Chaotic Systems Theory and Applications
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Chaotic Systems Theory and Applications Selected Papers from the 2nd Chaotic Modeling and Simulation International Conference (CHAOS2009) Chania, Crete, Greece
1 - 5 June 2009
editors
Christos H Skiadas Technical University of Crete, Greece
Ioannis Dimotikalis Technological Educational Institute of Crete, Greece
,~ World Scientific NEW JERSEY· LONDON· SINGAPORE· BEIJING· SHANGHAI· HONG KONG· TAIPEI· CHENNAI
Published by
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CHAOTIC SYSTEMS Theory and Applications Selected Papers from the 2nd Chaotic Modeling and Simulation International Conference (CHAOS2009) Copyright © 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN-13 978-981-4299-71-8 ISBN-IO 981-4299-7 1-5
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v
Preface This book contains a collection of papers suggested by the scientific committee including the best papers presented in the 2nd International Conference (CHAOS2009) on Chaotic Modeling, Simulation and Applications, Chania, Crete, Greece, June 1-5,2009. The aim of the conference was to invite and bring together people working in interesting topics of chaotic modeling, nonlinear and dynamical systems and chaotic simulation. The book includes papers on various important topics of Chaotic Modeling and Simulation as: • Bifurcation and chaos in multi machine power systems, Chaos in multiplicative systems, Deterministic chaos machine, Dynamics of a bouncing ball, N-body chaos. • Spatiotemporal chaos, Chaotic mixing in the system Earth, Multiple equilibria and endogenous cycles, Growth models, EnKF and particle filters for meteorological models, 3D steady flows, Local and global Lyapunov exponents, Routh-Hurwitz conditions and the Lyapunov second method, Chaotic behavior of plasma surface interaction, Simulation of geochemical self-organization, Atmospheric pressure plasma. • On the entropy flows to disorder, Quantum scattering and transport, Electron quantum transport, Qualitative dynamics of interacting classical spins, Atomic de Broglie-wave chaos, q-deformed nonlinear maps. • A two population model for the stock market problem, Complex dynamics in an asset pricing model, Hick Samuelson Keynes dynamic economic models, Maxwell-Bloch equations as predator-prey system. • Linear communication channels and synchronization, Identifying chaotic and quasiperiodic time-series, New model of nonlinear oscillations generators, Transmission line models, Optimum CSK communication, Symmetry-break in a minimal Lorenz-like system, Longwave Marangoni convection. • Nonlinear system's synthesis, Strategies of synergetics, Synergetics control, Synergetics and multi-machine power system modification. • Sub-critical transitions in coupled map lattices, Markov processes and transition probabilities. • Music composition from the cosine law of a frequency-amplitude triangle, Composing chaotic music from a varying second order recurrence equation. • Chaotic neural networks, A novel neuro-fuzzy algorithm. We thank all the contributors to the success of the CHAOS 2009 International Conference, the committees, the plenary speakers, the reviewers and especially the authors of this volume. Special thanks to the Conference Secretary Dr. Anthi Katsirikou for her work and assistance. Finally, we would like to thank Mary Karadima, Aggeliki Oikonomou and George Matalliotakis for their valuable support. October 30, 2009
Christos H. Skiadas, Technical,Dniversity of Crete, Greece Ioannis Dimotikalis, Technological Educational Institute of Crete, Greece
vi
Honorary Committee David Ruelle Academie des Sciences de Paris Honorary Professor at the Institut des Hautes Etudes Scientifiques of Bures-sur-Yvette, France Leon O. Chua EECS Department, University of California, Berkeley, USA Editor of the International Journal of Bifurcation and Chaos Ji-Huan He Donghua University, Shanghai, China Editor of Int. Journal of Nonlinear Sciences and Numerical Simulation
International Scientific Committee C. H. Skiadas (Technical University of Crete, Chania, Greece), Chair H. Adeli (The Ohio State University, USA) N. Akhmediev (Australian National University, Australia)
M. Amabili (McGill University, Montreal, Canada) J. Awrejcewicz (Technical University of Lodz, Poland) S. Bishop (University College London, UK)
T. Bountis (University of Patras, Greece) Y. S. Boutalis (Democritus University of Thrace, Greece)
C. Chandre (Centre de Physique Theorique, Marseille, France) M. Christodoulou (Technical University of Crete, Chania, Crete, Greece) P. Commendatore (Universita di Napoli 'Federico II', Italy) D. Dhar (Tata Institute of Fundamental Research, India) J. Dimotikalis (Technological Educational Institute, Crete, Greece)
B. Epureanu (University of Michigan, Ann Arbor, MI, USA) G. Fagiolo (Sant'Anna School of Advanced Studies, Pisa, Italy) V. Grigoras (University of Iasi, Romania) L. Hong (Xi'an Jiaotong University, Xi'an, Shaanxi, China) G. Hunt (Centre for Nonlinear Mechanics, University of Bath, Bath, UK) T. Kapitaniak (Technical University of Lodz, Lodz, Poland) A. Kolesnikov (Southern Federal University, Russia) G. P. Kapoor (Indian Institute of Technology Kanpur, Kanpur, India) V. Krysko (Dept. of Math. and Modeling, Saratov State Technical University, Russia) W. Li (Northwestern Poly technical University, China)
International Scientific Committee
B. L. Lan (School of Engineering, Monash University, Selangor, Malaysia) V. J. Law (Dublin City University, Glasnevin, Dublin, Ireland) V. Lucarini (University of Reading, UK)
J. A. T. Machado (ISEP-Institute of Engineering of Porto, Porto, Portugal) W. M. Macek (Cardinal Stefan Wyszynski University, Warsaw, Poland) P. Mahanti (University of New Brunswick, Saint John, Canada) G. M. Mahmoud (Assiut University, Assiut, Egypt) P. Manneville (Laboratoire d'Hydrodynamique, Ecole Polytechnique, France) N. Mastorakis (Techn. Univ. of Sofia, Bulgaria and ASEI, Greece) A. S. Mikhailov (Fritz Haber Institute of Max Planck Society, Berlin, Germany) M. S. M. Noorani (Universiti Kebangsaan Malaysia, Malaysia) G. V. Orman (Transilvania University of Brasov, Romania) S. Panchev (Bulgarian Academy of Sciences, Bulgaria) G. Pedrizzetti (University of Trieste, Trieste, Italy) F. Pellicano (Universita di Modena e Reggio Emilia, Italy) S. V. Prants (Pacific Oceanological Institute of RAS, Vladivostok, Russia) A. G. Ramm (Kansas State University, Kansas, USA) G. Rega (University of Rome "La Sapienza", Italy) H. Skiadas (Hanover College, Hanover, USA) V. Snasel (VSB-Technical University of Ostrava, Czech) D. Sotiropoulos (Technical University of Crete, Chania, Crete, Greece) B. Spagnolo (University of Palermo, Italy)
P. D. Spanos (Rice University, Houston, TX, USA)
