No

No

No

I-9,13 ; II-9,11,13 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1

245

10

80

No

No

No

I-9,13 ; II-11,13 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1,6.4

162

242 186

No

No

No

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1

163

114

58

No

No

No

I-10,11,14,17 ; II-12,14 ; III-4,5 ; IV-10,13,16 ; V-10

2,3,8,30

5.4

226

226 184

No

No

Yes

I-10,11,14,17 ; II-12,14 ; III-4,6,10 ; IV-10,13,16 ; V-9

2,3,7,27,30

5.4,6.4,7.1

77 170 1

Yes Yes

ε 1 ε 34

Yes Yes

ε

No

No

80 174 1

Yes Yes

3 4

1 20 2 25

81 175 1

Yes Yes

82 176 1

Yes Yes

ε 1 ε 32

Yes Yes

ε

3 4

Yes Yes

ε

3 9

83 177 3 84 184 3

VI-23

ε

Yes Yes

79 173 2

III-5

2 1

76 163 3

78 171 1

VI-29

1 9

6

VI-23

IV-4,10 ; V-3,5 ; VI-23,26,30,41,43

III-4,9 ; VI-29,32

Table A-7. κ


85 185 2

Yes Yes

ε192

227

98

56

No

No

Yes Yes

ε

242

162 176

No

243

34

48

230

n

N

86 186 1 87 187 1

Yes Yes

88 188 2

Yes Yes

89 189 2

Yes Yes

90 190 1

Yes Yes

ε ε 382 1 18

ε ε 342 2 5

91 191 1

Yes Yes

92 194 2

Yes Yes

ε ε 382

Yes Yes

ε

93 208 1 94 209 3

Yes Yes

95 211 2

No

96 212 1

No

Yes Yes

932

97 213 1

Yes Yes

98 214 2

No

No

1 3

1 28

ε ε132 3 3

ε

1 30

ε131 ε 362

99 215 1

No

100 226 3

Yes Yes

ε12 ε 93

Yes Yes

ε

No

No

ε

Yes Yes

ε

Yes Yes

ε

Yes Yes Yes Yes

101 227 2 102 229 2 103 230 2 104 231 2 105 240 1 106 241 1

No

107 242 1

Yes Yes

108 243 1

Yes Yes


No

I-10,14,17 ; II-12,14 ; III-3,5,30 ; IV-10,13,16 ; V-9

2,3,7,30

5.4

No

No

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1

No

No

No

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-7,9

2,3,5,30

5.1

194 152

No

No

No

I-10,14,17 ; II-12,14 ; III-3,5,30 ; IV-10,13,16 ; V-9

2,3,5,30

5.1

231

66

24

No

No

No

I-10,14,17 ; II-12,14 ; III-3,5 ; IV-10,13,16 ; V-10

2,3,5,30

5.1

246

130 144

No

No

No

I-10,14,17 ; II-10,12,14 ; III-3,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1

16

No

No

No

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1

152

188 230

No

No

No

I-10,14,17 ; II-8,11,13 ; III-3,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1

138

244 174

No

No

No

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1

139

116

46

No

No

No

I-10,11,14,17 ; II-12,14 ; III-4,5 ; IV-10,13,16 ; V-10

2,3,5,30

5.1

155

52

38

No

No

No

I-10,14,17 ; II-12,14 ; III-3,5,30 ; IV-11,14,16 ; V-10

30

6.4

142

212 142

No

Yes

No

I-9,13 ; II-12,14 ; III-2,6,9 ; IV-10,13,16 ; V-9

2,3,7,30

5.4,6.4

143

84

14

No

No

No

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,7,30

5.4

158

148 134

No

No

No

I-10,14,17 ; II-10,12,14 ; III-3,5 ; IV-11,14,16 ; V-9

30

5.4,6.4

159

20

No

No

No

I-10,14,17 ; II-12,14 ; III-3,5 ; IV-11,14,16 ; V-9

30

I-10,11,14,17 ; II-8,11,13 ; III-4,6,10 ; IV-10,13,16 ; V-9

2,3,7,30

5.4

247

2

6


III-9

184

184 226

No

No

Yes

185

56

98

No

No

No

I-10,14,17 ; II-11,13 ; III-3,5,30 ; IV-10,13,16 ; V-9

2,3,7,30

5.4

2 25

173

88

74

No

No

No

I-10,14,17 ; II-11,13 ; III-3,5,30 ; IV-12,15,16 ; V-9

30

6.4,8

2 38

188

152 194

No

No

No

I-10,14,17 ; II-9,11,13 ; III-3,5,30 ; IV-10,13,16 ; V-9

2,3,5,30

5.1

2 5

189

24

66

No

No

No

I-10,14,17 ; II-11,13 ; III-3,5 ; IV-10,13,16 ; V-10

2,3,5,30

5.1

ε

1 34

170

240 170

No

Yes

No

I-9,13 ; II-12,14 ; III-2,6,9 ; IV-10,13,16 ; V-3,4,9

2,3,5,24,30

5.1,5.2,6.4,7

ε

1 20

171

112

42

No

No

No

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1

186

176 162

No

No

No

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1

187

48

34

No

No

No

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1

174

208 138

No

No

No

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1

175

80

10

No

No

No

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1,6.4

190

144 130

No

No

No

I-10,14,17 ; II-10,12,14 ; III-3,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1