J. C. Sprott (University of Wisconsin, Madison, WI, USA) S. Thurner (Medical University of Vienna, Austria) D. Trigiante (Universita di Firenze, Firenze, Italy) G. Unal (Yeditepe University, Istanbul, Turkey) A. Valyaev (Nuclear Safety Institute of RAS, Russia) A. Vakakis (University of Illinois, Urbana, USA)
J. P. van der Weele (University ofPatras, Greece) M. Wiercigroch (University of Aberdeen, Aberdeen, Scotland, UK)
VII
viii
Keynote Talks Marco Amabili Department of Mechanical Engineering, McGill U ni versity, Montreal, Canada Chaotic vibrations of circular cylindrical shells: Garlekin versus reducedorder models Jan Awrejcewicz Department of Automatics and Biomechanics, Technical University of Lodz, Lodz, Poland Deterministic chaos machine: Experimental vs. numerical investigations Alfred Inselberg School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel Multidimensional visualization and its applications Pier A. Mello Institute of Physics, Universidad N acional Autonoma de Mexico, Mexico Quantum scattering and transport in classically chaotic cavities: An overview of old and new results Sergey V. Prants Pacific Oceanological Institute of the Russian Academy of Sciences, Vladivostok, Russia Quantum chaos with atoms in a laser field Alexander G. Ramm Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA http://www.math.ksu.edu/-ramm Acoustic and electromagnetic wave scattering by many small particles and creating materials with desired properties Giuseppe Rega Department of Structural Engineering and Geotechnical Engineering, University of Rome 'La Sapienza', Italy Experimental unfolding and theoretical model of the transition to complex dynamics in sagged cables Konstantinos S. Tsakalis Department of Electrical Engineering, Ira A. Fulton School of Engineering, Arizona State University, USA Chaos, brain and epilepsy: A bioengineering approach Alexander F. Vakakis Department of Mechanical Science and Engineering, University of Illinois, Urbana, USA Nonlinear targeted energy transfer in dynamical systems
ix
Contents Preface
v
Honorary Committee and International Scientific Committee
vi
Keynote Talks
viii
The Influence of Machine Saturation on Bifurcation and Chaos in Multimachine Power Systems
Majdi M. Alomari and Jian Cue Zhu Chaos in Multiplicative Systems
9
Dorota Aniszewska and Marek Rybaczuk Deterministic Chaos Machine: Experimental vs. Numerical Investigations
17
Jan Awrejcewicz and Crzegorz Kudra Some Issues and Results on the EnKF and Particle Filters for Meteorological Models
27
Christophe Baehr and Olivier Pannekoucke Local and Global Lyapunov Exponents in a Discrete Mass Waterwheel
35
David Becerra Alonso and Valery Tereshko Complex Dynamics in an Asset Pricing Model with Updating Wealth
43
Serena Brianzoni, Cristiana Mammana, and Elisabetta Michetti Chaotic Mixing in the System Earth: Mixing Granitic and Basaltic Liquids
51
Cristina De Campos, Werner Ertel-Ingrisch, Diego Perugini, Donald B. Dingwell, and Ciampero Poli Multiple Equilibria and Endogenous Cycles in a Non-Linear Harrodian Growth Model Pasquale Commendatore, Elisabetta Michetti, and Antonio Pinto Hick Samuelson Keynes Dynamic Economic Model with Discrete Time and Consumer Sentiment
59
67
Loretti 1. Dobrescu, Mihaela Neamtu, and Dumitru Opri§ On the Entropy Flows to Disorder
C. T. J. Dodson
75
x
Contents
Linear Communication Channel Based on Chaos Synchronization
85
Victor Grigoras and Carmen Grigoras Maxwell-Bloch Equations as Predator-Prey System A. S. Hacinliyan, 0. O. Aybar, 1. KU$beyzi, 1. Temizer, and E. E. Akkaya Identifying Chaotic and Quasiperiodic Time-Series Candidates for Efficient Nonlinear Projective Noise Reduction
93
101
Nada Jevtic and Jeffrey S. Schweitzer Chaos from the Observer's Mathematics Point of View
111
Dmitriy Khots and Boris Khots New Models of Nonlinear Oscillations Generators Alexander A. Kolesnikov Nonlinear System's Synthesis - The Central Problem of Modern Science and Technology: Synergetics Conception. Part II: Strategies of Synergetics Control Anatoly A. Kolesnikov Synergetic Approach to Traditional Control Laws Multi-Machine Power System Modification
119
128
134
Anatoly A. Kolesnikov and Andrew A. Kuzmenko Dynamic Stability Loss of Closed Circled Cylindrical Shells Estimation Using Wavelets
139
V. A. Krysko, J. Awrejcewicz, M. Zhigalov, V. Soldatov, E. S. Kuznetsova, and S. Mitskevich The Application of Multivariate Analysis Tools for Non-Invasive Performance Analysis of Atmospheric Pressure Plasma
147
V. J. Law, J. Tynan, G. Byrne, D. P. Dowling, and S. Daniels
Chaos Communication: An Overview of Exact, Optimum and Approximate Results Using Statistical Theory
155
Anthony J. Lawrance Dynamics of a Bouncing Ball
165
Shiuan-Ni Liang and Boon Leong Lan Symmetry-Break in a Minimal Lorenz-Like System Valerio Lucarini and Klaus Fraedrich
170
Contents
Sub-Critical Transitions in Coupled Map Lattices P. Manneville Quantum Scattering and Transport in Classically Chaotic Cavities: An Overview of Old and New Results
xi 185
191
Pier A. Mello, VIctor A. Gopar, and J. A. Mendez-Bermudez
Parametric Excitation of a Longwave Marangoni Convection
207
Alexander B. Mikishev, Alexander A. Nepomnyashchy, and Boris L. Smorodin
Symmetry Broken in Low Dimensional N-Body Chaos David C. Ni and Chou Hsin Chin On Markov Processes: A Survey of the Transition Probabilities and Markov Property
215
224
Gabriel V. Orman
Qualitative Dynamics of Interacting Classical Spins Leonidas Pantelidis
232
A Transmission Line Model for the Spherical Beltrami Problem
241
C. D. Papageorgiou and T. E. Raptis
Two Aspects of Optimum CSK Communication: Spreading and Decoding
249
Theodore Papamarkou
A Numerical Exploration of the Dynamical Behaviour of q-Deformed Nonlinear Maps Vinod Patidar, G. Purohit, and K. K. Sud Spatiotemporal Chaos in Distributed Systems: Theory and Practice George P. Pavlos, A. C. Iliopoulos, V. G. Tsoutsouras, L. P. Karakatsanis, and E. G. Pavlos An Enhanced Tree-Shaped Adachi-Like Chaotic Neural Network Requiring Linear-Time Computations Ke Qin and B. John Oommen Clustering of Inertial Particles in 3D Steady Flows Themistoklis Sapsis and George Haller
257
268
284
294
XII
Contents
A Two Population Model for the Stock Market Problem
302
Christos H. Skiadas
Electron Quantum Transport through a Mesoscopic Device: Dephasing and Absorption Induced by Interaction with a Complicated Background
309
Valentin V. Sokolov
Composing Chaotic Music from a Varying Second Order Recurrence Equation
320
Anastasios D. Sotiropoulos and Vaggelis D. Sotiropoulos
Music Composition from the Cosine Law of a Frequency-Amplitude Triangle
327
Vaggelis D. Sotiropoulos and Anastasios D. Sotiropoulos
Chaos in a Fractional-Order Jerk Model Using Tanh Nonlinearity
338
Banlue Srisuchinwong