191

16

No

No

No

I-9,13 ; II-12,14 ; III-2,5 ; IV-10,13,16 ; V-9

2,3,5,30

5.1

ε ε181

1 32

Yes Yes

110 245 1

Yes Yes

ε ε 91

Yes Yes

ε

Yes Yes

ε

112 247 1

Table Index Part VI

N = N*

2 19

109 244 1

111 246 2


ε mκ T † ( N ) T ( N ) T * ( N ) N = N † N = N

1 32

(Continued )

1 28

2 34 1 3

2

IV-4,20 ; V-4,6 ; VI-23,26,31,42,44

Table A-8.

Tables which show all 256 local rules.

933

♦

ERRATA FOR VOLUME I ♦

Ë 1. Page 76 There is a typo in equation (9): change “fof” to “for” 2. Page 86 Equations (22) and (23) should read: w(σ) = (σ − 3)(σ + 1) = σ 2 − 2σ − 3

(22)

x˙ i = g(xi ) + (ui−1 + 2ui − 3ui+1 )2 − 2(ui−1 + 2ui − 3ui+1 ) − 3

(23)

3. Page 141 The equation for rule 63 should read: = sgn[−uti−1 − uti + 1] ut+1 i 4. Page 151 The equation for rule 101 should read: = sgn[2 − |uti−1 − uti − 2uti+1 − 1|] ut+1 i

934

♦

EPILOGUE ♦

Ë This book heralds the end of the beginning of our analytic approach to cellular automata and complexity theory. We hope that it will usher in a new era of in-depth and synergetic research by generalizing our new paradigm not only to infinite length and two-dimensional cases (with continuous rather than discrete states), but also to uncover nature’s secret on countless complexity phenomena reported from numerous disciplines, by exploiting the deep implications of the author’s local activity principle, and his technically precise concept of “edge of chaos” [Chua, 1998]. After all, local activity is the origin of complexity, and its associated edge of chaos is where it all begins. We have deliberately restricted ourselves in this elementary book to a tiny nook of one-dimensional binary two-neighbor dynamical systems where the majority of the phenomena and concepts articulated can be, or perhaps should be, taught in high schools. Indeed, new paradigms need not be abstruse, though they could appear to be so for those addicted to the bad habit of brute force computer simulations. CA local rules are not ordained by God; they are manifestations of simple dynamical systems, such as neurons, described by a scalar nonlinear differential equation. The dynamical complexity endowed on each local rule is coded in the “number” of coefficients needed to specify the neuron. In particular, a local rule has an index of complexity equal to one, two, or three, if the color of the 8 vertices in its associated Boolean cube can be segregated into a common color by one, two, or three parallel separating planes, respectively. Classifying the 256 local rules via this recipe would constitute an interesting exercise for high school kids. Our recipe for partitioning the 256 local rules into 88 equivalence classes should appeal to the high school kids’ aesthetic sense of symmetry. Every kid would appreciate the “equivalence class” theorem from Chapter 3 allowing him to examine only 88 rules, out of 256. Once hooked, many kids would want to glance at the dynamical evolution of the local rules. They would therefore appreciate learning the simple recipes for predicting the long-term steady state dynamics of all but 50 local rules belonging to the 18 equivalence classes listed in Tables 17 and 18 of Chapter 4 (p. 500) that are yet to be derived. In particular, they would relish in predicting the evolution of all 112 Bernoulli στ -shift rules from any initial bit-string configuration living on an attractor, by extracting only three pieces of information (α, β, τ ) from Table 16 of Chapter 4 (p. 483). The recipe calls for merely shifting the bit-string to the left (if σ > 0), or right (if σ < 0), by |σ| pixels every τ iterations, and complementing it (by changing each pixel color) if β < 0. Perhaps the funniest game yet would be the realization of “time travel” on cellular space via any one of 170 distinct time machines he or she can select from the 170 time-reversible rules listed in Table 2 of Chapter 6 (p. 711). How nice it is to be able to relive those fond birthday parties of bygone days! But the most challenging of all would be to hunt for new “isles of Eden” from the local rules. Hunting for a “period-T ” isle of Eden would be a real quest specially for large period T and long bit strings. They are like well-hidden Easter eggs, and every such new isle of Eden will be a gem to be cherished forever, even for adults. 935


♦

REFERENCES ♦

Ë Alligood, K. T., Sauer, T. D. & Yorke, J. A. [1996] Chaos: An Introduction to Dynamical Systems (Springer-Verlag NY). Barnsley, M. F. [1988] Fractals Everywhere (Academic Press). Billingsley, P. [1978] Ergodic Theory and Information (Robert Keirger Publishing Company, Huntington). Blanchard, F., Maass, A. & Nogueira, A. [2000] Topics in Symbolic Dynamics and Applications (Cambridge University Press, Cambridge). Boyd, S. & Chua, L. O. [1985] “Fading memory and the problem of approximating nonlinear operators with Volterra series,” IEEE Trans. Circuits Syst. CAS-32, 1150–1161. Chua, L. O. [1969] Introduction to Nonlinear Network Theory (McGraw-Hill Book Company, NY). Chua, L. O. & Kang, S. M. [1977] “Section-wise piecewise-linear functions: Canonical representation, properties and applications,” Proc. IEEE 65, 915–929. Chua, L. O., Desoer, C. A. & Kuh, E. S. [1987] Linear and Nonlinear Circuits (McGraw-Hill, NY). Chua, L. O. [1998] CNN: A Paradigm for Complexity (World Scientific, Singapore). Chua, L. O. & Roska, T. [2002] Cellular Neural Networks and Visual Computing (Cambridge University Press, Cambridge). Chua, L. O., Yoon, S. & Dogaru, R. [2002] “A nonlinear dynamics perspective of Wolfram’s new kind of science. Part I: Threshold of complexity,” Int. J. Bifurcation and Chaos 12, 2655–2766. Chua, L. O., Sbitnev, V. I. & Yoon, S. [2003] “A nonlinear dynamics perspective of Wolfram’s new kind of science. Part II: Universal neuron,” Int. J. Bifurcation and Chaos 13, 2377–2491.

Chua, L. O., Sbitnev, V. I. & Yoon, S. [2004] “A nonlinear dynamics perspective of Wolfram’s new kind of science. Part III: Predicting the unpredictable,” Int. J. Bifurcation and Chaos 14, 3689–3820. Chua, L. O. [2005] “Local activity is the origin of complexity,” Int. J. Bifurcation and Chaos 15, 3435–3456. Chua, L. O., Sbitnev, V. I. & Yoon, S. [2005a] “A nonlinear dynamics perspective of Wolfram’s new kind of science. Part IV: From Bernoulli shift to 1/f spectrum,” Int. J. Bifurcation and Chaos 15, 1045– 1183. Chua, L. O., Sbitnev, V. I. & Yoon, S. [2005b] “A nonlinear dynamics perspective of Wolfram’s new kind of science. Part V: Fractals everywhere,” Int. J. Bifurcation and Chaos 15, 3701–3849. Cornfeld, I. P., Fomin, S. V. & Sinai, Ya. G. [1982] Ergodic Theory (Springer-Verlag, Berlin). Davies, P. [1995] About Time (Simon & Schuster, NY). Davies, P. [2003] How to Build a Time Machine (Penguin Books). Devaney, R. L. [1992] A First Course in Chaotic Dynamic Systems: Theory and Experiment (AddisonWesley, Reading, MA). Feller, W. [1950] An Introduction to Probability Theory and Its Applications, Vol. 1 (John Wiley, NY). Feynman, R. P. [1949] “The theory of positrons,” Phys. Rev. 76, 749–759. Gerstner, W. & Kistler, W. [2002] Spiking Neuron Models (Cambridge University Press, Cambridge). Hawking, S. [2005] A Briefer History of Time (Bantam Books). Hirsch, M. W. & Smale, S. [1974] Differential Equations, Dynamical Systems, and Linear Algebra (Academic Press, NY). 937

938 References

Kitano, H. [2001] Foundations of Systems Biology (The MIT Press, Cambridge). Kunstsammlung, N.-W. [2005] Gerhard Ritcher (Ritcher Verlag, K¨ oln, Germany). Kurka, P. [2003] Topological and Symbolic Dynamics (Societie Mathematique de France, Paris). Mackey, M. C. [1992] Time’s Arrow: The Origin of Thermodynamic Behavior (Springer-Verlag, NY). Mathcad User’s Guide [2000] Math Soft, Inc., Cambridge. Moore, E. F. [1962] “Machine models of selfreproduction,” Proc. Symp. Appl. Math. 14, 17–33. Nagashima, H. & Baba, Y. [1999] Introduction to Chaos (Institute of Physics Publishing, Bristol). Niven, I. [1967] Irrational Numbers (The Mathematical Association of America, NJ). Novikov, I. [1998] The River of Time (Cambridge University Press).

Poincare, H. [1897] Les Methods Nouvelles de la Mechanique Celeste, Vol. I and II (Gauther-Villars, Paris). Sachs, R. G. [1987] The Physics of Time Reversal (University of Chicago Press, Chicago). Schroeder, M. [1991] Fractals, Chaos, Power Laws (W. H. Freeman and Company, NY). Shilnikov, L. P., Shilnikov, A. L., Turaev, D. V. & Chua, L. O. [2001] Methods of Qualitative Theory in Nonlinear Dynamics [Part II ] (World Scientific, Singapore). Tang, Y. S., Mees, A. I. & Chua, L. O. [1983] “Synchronization and chaos,” IEEE Trans. Circuits Syst. 9, 620–626. Wells, H. G. [1895] The Time Machine (Henry Holt, NY). Wiener, N. [1958] Nonlinear Problems in Random Theory (John Wiley, NY). Wolfram, S. [2002] A New Kind of Science (Wolfram Media, Inc. Champaign, IL).