Nonlinear Systems in Brunovsky Canonical Form: A Novel Neuro-Fuzzy Algorithm for Direct Adaptive Regulation with Robustness Analysis Dimitris C. Theodoridis, Yiannis Boutalis, and Manolis A. Christodoulou
346
Routh-Hurwitz Conditions and Lyapunov Second Method for a Nonlinear System Omiir Umut
362
Atomic de Broglie-Wave Chaos V. o. Vitkovsky, L. E. Konkov, and S. V. Prants Chaotic Behavior of Plasma Surface Interaction. A Table of Plasma Treatment Parameters Useful to the Restoration of Metallic Archaeological Objects
369
377
C. L. Xaplanteris and E. Filippaki Simulation of Geochemical Self-Organization: Acid Infiltration and Mineral Deposition in a Porous Ferruginous Limestone Rock
385
Farah Zaknoun, Houssam El-Rassy, Mazen AI-Ghoul, Samia Al-Joubeily, Tharwat Mokalled, and Rabih Sultan
Author Index
395
The Influence of Machine Saturation on Bifurcation and Chaos in Multimachine Power Systems Majdi M. Alomari and Jian Gue Zhu University of Technology, Sydney (UTS) P.O. Box 123, Broadway NSW 2007, Australia (e-mail:
[email protected]) Abstract: A bifurcation theory is applied to the multimachine power system to investigate the effect of iron saturation on the complex dynamics of the system. The second system of the IEEE second benchmark model of Subsynchronous Resonance (SSR) is considered. The system studied can be mathematically modeled as a set of first order nonlinear ordinary differential equations with (p=X/XL ) as a bifurcation parameter. Hence, bifurcation theory can be applied to nonlinear dynamical systems, which can be written as dxldt=F(x;p). The results show that the influence of machine saturation expands the unstable region when the system loses stability at the Hopf bifurcation point at a less value of compensation. Keywords: machine saturation , Hopf bifurcation, chaos, subsynchronous resonance, damper windings.
1. Introduction Addition of series compensation in power systems, which is considered for increasing effectively the power transfer capability as well as improving the stability, introduces a problem called subsynchronous resonance (SSR). Subsynchronous resonance can be defined as electromechanical interaction between electrical resonant circuits of the transmission system and the torsional natural frequencies of the turbine-generator rotor. Where the torques produced by the interaction of the subharmonic current in the stator RLC circuit and the rotor circuits of the synchronous generator may excite one of the natural torsional modes of oscillation of the turbine-generator masses leading to shaft failure. Three types of SSR can identify the interaction of the system and the generator under the subsynchronous resonance. They have been called torsional interaction effect, induction generator effect, and transient torque effect. In this research, we focus on the torsional interaction effect, which results from the interaction of the electrical subsynchronous mode with the torsional mode. On the other hand, several methods have been used in SSR study. The most common of these methods are eigenvalue analysis, frequency scanning, and time-domain analysis. The eigenvalue analysis is used in this study. It is a very valuable technique because it provides both the frequencies of oscillation and the damping at each frequency. Abed and Varaiya [1] employed Hopf bifurcation theory to explain nonlinear oscillatory behavior in power systems. The bifurcation theorem was used by Zhu et al [2] to demonstrate the existence of a Hopf bifurcation in a single machine infinite busbar (SMIB) power system, in which the dynamics of
2
M. M. Alomari and 1. G. Zhu
the damper windings and the AVR are neglected. Tomim et al [3] proposed an index that identifies Hopf bifurcation points in power systems susceptible to subsynchronous resonance. N ayfeh et al [4] applied the bifurcation theory to a practical series capacitor compensated single machine power system, the BOARDMAN turbine-generator system. To make the analysis simple and manageable, the effect of stator and rotor iron saturation in synchronous machines was neglected in all of the bifurcation analysis given in the literature. However, the importance of saturation has been recognized by many authors [5, 6, 7]. The effect of electrical machine saturation on SSR was studied by Harb et al [8]. They concluded that the generator saturation slightly shrinks the positively damped region by shifting the Hopf bifurcation point, secondary Hopf bifurcation and bluesky catastrophe to smaller compensation level. In this research, power system dynamics has been studied using the nonlinear dynamics point of view, which utilizes the bifurcation theory. Bifurcation theory and chaos is used to investigate the complex dynamics of the considered system. The system loses stability via Hopf bifurcation point (H) and a limit cycle is born at /1=H. This limit cycle is stable if the bifurcation is supercritical and unstable if it is subcritical. The type of the Hopf bifurcation is determined by numerical integration of the system, with specific amount of initial disturbances, slightly before and after the bifurcation value. On further increase of the compensation factor, the system experiences chaos via torus attractor. Chaos is a bounded steady-state behavior that is not an equilibrium solution or a periodic solution or a quasiperiodic solution [9]. In this paper, we study the effect of iron saturation on Bifurcation and Chaos.
2. System Description The system considered is the two different machine infinite bus system, shown in Figure 1. The two machines have a common torsional mode connected to a single series compensated transmission line. The model and the parameters are provided in the second system of the IEEE second benchmark model. The electromechanical systems for the first and second units are shown in Figure. 2. The first unit consists of exciter (EX.), generator (Gen. I), low-pressure (LPl) and highpressure (HPl) turbine sections. And the second unit consists of generator (Gen.2), low-pressure (LP2) and high-pressure (HP2) turbine sections. Every LPl
M4 D4
Grn
Tran.-fonnm
Inllnite Bus
LP2 ~ K12b
~
~ MJb DJb
Fig. 1. Electrical system (Two different machine infinite bus system).
K23b
Ml Dl
D2
DJ
Iilin2l ~nth.
M2
MJ
M2b
Mlb
D2b
Dlb
Fig. 2. Electro-mechanical systems for the first and second units.
Influence of Machine Saturation on Bifurcation and Chaos
3
section has its own angular momentum constant M and damping coefficient D, and every pair of successive masses have their own shaft stiffness constant K, as shown in Figure 2. The data for electrical and mechanical system are provided in [10]. Replacement of these generators with a single equivalent generator will change the resonance characteristics and therefore is not justified. Consequently, each generator is represented in its own rotor frame of reference and suitable transformation is made.