♦

INDEX ♦

Ë π-rotation mapping symmetry, 452, 459 xI -insensitive local rules, 633

1/f power-frequency, 498 1/f power-frequency characteristics, 499 1/f spectrum, vii, 369 1/f -spectrum, 452 1D CA, 115, 119 16 gene siblings, 210 180◦-rotation, 192 256 CA rules, 501 256 local rules, 933 8/24 alternating periodicity, 205 στ -shift dynamics, 483 στ -shift rule, 491 σ τ -shift theorem, 475 σ τ -shifting dynamics, 475 στ -shifting rule, 475, 487, 491 σ τ -shifting rule theorem, 475 τ th-iterated characteristic function χτN , 494 χ130 , 379 χ151 , 379 χ251 , 379 χ162 , 379 χ362 , 379 χ1110 , 379 χ1128 , 373 χ1170 , 379 χ1200 , 379 χ1240 , 379

Abelian group, 248, 270 absolute-value function, 6, 190, 270, 387 absolute-value functions, 232 abstract group, 359 abstract groups, 270 action potentials, 233 affine (mod 1) local rules, 594 affine (mod 1) rules, 595 algorithm, 4, 198 analog computation, 232 analytical approach, 657 analytical theory, 914 AND, 111 annihilation, 657 arrow of time, vii, 657, 899 attractor, 2, 77–79, 81, 229, 379, 765 attractor bit-string generator, 662, 766 attractor color code, 77 attractor time-τ maps, 369 attractor time-1 maps, 596 attractor vignettes, 387 attractors, 1, 6, 79–82, 386 backward Boolean string, 372, 382, 658 backward time, 386 backward time series, 382, 383 939

940

Index

backward time-τ map, 383 backward time-τ map ρ†τ [N ], 453 backward time-τ return map, 660 backward time-1 map ρ†1 [N ], 387, 454 backward time-1 map ρ†1 [N † ], 452 backward time-1 map, 384 backward time-1 maps ρ†1 [N ], 388 backward time-2 map, 384 backward time-3 map, 384 basin of attraction, 379–381, 765 basins of attraction, 658 basis function, 6 Bernoulli στ -shift attractor, 383, 694–699, 701, 702, 817 Bernoulli στ -shift formula, 911 Bernoulli στ -shift map, 463, 465, 466, 475, 495, 868 Bernoulli στ -shift rule, 369, 463 Bernoulli attractor, 463 Bernoulli attractors, 463–466, 476 Bernoulli coin toss, xvii Bernoulli constant ν(I), 872 Bernoulli copycat law, 910 Bernoulli map, xi, 463, 509, 528, 593 Bernoulli map intercept µ(I), 872 Bernoulli orbits in complex plane, 816 Bernoulli rule 240 , 493 Bernoulli rule, 465 Bernoulli rules, 483, 487, 488, 491, 494 Bernoulli shift map ρ 170 , 730 Bernoulli shift map ρ 240 , 730 Bernoulli shift map, 463 Bernoulli shift, vii, 369, 379, 492 bifurcation point, 499 Big Bang, 657 bilateral, 452, 455, 457–459, 461, 499 bilateral image, 372 bilateral mapping invariance, 452, 459 bilateral period-2 rules, 459 bilateral reflection, 259 bilateral rule, 714 bilateral rules, 460 bilateral twin, 700 bilateral twin rule N † , 657 binary, 2, 113 binary code, 5 binary inputs, 114 binary number, 115 binary signals, 2 binary string, 230, 370 binary variable, 5 binary variables, 6 bit-string laws, 765, 766 blue background, 210

blue flag, xi, xiv blue gene family group, 198, 199, 204, 207 blue group, 373 Boolean code, 120 Boolean computations, 369 Boolean cube, 1, 5, 6, 11, 79, 81, 82, 103, 113, 114, 119, 190–192, 206, 230–233, 238, 241, 253, 264, 266, 273, 279, 369, 370 Boolean cubes, 11, 113, 115, 190, 198, 206, 241, 284, 359 Boolean cube N , 119 Boolean cube families, 233 Boolean cube family, 211 Boolean cube representation, 231 Boolean function, 5, 6, 104, 229 Boolean function rules, 103, 105 Boolean functions, 371 Boolean local rule, 1 Boolean rule, 6 Boolean rules, 110 Boolean string, 279, 371 Boolean variable, 371 Boolean variables, 76 brain science, 3 Brownian motion, 454 CA, 259, 261 CA attractors, 382 CA characteristic functions, 369 CA difference equation, 371 CA gene families, 190 CA gene family, 193, 195 CA gene family code, 193 CA gene siblings, 190 CA characteristic function, 372, 385, 509 Cantor sets, 494 Cantor space, 914 cartoons, vii, xi cells, 4, 114 cellular automata, vii, xi, 1–3, 6, 78, 82, 83, 112–114, 121, 229, 230, 232, 252, 348 cellular automaton, 371 cellular neural network, 3, 231 cellular nonlinear networks, 3, 359 chaos, 386 chaotic, 386, 492 characteristic function, 379, 383, 492 characteristic functions, 371–373, 593, 595, 610, 917 characteristic function χτN , 494 characteristic functions χτN , 512 characteristic function χ114 , 496 checkerboard attractor, 877 chip, 4

Index

chiral number, 270, 284 chiral numbers, 338 clock, xv clusters of period-1 points, 458 CMOS transistors, 4 CNN, 3, 4, 229, 232, 259, 359 CNN cell, 112 CNN chip, 4, 232, 265 CNN universal chip, 4, 110 CNN universal chips, 110 coarse grain, 373 cobweb diagram, 385, 386, 492, 493, 596 code, 114 coin toss, 492 coin-toss experiments, 493 color code, 79 colored vertices, 231 commutation diagram, 358 commutation property, 259 commutative, 372 commutative diagram, 359, 372, 511 complementary bilateral initial pattern, 239 complementary rules, 206 complementation, 240 complementation invariance theorem, 259 complementation of the vertex colors, 233 complementation operation, 238 complemented Bernoulli shift map ρ 15 , 731 complemented Bernoulli shift map ρ 85 , 731 complements, 206 complex of systems, xi complex patterns, 112 complex phenomena, 112 complexity, 112 complexity index, 1, 2, 83, 96, 102, 121, 190, 232, 241, 246, 248, 270, 284, 454, 463 complexity patterns, 112 composite structure of nested absolute value functions, 89 composition, 258 composition operation, 372 continuous-time neuron, 121 coupling law, 4 coupling laws, 2 cross index, 926, 928, 929 cyclic group, 248, 270 cyclic subgroups, 272 decimal code, 5, 120 decimal equilvalent, 115 decimal number, 5, 230 degenerate rules, 265

941

dense period-1 fixed points, 792, 794 dense period-1 points, 796 dense period-2 points, 814 dense period-3 points, 816 dense, 792 difference equation, 4, 126, 190, 233–236, 237, 239, 265, 346, 371, 922 difference equations, 348 differential equation, 4, 12 directed graph, 660 directed graph ( N ), 765 discrete map, 121, 190 discrete time evolution, 120, 126 discrete time neuron, 121 discriminant, 11, 78, 79, 83, 84 discriminant function, 11, 76, 77, 83, 84, 86, 89 “do loop”, 232 double helix, 206 double-helix torus, 113, 198, 206, 209, 210 double-valued function, 593 driving-point function, 6, 76, 77, 79 driving-point functions, 76 dual mapping correspondence, 452, 459 dynamic organization principle, 210 dynamic pattern, 379, 380 dynamic patterns, 488 dynamic route, 76, 77 dynamic routes, 80 dynamic systems, 76, 102 dynamical system, 1, 2, 4, 6, 11, 12, 78, 82, 83, 231, 382 emergence, 112 empirical observations, 2, 3 equilibrium point, 76 equilibrium solution, 78 equilibrium state, 11 equilibrium value, 11 equivalence class, 111 equivalence classes, 241 equivalent class, 104 equivalent classes, 105, 914 equivalent map χ N , 372 ergodic, 493 ergodic Bernoulli rules, 494 ergodic over, 494 even gene families, 202 evolution law, 4 evolved pattern, 193 exact Bernoulli formula, 871, 873 exact formula, 875, 876, 884 excitatory, 119 excitatory synapse, 914

942

Index

exclusive global relationships, 264 explicit formulas, 512, 594 feedbacks, 232 Felix Klein, 229, 248 fine grain characteristic function χτN , 373 finite abstract groups, 248 firing pattern, xi, 119, 120, 193, 236, 633, 634 firing patterns, xi, xvi, xviii–xxii, xxiv–xxvii, 113, 119, 120, 126, 190, 191, 210, 211, 233–236, 662, 766 fixed point, 456 fixed points, 241, 251, 792 fixed-point, 491 flow, 76 forward Boolean string, 372, 382, 658 forward time, 386 forward time series, 382, 383 forward time-τ map, 383 forward time-τ map ρτ [N ], 453 forward time-τ return map, 660 forward time-1 map ρ1 [N ], 387, 452, 454 forward time-1 map, 384, 465 forward time-1 return map, 765 forward time-2 map, 384, 465 forward time-3 map, 384 four group, 248 four-element Abelian noncyclic group, xxiii fractal, 611–617 fractals, vii, 509, 610 fractal stratifications, 618 fractal structures, 618, 630 fundamental geometric property, 86 gallery, 121 gallery of characteristic functions, 529 gallery of forward time-1 maps ρ1 [N ], 388 gallery of gene family patterns, 210 gallery of time-1 maps, 387 game of life, 78 gaps, 494 garden of Eden, 382, 765 gardens of Eden, 509, 634, 635, 656 gene decoding book, 78 gene decoding tape, 78 gene families, 113, 192, 198 gene family, 195, 198–200, 204, 207, 208 gene sibling number, 206 gene siblings, 113, 192, 195, 198–200, 202, 203, 207, 208 genealogic classification, 190 generalized Bernoulli map, 657 geometrical approach, 2 geometrical interpretation, 876