3. Mathematical Model The mathematical model of the electrical and mechanical system will be presented in this section. Actually, the electrical system includes the dynamic nonlinear mathematical model of a synchronous generator and that of the transmission line. The generator model considered in this study includes five equations, d-axis stator winding, q-axis stator winding, d-axis rotor field winding, q-axis rotor damper winding and d-axis rotor damper winding equations. Each mass of the mechanical system can be modeled by a second order ordinary differential equation (swing equation), which is presented in state space model as two first order ordinary differential equations. Using the direct and quadrature d-q axes and Park's transformation, we can write the complete mathematical model that describes the dynamics of the system. The mathematical model of the electrical and mechanical system is provided in [II].
4. System Response in Case of Absence of Saturation In this section, we study the case of adding damper winding on sub synchronous resonance of the first generator but neglecting saturation. In case of no saturation, we set do, d l , d2, d3, qo, q" q2, and q3 equal to zero in equations (1) and (2). In this case we have 23 ordinary nonlinear differential equations. The equilibrium solution is obtained by setting the derivatives of the 23 state variables in the system equal to zero. The stability of the operating point is studied by examination of the eigenvalues of the linearized model evaluated at the operating point. The operating point stability regions in the brd1 plane together with two Hopf bifurcation points are depicted in Figure 3. We observe that the power system has a stable operating point to the left of HI::::0.212486 and to the right of H2::::0.880735, and has an unstable operating point between HI and H 2• The operating point loses stability at a Hopf bifurcation point, namely f1=H I . It regains stability at a reverse Hopf bifurcation, namely f1=H2 . In this case a pair of complex conjugate eigenvalues will transversally cross from left half to right half of the complex plane, and then back to the left half. To determine whether the limit cycles created due to the Hopf bifurcation are stable or unstable, we obtain the time response of the system by numerical integration with small disturbance slightly before HI. Figure 4 shows the response of the system with 7% initial disturbance on the speed of the generator at 1'=0.199135, which is less than HI. It can be observed that the system is unstable. Therefore, the type of this Hopf bifurcation is subcritical. So, the periodic solution emanating at the bifurcation point is unstable.
4
M. M. Alomari and 1. G. Zhu
: Stable operating point
!::\
r: ~
)I~ ~--'H-,
1-,-,
~
"'j
B~"nZ
~ 0.9
a:
: Unstable operating point Hi, H2 : Hopf bifurcation points
0.8 0.7
0 .6'----~-~-~-~-~-~-__'c=-~-~,__---'
o
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Compensation factor ~
• • • : Stable periodic orbits
1.22
o0 0
:0 1.21
: Unstable periodic orbits
IE
£
i
CF : Cyclic-fold bifurcation point
1.2
SH : Secondary Hopf bifurcation poinl
1.19
~ 1.18 ~ 1.17
H1
£ 1.16 15 ~
g>
1. 15
~ 1.14
~
CF
1.13
1.12 0.17
0.18
0.19
0.2
0.21
0.22
0.23
0.24
0.25
Compensation factor V
Fig. 3. Bifurcation diagram showing variation of the first generator rotor angle 0,., with the compensation factor f1 (for the case of absence of Saturation).
90
eo
10
°O:='--'O:'":,c--c. .8-~----""::.'-----C1.4 OAC --:CO""::.6- O':: lime (sec)
1.6
1.8
Time (sec)
Fig. 4. Response of the system. Rotor speed of the generator (left) and rotor angle of the generator (right) at f1=0.199135 with 7% initial di sturbance in rotor speed of generator (for the case of absence of Saturation).
Influence of Machine Saturation on Bifurcation and Chaos
5
Locally, the system has a stable equilibrium solution and a small unstable limit cycle for values of 11 slightly before HI' As 11 increases past HI. the equilibrium solution loses stability and the small unstable limit cycle disappears. As 11 decreases below HI. the unstable limit cycle deforms and increases in size until the branch of unstable limit cycles meets a branch of stable large-amplitude limit cycles at 11=0.198526, corresponding to point CF. As the value of compensation factors decreases more, the stable and unstable limit cycles collide and exterminate each other in a cyclic-fold bifurcation. As 11 increases from CF, the stable limit cycle grows in size while remaining stable. The limit cycle undergoes a secondary Hopf bifurcation at I1=SH;.o:oO.201325, resulting in a twoperiod quasiperiodic attractor. While it oscillates with more than one incommensurate frequency slightly after the secondary Hopf bifurcation point at 11=0.201350 and obtain a period quasiperiodic (torus) attractor. As 11 increases slightly after SH up to 11=0.206433, the chaotic attractor is obtained. Figure 5 shows the two-dimensional projections and time histories of the system at different compensation factors. ~ 10015
~
(a)
1.001
iU)OO5 .; ~
1
i
~O.9995
•
If
0 .999
1.208
1.209
1.21
1. 2 11
1.212
1.2131.214
1,2151 .2161.217 Tome (.ee)
Time (.eel
(b)
"
flolor..-.gleolll'l9f1rSI9_"'W!;r I I'OO)
~1_00
~
~
1.04
1
102
(c)
;
1
~
F:
"'0.94
RolC>ran 9leoflheir 24.84 for a = 10 and b = ~. Trajectory spirals out two equilibria and creates butterfly shape attractor, which is visible in logarithmic scale. In case of multi-dimensional dynamical systems calculating Lyapunov exponents is more complicated, because local behaviour of close trajectories may vary with the direction. For calculation of the largest Lyapunov exponent of the multiplicative Lorenz system we employed multiplicative version of method introduced in [6]. For classical dynamical system the largest Lyapunov exponent equals: . 1 L(t) (16) A = t-Hx) lun -In-(-), t L 0 . where L( t) is linear extent of ellipsoid created in space by two trajectories. For multiplicative dynamical systems the largest Lyapunov exponent should be calculated as: . 1 L(t) (17) A = t~oo lun -1n- t In -( -) , L 0 . because time shift is multiplicative. Results of the largest Lyapunov exponent computation for the multiplicative Lorenz system are given in Table 1.
11-0.93
1-0.59
1- 0.34
1- 0 . 12
1
0.82
1
0.91
Table 1. The largest Lyapunov exponent of the multiplicative Lorenz system for IJ = 10, b = ~, 40 000 steps, starting time 0.01 and time step 0.01.
6
Discussion
The results presented in the paper proof that all classical conditions concerning chaotic behavior can be extended onto multiplicative systems. The paper presents exemplary multiplicative systems with chaotic behaviour. Similarly like in case of dynamical system described with classical derivatives, nonlinearity and three indepenent variables are necessary for chaos occuring in systems described with multiplicative derivatives.
References l.D.Aniszewska. Multiplicative runge-kutta methods. Nonlinear Dynamics, 50:265 - 272, 2007.
16
D. Aniszewska and M. Rybaczuk
2.D.Aniszewska and M. Rybaczuk. Lyapunov type stability and lyapunov exponent for exemplary multiplicative dynamical systems. Nonlinear Dynamics, 54:345 ~ 354, 2008. 3.M.Rybaczuk, A. Kedzia, and W. Zielinski. The concept of physical and fractal dimension II. the differential calculus in dimensional spaces. Chaos, Solitons E'3 Fractals, 12:2537~2552, 200l. 4.M.Rybaczuk and P .Stoppel. The fractal growth of fatigue defects in m aterials. International Journal of Fracture, 103:71 ~94 , 2000. 5.V. Volterra and B.Hostinsky. Herman, Paris, 1938. 6.A. Wolf, J.B. Swift, H.L. Swinney, and J.A. Vastano . Determining lyapunov exponents from a time series. Physica 16 D, pages 285 ~ 317, 1985.