geometrical interpretation of T ∗ , 256 geometrical interpretation of T † , 254 geometrical interpretation of T , 255 geometrical interpretation of Tu∗ , 253, 257 geometrical structure, 1 Gerhard Richter, 916 germinating local rule, 198 global complementary bilateral relationship, 239 global complementary relationship, 237 global complementation, 244, 246, 248, 252, 255, 358, 359 global complementation T , 240, 241, 248, 257, 259 global complementation operator, 453 global dynamics, 596 global equivalence, xxiii, 233, 241 global equivalence class, 260, 383 global equivalence classes, 229, 241, 246, 248, 359 global equivalence group, 252 global equivalence principle, 499 global equivalence-class, 358 global equivalent rules, 498 global left-right (bilateral) symmetrical relationship, 239 global left-right complementation T ∗ , 257 global map, 372 global patterns of evolution, 3 global representation of N , 385 global result, 241 global transformations, 260, 342 globally equivalent, 240, 264, 452 globally equivalent local rules {149, 135, 86, 30}, 242 globally equivalent local rules {226, 184}, 243 globally equivalent local rules, 244 globally equivalent rules, 462 globally-equivalent local rule, 250 globally-equivalent local rules, 249 globally-equivalent rules, 827 graph ( N ), 765 graph of the characteristic function, 372, 596 graph of the time-1 map, 494 graphical composition, 878–882 graphs, 494, 507 graphs of ρ1 [N ], 453 graphs of characteristic functions χ1N , 528 Green’s function, 454 group-theoretic perspective, 192 hardware, 232 homogeneous, 457 homogeneous period-1 rules, 459 homogeneous “0” rules, 458 homogeneous “1” rules, 458

Index

homogeneous rules, 456 Hopfield network, 4 ID number, 119 ideal coin toss, xi, xvii identification (ID) number, 114 identification number, 6 identification process, 87 independent linearly separable, 104 index of complexity, 232 inhibitory, 119 inhibitory synapse, 914 initial, 210 initial bit-string configuration, xi initial condition, 12, 76, 77, 79 initial conditions, 76 initial configuration, 380, 452 initial pattern, 190, 191, 193, 233 initial state, 4, 11 input, 4, 79, 114, 230 inputs, 2, 119 input space, 84 intrinsic state, 2 invariant, 79, 104 invariant orbit, 369, 381, 660 invariant orbits, 382, 386, 387, 463, 494, 662, 825 inverse Bernoulli map, 593 inverse star transition graph, 347 invertible, 457, 459, 499, 461, 715 invertible Bernoulli rules, 466 invertible Bernoulli στ -shift rules, 464 invertible function, 454 invertible period-1 rules, 455 invertible period-2 rules, 459, 460 invertible rules, 454, 715 invertible time-τ maps, 453 isle of Eden, 382, 509, 655, 656, 660, 672, 700 isolated, 792 isolated period-1 fixed points, 792, 793 isolated period-1 points, 795 isolated period-2 points, 813 isolated period-3 points, 816 iterative map, 4 Klein Vierergruppe, xxiii, 596, 700 Lameray diagram, 385, 492, 596–609, 671 lateral twin, 452 lateral twin vignette, 452 laterally symmetric groupings, 343 laterally symmetric interaction group, 338 lattice, 4

943

law, 897, 910 laws governing generic attractor bit-strings, 662 laws governing period-1 bit strings, 765 left ↔ right complementation, 244, 246, 248 left ↔ right symmetrical pairs, 104 left ↔ right symmetrical transformation, 102 left ↔ right transformation, 244, 246, 248 left ↔ right vertex transformation, 105 left–right transformation operator, 372, 383 left-copycat, 897, 910 left-copycat local rule, 509 left-copycat rule, xii, xviii, 593 left-right complementation, 251, 252, 256, 258, 358, 359 left-right complementation Tu∗ , 257 left-right complementation T ∗ , 248, 259 left-right transformation, 251, 252, 254, 358, 359, 382 left-right transformation T † , 240, 241, 248, 259, 338, 452, 700 linear separability, 1 linearly non-separable Boolean function rules, 101, 107, 108 linearly non-separable local rules, 111 linearly non-separable rule, 6 linearly non-separable rule, 102 linearly-non-separable rules, 96 linearly separable, 114, 190 linearly-separable Boolean functions, 232 linearly-separable rules, 102, 108 linearly separable Boolean function rules, 101, 107 linearly separable rule, 6 linearly separable rules, 95, 96, 102, 111 local complementation operation, 233 local connectivity, 4 local equivalence, 238 local equivalence class, 286, 289, 295, 347, 348 local equivalence classes, 229, 270, 273, 284, 297, 342, 359 local equivalent class, 382 “local in iteration time”, 238 local rule, 1–3, 11, 82, 83, 86, 87, 102, 113, 119, 190, 201, 204, 206, 231, 232, 286, 348 local rules, 2, 79, 102, 104, 113, 115, 126, 190, 192, 198, 206, 229, 264, 273, 284, 371 local rule 62 , 660 local rule 110 , 80, 119, 120, 235 local rule 184 , 236 local rule 232 , 234 local rule N , 119, 348 local rule number N , 121 local rule number, 229, 231 locally-equivalent rules, 338