17
Deterministic Chaos Machine: Experimental vs. Numerical Investigation Jan Awrejcewicz and Grzegorz Kudra Department of Automatics and Biomechanics Technical University of Lodz , Poland (e-mail:
[email protected])
Abstract: Detenninistic chaos machine consisting of the plane triple pendulum as well as of driving and measurement subsystems is presented and studied. The pendulum-driving subsystem consists of two engines of slow alternating currents and optoelectronic commutation. In addition, a mathematical model of the experimental rig is derived as a system of three second order strongly nonlinear ODEs. Mathematical modeling includes details, taking into account some characteristic features (for example, real characteristics of joints built by the use of roller bearings) as well as some imperfections (asymmetry of the forcing) of the real system. Parameters of the model are obtained by a combination of the estimation from experimental data and direct measurements of the system's geometric and physical parameters. A few versions of the model of resistance in the joints are tested in the identification process. Good agreement between both numerical simulation results and experimental measurements have been obtained and presented. Keywords: triple pendulum, experiment, chaos, mathematical modeling, parameter estimation.
1. Introduction Development of mathematics, mechanics and related numerical methods allow more exactly model the real dynamic phenomena that are exhibited by various physical objects. Following historical overview of the natural sciences mentioned, a significant role is played by a physical pendulum, which is the very useful mechanism used in the design of various real processes. As a single or a double pendulum are often studied experimentally [5,6], a triple physical pendulum is rarely presented in literature from a point of view of real experimental object. For example, in the work [7] the triple pendulum excited by horizontal harmonic motion of the pendulum frame is presented and a few examples of chaotic attractors are reported. Experimental rigs of any pendulums are still of interest of many researchers dealing with dynamics of continuous multi degrees-of-freedom mechanical systems. The model having such a properties has been analyzed in work [3]. It consists of a chain of N identical pendulums coupled by dumped elastic joints subject to vertical sinusoidal forcing on its base. Lately, also the monograph on the pendulum has been published [4]. This is a large study on this simple system also from the historical point of view. In February, 2005, in the Department of Automatics and Biomechanics, the experimental rig of triple physical pendulum was finished and activated. This stand has been constructed and built in order to investigate experimentally various phenomena of nonlinear dynamics, including regular and chaotic motions, bifurcations, coexisting attractors, etc. In order to have more deep
18
1. Awrejcewicz and G. Kudra
insight into dynamics of the real pendulum, the corresponding mathematical model is required. In the work [1] the suitable mathematical modeling and numerical analysis have been performed, where the viscous damping in the pendulum joints (constructed by the use of rolling bearings) has been assumed. In the next step [2], we have also taken into account the dry friction in the joints with many details and variants. Here we present the model of friction taking into account only essential details.
Figure 1. Experimental rig: 1, 2,3 - links; 4 - stand; 5- rotors ; 6 - stators; 7, 8, 9 rotational potentiometers
2. Experimental setup The experimental model (see Fig. 1) of the triple physical pendulum is composed of the following main modules: pendulum, drivin·g subsystem and the measurement subsystem. The construction of the pendulum ensures the plane motion of the dynamical system. The links (1 , 2, 3) are suspended on the frame (4) and joined by the use of radial and axial needle bearings. The fust link is forced by a special directcurrent motor of our own construction with optical commutation consisting of two stators (6) and two rotors (5). The construction ensures avoiding the skewing of the structure and forming the forces and moments in planes different
Deterministic Chaos Machine
19
that the plane of the assumed pendulum motion. On the other hand the construction allows the full rotations of all the links of the pendulum. The voltage conveyed to the DC motor inductors is controlled by the use of special digital system of our own construction together with precise signal generator HAMEG. As a result the square-shape in time of the voltage signal (but with some asymmetry - see the next sections) with adjustable frequency is obtained. The measurement of the angular position of the three links is realized by the use of the precise rotational potentiometers (7, 8, 9). Then the LabView measure-programming system is used for experimental data acquisition and presentation on a computer.
y
Figure 2. Physical model of the pendulum.
3. Mathematical model In the Figure 2 the idealized physical concept of real pendulum is presented. The system is idealized since it is assumed that it is an ideally plane system of coupled links, moving in the vacuum with the assumed model of friction in joints. Each of the pendulums has a mass center lying in the lines including the joints and one of the principal central inertia axes (Zei) of each link is
20
1. Awrejcewicz and G. Kudra
perpendicular to the movement plane. The system is governed by the following set of differential equations:
where the pendulum position is described by the use of three angles If/i (i = 1,2, 3), Mit) is the external forcing of the first link, MRi (i = 1, 2, 3) are friction torques generated in the joints and where the following notation is used
M, =m,ge, +(m2 +m3)gl, ' M2 M3
= m2ge 2 + m3g12, = m3ge3,
= I, +e,2m, +1,2 (m2 +m3)' 2 2 B2 = 12 + e2 m2 + 12 m3 ' B,
B3 =13 +e32m3 ' N'2 = m2e21, + mi,12' N'3 = m3 e31, ' N 23 = m3e312,
where mj and Ii (i = 1, 2, 3) are the masses and inertia moments with respect to the central axes perpendicular to the movement plane of the corresponding links and g is gravitational acceleration. Other geometrical parameters are seen in the Figure 2. Note that there is the following dependence between introduced quantities
Deterministic Chaos Machine
21
and
The resistance torques in the pendulum joints are modeled with the assumption that they are the function s of the relative velocities of the corresponding links only (among others they are not dependent on the bearing loadings):
M RI
=I; 3.. arctan (£I'ifl) + CI'if1 ' Jr
M R2
= T2 3.. arctan (£2 ('if2 - 'ifl)) + C2('if2 - 'ifl) ,
M R3
= T33.. arctan ( £ 3 ( 'if3 -
Jr
Jr
'if2 ) ) + C3 ( 'if3 - 'if2 ) ,
and they are composed of two parts: linear (viscous) part and nonlinear part in the form of arctan function as seen above. The arctan (or tanh) function is usually used in modeling of dry friction with high value of the c parameter and its classical objective is to obtain a smooth approximation of the sign function. Although our initial goal was just a smooth approximation of sign function, the parameter estimation results (see the next section) have been shown, that smaller values of the c parameter can give better results of modeling of the investigated triple pendulum. The external input to the system is the voltage signal u controlling the direction of the constant voltage applied to the DC motor of the pendulum. The signal u in the pendulum model can be an arbitrary function of time, and in particular, it can be the same function as applied (and recorded in a file) in real system (it is useful in the parameter estimation process). On the other hand, it is possible to apply the signal u due to the following mathematical description:
U(f)={1
-I
for mod ( lOt + 90' 2Jr) ::; 2Jra for mod(lOt+90,2Jr»2Jra'
which imitates the signal applied in the real pendulum, with adjustable angular velocity w, initial phase Wo and coefficient a (in the investigated experimental pendulum there is a = 0.4927 which means an asymmetry in the forcing, as mentioned in section 2).