944

Index

logic state, 113 Lyapunov exponent, 492 majority rule, 359 mathematical foundation, 867 matter–antimatter, 657 matter–antimatter pair annihilation, 921 matter–antimatter production, 921 metaphors, vii mirror images, 453 monkey, vii, xi monkey-time keeper, xv monkey time keeper, xi monkey “ring”, xi monkey’s neuron, xvi monkey’s neurons, xi monotone increasing state, 78 Monte-Carlo, xvii multiple attractors, 452 multiple universes, 657 multiplication table, 252, 272, 358, 359 multivalued paradox, 491 nano seconds, 232 nanoseconds, 11 nearest neighbors, 2, 4, 113 neighborhood pattern, 119 neighborhood patterns, 78 nested absolute-value functions, 89, 113 neural network, 119 neural networks, 113 neuron, 113, 119, 121 no fractal stratifications, 630 non-Abelian group, 270 non-bilateral rules, 715 non-invertible rules, 715 nonbilateral, 458, 459, 461, 499 nonbilateral rule, 714 nonbilateral rules, 459–461, 715 noninvertible, 459, 461, 499 noninvariant Bernoulli rules, 494 noninvertible Bernoulli στ -shift rules, 463, 465 noninvertible Bernoulli rules, 476 noninvertible nonbilateral rules, 500 noninvertible period-1 rules, 454, 456 noninvertible bilateral rules, 500 noninvertible period-2, 460 noninvertible period-3 rules, 461 nonlinear circuit theory, 6 nonlinear difference equation, 121, 126, 232 nonlinear difference equations, 229, 349 nonlinear differential equation, 119

nonlinear dynamical system, 3 nonlinear dynamics, vii, 1, 2, 231, 241, 265, 359, 913 nonlinear dynamics foundation, 914 nonlinear map, 2 numeric binary output, 231 numeric Boolean vector, 231 numeric input, 231 numeric truth table, 230, 231, 263, 370, 510, 659 numeric truth tables, 231 numeric variable, 371 odd gene families, 203 offset level, 76, 77 one-dimensional cellular automata, vii, 369, 370, 510 one-iteration complementation property, 238 operating mode, 382 OR, 111 OR operations, 111 orbit-invariant rules, 507 ordinary differential equation, 2 orientation vector, 78, 82, 84, 86–88, 90, 94, 97 output, 2, 4, 114, 119, 230 output equation, 2, 11, 12, 231 output variable, 114 pair-production, 657 palindrome rules, 487 paradigm shift, 913 parallel computer, 4 pattern features, 113 period-k isle of Eden, 382 period-k time-1 maps, 454 period-TΛ attractor, 380, 386 period-1 attractor, 383, 491 period-1 attractors, 379 period-1 fixed points, 379 period-1 rules, 454, 596 period-10 isles of Eden, 651 period-11 isles of Eden, 653 period-2 attractor, 383 period-2 attractors, 459 period-2 isles of Eden, 641 period-2 palindromes, 491 period-2 rules, 459, 596 period-3 attractor, 383 period-3 attractors, 461 period-3 isle of Eden, 701 period-3 isles of Eden, 642 period-3 isle of Eden of 62 , 700 period-3 rules, 460, 596 period-4 attractor, 499 period-4 fixed points, 495

Index

period-4 isles of Eden, 643 period-5 isles of Eden, 644, 645 period-6 isles of Eden, 646 period-7 isles of Eden, 647 period-8 isles of Eden, 648 period-9 isles of Eden, 650 periodic boundary condition, 2, 3, 114, 230, 370, 511 periodicity property, 198 “periodicity” property, 198 piecewise-linear driving-point function, 6 Poincare cross-section, 383 Poincare return map, 385 Poincare return maps, 383 polynomial discriminant, 89 power spectrum, 369, 387, 461, 466, 494, 495, 498 predecessors, 509 primary firing pattern, 193, 206 primary firing patterns, 190, 192, 210 primary firing vertices, 191 primary trihedron, 191, 192 primary vertex weight, 193, 195 probing inputs, 494 probing signals, 259 projection, 78, 81, 84, 85, 90, 91, 94, 97 projection axis, 82, 84, 87, 88 qualitative characterization, 232 qualitative global dynamics, 463 quenching pattern, 119, 120 quenching patterns, 113, 120 random bit strings, 452 random configuration “probing” string, 453 random initial conditions, 112 random initial pattern, 234–236 random input patterns, 259 random signals, 453 randomly-generated bit strings, 494 real numbers, 76 red ↔ blue complementary pairs, 103 red ↔ blue complementary transformation, 102 red ↔ blue vertex transformation, 105, 192, 233 red background, 227 red center-pixel initial pattern, 237, 239 red flag, xi, xiv red gene family group, 198, 200, 205, 208 red gene family R, 206 red gene family group R, 206 red group, 373 reflection invariance theorem, 259 reflection plus complementation invariance theorem, 259