22
1. Awrejcewicz and C. Kudra
The relation between the control signal u and torque Me acting on the first link is modeled by the following equation
Me (t) + TM dM e = q U (t) dt
,
where q is the amplitude of the external torque corresponding to the constant voltage applied in DC motor and TM is the time-constant of the DC motor.
4. Parameter estimation and experimental vs. numerical results Although some parameters of the system (for example masses, some geometrical lengths) can be obtained from direct measurements or calculated from such measurements, it can be difficult, labour-consuming or inaccurate for some of them. Because of that we have used only values that can be got easily and accurately, that is g = 9.812 rnIs 2, II = 0.17418 m and 12 = 0.22405 m. Other model parameters are obtained from identification process, that is from experimental data of input and output signals, where the input signal is the voltage signal u controlling the direction of the constant voltage applied to the DC motor of the pendulum and the outputs are the angular positions of the pendulum links lfIi (i = 1, 2, 3). One can also note that some other special experiments might be performed in order to identify particular model parameters, but here we assume that only above mentioned experimental data from the triple pendulum is available. The model parameters are estimated by the global minimum searching of the error-function of the model and real system matching. The matching of model and real system is understood as the matching of the corresponding output signals lfIi (i = 1, 2, 3) from model integrated numerically and from the real pendulum, assuming the same inputs to both the model and real system. The sum of squares of deviations between corresponding samples of signals from model and experiment, for few different solutions, serves as a criterion function. A minimum is searched by applying the simplex method. In order to avoid the local minima, the simplex method is stopped from time to time and a random searching is then applied. After random searching the simplex method is restarted again. If we divide final value of criterion-function by the number of samples used in calculation of the error -function, we obtain average square of deviation between two signals (obtained from the model and the experiment) let us denote this parameter as Fer- Now this parameter can be used for comparison of matching of different sets of experimental data and corresponding numerical solutions.
Deterministic Chaos Machine
23
Since the amplitude of the forcing is assumed to be known before identification process (q = 1.718 Nm), the full vector of the parameters to be estimated is
and other parameters are dependent since
and
0.2 0.103 0.1 ~
0.05
~
0
eel
,-
O. Therefore the demand functions are given by: Zl,t
=
1 A0'2 (1
+ r)(Xt
- 1),
Z2 ,t
=
1 A0'2 [a - 1 + r(Xt - 1)],
where Xt = ~ is the fundamental price ratio. p, The final nonlinear dynamical system T is written in t erms of the state variables Xt , mt and Wt:
= h(Xt, Wt) = tanh {,6/2 [,\~2 (Xt - a) [2'\"'0'2 (Xt - 2 + a + 2r(Xt - 1) +(Xt - a)Wt ) + r(Xt - 1)]- en,
mt+1
Complex Dynamics in Asset Pricing Model with Updating Wealth
with:
F _ 1 -
p.2
-
-4(1- w t){R+6 [a-Hr(xt- 1)][ ~ (Xt -2+a+2r(xt- 1)+(Xt -a)w,)+r(xt- 1)]} (1-m,)[exp{!3[",!2 (Xt-a)[2",Qa2 (xt-2+a+2r(xt-1)+(xt-a)w,)+r(xt-1)]-C]}+1]' 4(1+wt){ R+~ (l+r )(xt-1) [~(Xt -2+a+2r(xt -1 )+(Xt -a)w')+r(xt -1)]} /\17 2,\q (l+m,)[exp{ -!3[ (Xt- a )[ (Xt -2+a+2r(xt-1)+(xt -a)w,)+r(xt-1)]-C]}+1] '
2>.:2
",!2
G = 2R+ >'~2 [xt-2+a+2r(xt-l)+(xt-a)Wt]· +(Xt - a)Wt) + r(xt - 1)].
3
47
[2>''''0"2
(Xt - 2 + a + 2r(xt - 1)
Results
Our model admits two types of steady states:
- fundamental steady states characterized by x = 1, i.e. by the price being at the fundamental value, - non fundamental steady states for which x =I 1. More precisely, for a =I 1 the fundamental steady state E f of the system is such that wf = 1 and there exists a non fundamental steady state Enf such that wnf = -1, xnf = l-;:-a + 1. Observe that at the fundamental (non-fundamental) equilibrium the total wealth is owned by fundamentalists (chartists). When a = 1 the fixed point Enf becomes a fundamental steady state. More precisely, every point E = (1, tanh{ - cf}, w) is a fundamental equilibrium, i.e. the long-run wealth distribution at a fundamental steady state is given by any constant w E [-1,1]. In other words, a continuum of steady states exists: they are located in a one-dimensional subset (a straight line) of the phase space. In this case the steady state wealth distribution which is reached in the long run by the system strictly depends on the initial condition. Summarizing, the following lemma deals with the existence of the steady states: Lemma 1. The number of the steady states of the system T depends on the parameter a:
1. Let a =I 1, then: • for a < 1+r there exist two steady states: the fundamental equilibrium Ef =
(X f
=
1, rn
f = tanh { -
c:} ,W 1) f =
and the non-fundamental equilibrium l-a Enf= ( --+I,tanh
r
{f3[ l+r - -1- ( I - a ) 2]}) -C ,-1 2
'\0'2
r
48
S. Brianzoni, C. Mammana, and E. Michetti
1
(a)
(b)
1\~~~\ o
-1L-______________~
60
o
60
Fig. 1. State variable Wt versus time for an i.c. Xo = 0.8, rna = 1 and Wo = 1 and parameter values a = 0.5, ).. = 1, (J2 = 1, r = 0.02, C = 0.5, (3 = 1. (a) a = 0.1. Periodical fluctuations. (b) a = 0.5. Convergence .
• for a ~ 1 + r the fundamental steady state E f is unique 2. Let a = 1, then: • EveTY point E = (1, tanh { - Cf}, ill) is a fundamental equilibrium. Now we move to the study of the asymptotic dynamics of the system by using numerical simulations. We first consider the extreme case in which, at the initial time, all agents are fundamentalists and they own the total wealth. Moreover, we focus on the more interesting a's range of values, i.e. a < 1 + r. In Figure 1 (a) we present the trajectory of the state variable Wt while assuming that at the initial time 'rna = Wa = 1 and the price is above its fundamental value. We choose two different values of a and observe that if a is low enough the system fluctuates (a 12-period cycle is observed) while, as a increases the long run dynamics becomes simpler (Wt --+ 1 after a long transient) . Numerical simulations show the existence of a value ii such that for all a E (ii, 1 +r) the system converges to Ef, while more complex dynamics is exhibited if a < ii, i.e. for a low enough (see Figure 2). Notice that the system preserves similar features, i.e. the stabilizing effect of a in the long run, also in the other extreme case Wa = 'rna = -1, i.e. at the initial time the market is dominated by chartists who own all the wealth. In Figure 2 we observe a succession of periodic windows which occur at border collision bifurcations (see Hommes and Nusse[5] and Nusse and Yorke[6]). These periodic windows have the following properties: (i) the width of the periodic window reduces monotonically as the parameter a increases; (ii) the windows are characterized by the same periodicity; (iii) for a > ii, the periodic windows terminate in convergence. In order to consider also the role of the parameter (3, in Figure 3 we present a two-dimensional bifurcation diagram in the parameters' plane (a, (3) in the extreme case 'rna = Wa = 1 and the price .TO above the fundamental value.
Complex Dynamics ill Asset Pricing Model with Updating Wealth
a
49
a
a
Fig. 2. Bifurcation diagrams of the state variables w.r.L a. Initial condition and parameter values as in Figure 1.
Rkh dynamics is exhibited and the final behaviour increases in complexity as (3 increases (according to what happens in asset pricing models with heterogeneous agents and adapti veness). Other numerical simulations confirm the evidence that the systems is lIlore cornplicated for high values of /3 and low values of a.
.~jg~
I Jj:;
I.
.
G-~'
c--';r. ~
oUl=1440°.
3. Sample Preparation and Preliminary Electron Microprobe (EMP) Data The experiment was stopped by termination of the stirring protocol, switching off furnace power and cooling to room temperature. Crucible and spindle were removed from the furnace, and a cylinder of the resultant mixed glassy sample was recovered by coring. Horizontal equidistant sections (5 mm intervals) of this cylinder were prepared for
56
C. De Campos et al.
electron microprobe (EMP) analyses (Figures 3 and 5). The location of the EMP profile is along the black line in Figure 5. Optical and EMP micro-chemical studies revealed crystal-free filaments of intermediary compositions, changing with depth in complex patterns, with chaotic attractors showing slight variations in vortex strength. Here we present the results for the distribution of CaOand K20-contents (Figures 6 and 7) from a single analytical profile along the black line in section 5 (sample CD-25) at 20 mm depth depicted on Figure 5. This image is a false colour image from a mosaic consisting of over 150 photomicrographs from section 5 on Figure 3. This section crosses several filaments along a striking, stretched vortex structure. In Figure 6 the bottom white line represents the maximum error in CaO-content of the starting materials. This means that all points above this line are due to non-ideal mixing. Two main CaOenriched filaments are visible at about 2.5 mm and 3.5 mm distance. In Figure 7 these points are not correlated with the K20-distribution along the same line. This clearly shows that the mobility of both elements is not correlated. Another way to demonstrate this lack of correlation is shown in Figure 8: ratios of LlC MgO and LlCcao between ideal hybrid concentrations and measured partially mixed concentrations, for MgO and CaO are plotted along the same line. For this plot we assumed an error of 0.02 in the MgO- and CaO-contents due to background signal and sample preparation.
0.6
~
cf
"
()
0.4
8.8 0.0
0.2
0.4
0.6
0.8
1.0
1.2
distance in mm
Figure 6. CaO distribution profile plotted against distance, along a line across slice CD-25 on Figure 5.
~-
0, where y";;P is the potential output, yt the current output, K t the stock of capital, L t the amount of labour employed in production all evaluated at time t (no production lag). Parameters a and b are the reciprocal of the capital and labour coefficients respectively. In each period the capital stock is not fully utilized, so that the potential and current outputs do not coincide. We define the degree of capacity utilisation as the ratio between current and potential output: Ut = yt / Y";;p. Saving corresponds to a constant fraction of the income of the economy:
(1) where St is the ratio saving to capital and s is the (constant) propensity to save out of income. Investment is determined as follows:
(2) The component G in equation (2) is interpreted as the Harrodian warranted rate of growth that corresponds to the saving generated at normal capacity utilisation: G = sail. (3) The value il represents the normal degree of capacity utilisation, that is, the optimal degree of capacity utilization given the existing technology. Capital accumulation G t also depends nonlinearly on the current degree of capacity utilization Ut. On the nonlinear component ¢(Ut) of the investment function we make the following assumption, which assigns to it a sigmoidal shape (see Kaldor[4J; and Bischi et al.[l]):
Multiple Equilibria and Endogenous Cycles
61
Assumption 1. Function ¢ is such that ¢( u) = O. Furthermore ¢ is strictly increasing with respect to Ut (i. e. ¢' (Ut) > 0) and a u E (0, 1] exists such that ¢//(Ut) ~ «)0 iff Ut ::; (> )u.
Investment increases with the current degree of capacity utilisation. However, when the degree of utilisation is substantially below the normal value, the entrepreneurs' incentive for capital accumulation is small; when the degree of utilisation is substantially above the normal value, investment slows down again, due to the increasing installation costs and to the increasing risks of entering other projects. In order to guarantee positive capital accumulation, we also make the following assumption. Assumption 2. We assume G + ¢(O) > 0 , that is the long run expected growth of demand G is always high enough to induce positive investments even in correspondence of a low capacity utiK5ation in the current period.
The dynamics of the system is generated by the variations in the degree of capital utilisation in the face of discrepancies between demand and supply as follows : (4) where Ut+l denotes the one-period forwarded value of the state variable Ut while parameter 8 is the adjustment speed of capacity utilization. By substituting equations (1), (2) and (3) into (4), we obtain the general model for the evolution of the state variable Ut describing the degree of capacity utilisation :
(5)
3
Fixed points and their stability
Concerning the fixed points of (5) the following proposition holds . Proposition 1. Let u* be a steady state of map (5), then the following relat'ion holds: u* = u + ¢(u*)/sa.
Observe that the corresponding equilibrium investment (and saving) is given by G* = S* = G + ¢( U *). It is easy to verify that the following proposition holds: Proposition 2. u* ters.
=u
is a fixed point of (5) for any choice of the parame-
This proposition follows directly from the asumption ¢(u) = 0, hence Proposition 1 is verified for u* = u. As a consequence, the equilibrium . corresponding to the warranted rate of growth always exists. We need to verify the existence of other equilibria. Observe that map (5) can be expressed as the sum of a linear function and a sigmoidal function, that is , Ut+l = f(ut) = g(Ut) + 8¢('ut) = (hUt + m) + 8¢(ut), where h = (1 - 8sa) and m = 8sau. Then we prove the following proposition about the number of fixed points of f.
62
P. Commendatore, E. Michetti, and A. Pinto
Proposition 3. If 1'Cu) :::; 1 then u* = u is the unique fixed point of f , otherwise, if l' (u) > 1 then f has three fixed points (Ul < u < u r ).