945

reflection, 192, 238, 240 resets, 2 return maps, 658 reversible catalog, 297 right-copycat rule, xi, xvi, 528, 593 right-copycat local rule, 509 right-shift, 910 right-shift law, 897 ring of coupled cells Ci , 2 robust attractor, 499 robust CA attractors, 387 robust periodic modes, 494 rotated cube, 266 rotation group, 229, 265, 270, 272, 296 rotation group R, 338 rotation matrices, 289, 295, 297, 348 rotation matrix, 271, 279, 347 rotations, 266, 288 rule N , 115 rule number, 6 rule of interaction, 2 rule 108 , 100 rule 110 , xix, 78, 79, 81, 83, 84, 88, 96, 235, 237–239 rule 124 , xx, 237, 239 rule 135 , xxvi rule 137 , xxi, 238 rule 145 , 238 rule 149 , xxvii rule 150 , 89–93, 96, 111 rule 170 , xvi rule 193 , xxii, 237–239 rule 20 , 98 rule 22 , 99 rule 232 , 93–96, 234 rule 240 , xviii rule 250 , 96–98 rule 254 , 111 rule 30 , xxiv, 99 rule 86 , xxv rule 90 , 100 rules {193, 137, 124, 110}, 240 scanning window, 191 secondary firing pattern, 190, 195 secondary firing patterns, 192, 210 secondary firing vertices, 191 secondary trihedron, 191, 192 secondary vertex weight, 195 seed, 273 “seed” Boolean cube, 295 self loop, 765 semigroup, 4

946

Index

separable rules, 110 separating plane, 97 separating planes, 85, 90, 91 separation plane, 84 separation planes, 86, 87 sgn[x], 120 shifting mode, 465 Shilnikov, 3 siblings, 198 sign-alternating property, 491 simplified differential equation, 79 single red center-pixel, 191, 192, 210, 211 single-valued inverse map, 453 space-time diagrams, 921 sphere of influence, 4 star transition graph, 347, 348 state, 4, 114 state equation, 11, 12, 76, 89, 231 state space, 79, 81 state space Σ, 372, 373 state transition diagrams, 672 state variable, 114 state variables, 80 steady state, 765 steady state regimes, 507 stratification of characteristic functions, 619 stratification prediction procedure, 623 stratification, 624–628 striped (alternating red-blue) background, 227 structural complexity, 2, 6 subshift, 893 symbolic and numeric truth tables, 231, 261 symbolic binary output, 231 symbolic Boolean vector, 229, 231 symbolic code, 338 symbolic truth table, 230, 231, 370, 510, 659 symmetric and self-similar, 262 synapses, 121 synaptic coefficient, 509 synaptic weights, 113, 121, 122, 190 synchronization, xv table of time-reversible rules, 707 the 4 musketeers, xxiii threshold of complexity, 1, 2, 111, 112, 232 threshold, 4, 112 time machines, 657 time reversal comparison pattern, 700 time reversal test, 702, 900, 902, 904, 907 time reversible, 715 time reversality, 911 time scale, 2

time scales, 3 time series, 382 time travel, 657 time-τ map, 465 time-τ maps, 382, 383 time-τ CA characteristic function, 373 time-τ characteristic function, 658 time-τ characteristic function χτ62 , 661 time-τ characteristic function χτN , 373 time-1 maps, 452, 459, 463 time-1 characteristic function, 509, 511 time-2 Bernoulli rules, 738 time-2 characteristic functions, 809 time-3 characteristic functions, 815 time-irreversible and noninvertible period-2 rules, 806 time-irreversible attractors, 657 time-irreversible period-3 attractor, 701 time-irreversible rules, 706, 715 time-reversal test, 657, 701 time-reversible and invertible period-2 rules, 797 time-reversible attractor, 700, 711 time-reversible attractors, vii, 657 time-reversible Bernoulli rules, 733 time-reversible rules, 705, 706, 715 topological dynamics, 660 topologically conjugate, 265 topologically-conjugate, 914 trajectory, 81 transient duration, 380, 381 transient period, 765 transient regime, 380, 665 transient regimes, 379, 658, 765, 769 transient, 507 transistors, 3 transition point, 84, 94, 96, 97 transition points, 84–86, 88, 90, 91 truth table, xiv, 1–3, 5, 11, 78, 114, 119, 233, 348, 370 truth table transformation matrices, 339 truth-table mapping matrices, 296 turing machine, 358 turing-universal rules, 369, 499 unimodular matrices, 264 unitary matrices, 348 universal CA map, 121, 122 universal computation, 121, 241, 264, 358, 498 universal computations, 229 universal neuron, vii, 113, 119, 121 universal turing machine, xi, 78, 121

Index

unpredictable, 229, 241, 264 user-friendly language, 4 Valery Sbitnev, vii vertex, 79, 80, 85, 90, 91, 94, 97, 103, 115, 191 vertex number, 120 vertex weight, 193 vertex weights, 231 vertices, 81, 82, 113, 114 Vierergruppe, 229, 248, 252, 338, 358, 658 vignette 3 , 453 vignette 62 , 454 vignette, 387, 459, 465 vignettes, 452

947

weights of vertices, 231 Wiener, 453, 454, 494 Wolfram, vii, 1, 2, 78, 112–114, 119, 121, 190, 232, 233, 241, 359 Wolfram’s classes 3 and 4, 382 Wolfram’s class 1 rules, 379 Wolfram’s class 2 rules, 379 Wolfram’s class 3 CA rules, 379 Wolfram’s class 4 rules, 379 zero-crossing, 84

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