Proof. Assume 0 :::; 1'(u) :::; 1 and observe that !"(Ut) 2: «)0 iff Ut :::; (> )u, being f"(ud = eq/'(Ut). Simple geometrical considerations prove that u* = u is the unique fixed point of f. Consider now the case in which l' (u) < O. Observe that 1'(Ut) = h + eq/(ud, \lUt > 0 and that eq/(Ut) :::; eq/(u) , hence 1'(Ut) :::; h + eq/(u) = 1'(u), \:IUt > O. Since 1'(u) < 0 thcn 1'(Ut) < 0, \:IUt > 0, i.e. f is strictly decreasing and consequently is has at most one fixed point. Finally consider the case 1'(u) > 1 and remember that f(O) 2: 0 and that f is convex (concave) for U :::; (> )u. Trivially f must intersect the 45 degree line in two more points, namely Ul and U r such that Ul < u < Ur . Observe that when 1'(u) = 1 the map passes from a fixed point to three fixed points via tangent bifurcation. When the fixed point u* = 'u is unique, we can draw some general conclusions about its stability and the final dynamics of the system, as stated in the following proposition. Proposition 4. Assume that u* = u is the unique fixed point of map (5). (i) If 0 :::; 1'(u) :::; 1 then u is globally stable; (ii) ~f - 1 < 1'(u) < 0 then Ul < u < U2 exist such that u attracts any trajectory starting from an i. c. Uo
E (V'1,U2); (iii) if 1'(u) :::; -1 then the generic trajectory diverges in
modulus. Proof. (i) Trivial. (ii) If 1'(u) < 0 the generic trajectory oscillates. In order to study the asymptotic behaviour of Ut, remember that only simple dynamics can be produced and consider the second iterate of f, namely f2. Then f2 is strictly increasing, it is concave (convex) if Ut :::; (> u) and remember that (j2(u))' = (j'(U))2. Then, - 1 < 1'(u) < 0 =} (j2(u))' E (0,1) and f2 has two more fixed points Ul and U2 such that Ul < U < U2 that are points of an unstable two-period cycle of f. This cycle separates the basin of attraction ofu, given by B(u) = (Ul,U2), from the basin B(oo) = (-oo,ud U (U2'+OO) whose trajectories diverges in modulus. (iii) Otherwise, 1'( u) :::; -1 =} (j2(u))' 2: 1 and, given its shape, j2 cannot have other fixed points, so that every trajectory diverges in modulus. About the case in which 1'(u) > I, and the map admits three equilibria, we can only draw conclusions about the instability of U. Anyway, in order to study the local stability of the other fixed points and to consider the possibility of complex dynamics to be exhibited, we have to specify function 4>. In particular we focus on the case in which 4> is simmetrc around the equilibrium U, i.e. 4>(- (Ut - u)) = -4>( Ut - u). A sigmoidal function enjoying this property is given by: (6)
Assume l' (u) > 1, then two more distinct equilibria are present , as proved in Proposition 3, namely Ul and Ur, which lie symmetrically around u i.c. Ur -
fl
= 'U - Ul.
Multiple Equilibria and Endogenous Cycles
63
Fig.L Two dimensional bifurcation diagram in parameters' plane (Z,8) being (31 = 0.08, (32 = 7.5 and u = 0.5. Curves Gi , i = 1, ... , 5 separate the plane into different regimes as analytically proved.
4
Local and global bifurcation analysis
We now present some local and global bifurcations for the map (5) when ¢(Ut) is given by (6). Define Z = sa, the map (5) becomes:
. F(u.t) = (1- eZ)'Ut
+ gZu + epi arctan(P2(Ut -
n).
(7)
We focus first on the case F ' (u) > 1, which implies P1P2 > Z . In such a case F(ut) may be non monotonic as stated in the following proposition. Proposition 5. Assume
pIfh >
Z and
ez > 1.
Then map F is bimodal.
Proof F is not monotonic iff F' changes sign. In order to have two distinct solutions for the equation [P2(Ut - u)]2(1- eZ) = -(1 + e(p1P2 - Z» being P1P2 > Z, it must be gZ > 1.
If Proposition 5 holds then em and eM are respectively the minimum and the maximum points of F , where em < u < eM and eM-n, = 'tt-e m · Observe that for a given Z , F is bimodal if the parameter () is sufficiently high. Let P1P2 > Z so that F admits three fixed points, then the curve () = l iZ, namely G1 , separates the parameter plane (Z, () into two regions (see Figure 1), For points below such a curve F is strictly increasing so that only simple dynamics can be exhibited, more precisely, every initial condition belonging to B(uI) = (-00, 'II) (B(u r ) = (u, +00» generates trajectories converging to UI (u r ). Otherwise, for points above G1 , F is bimodal and more complex dynamics can be produced, We focus on such a case. In the following discussion we take into account that, by simmetry, the properties of the local dynamics around the fixed point '/II also apply to '11 ' " and, similarly, the results concerning the minimum point also apply to the maximum point. We now distinguish between two cases. If em :s; F(c,n), that is, the minimum point is above the 45 degree line, then the fixed points UI and ttl' are locally stable and no complex dynamics
64
P. Commendatore, E. Michetti, and A.
Pinto
", Fig. 2. Staircase diagram of the map F for
/31 =
(l.OS,
/32 =
7.5, Z
=
0.4 and
u = 0.5. (a) B = 12.9: the attractor Al is chaotic; (b) B = 13.4: the unique attractor A after the contact bifurcation (occurring at B c:::'. 13.379).
can emerge. Condition C m = P(c",) defines a curve C 2 in the parameters' plane (Z, ()) such that for points above such a curve the minimum point is below the 45 degree line (see again Figure 1) . In fact a necessary condition for cycles or complex dynamics to be exhibited is that Cm > P(Cm), i.e. pl(nd < O. According to the previous consideration, in what follows we assume em > F(c,m). If PI(ud > -1, the only way nl may lose stability is via a flip bifurcation, hence the following proposition trivially holds. Proposition 6. Let
Ul
be a fixed point of F. Then for each (Z, ()) in the
region defined as .
J!(Ul)
the point
Ul
=
{
(B, Z) E 1R+ x (O,lh,82 ) : (}Z < 2 +
- B{"h[).,
1+
[,8 ( . ~ . )]2 2 lil -
}
11,
is locally stable.
From the previous result it is easy to determine the flip bifurcation curve C3 in the parameter plane (Z, B) (see Figure 1). Assume Cm > P(cm ), then map F is a non invertible Zl Z3 - Zl map (see Mira et al.[5]) and the set S = [F(cm) ,P(CM)] is trapping, i.e. pt(S) ~ S, 'It E Z+. As a consequence we focus our analysis on the restriction of F to the set S, that is, we consider the structure of the attractor A ~ S owned by P for any initia.l condit ion 110 E 8. The main results are summarized in the following Proposition. Proposition 7. Let Cm > P(cm ). (i) If p 2(c m ) < u then P has two coexist'ing attractors Al F( CM) almost all trajectories diverge. Consider Propositon 7. In case (i) map F has two coexisting attractors At and A,. having the same structure. They may both consist of fixed points,
Multiple Equilibria and Endogenous Cycles
65
O.
(1)
Here iL is the mean, and the standard deviation 17, given by K (;;:)2, is proportional to the mean. Hence the coefficient of variation is unity in the case that (1) reduces to the exponential distribution. Thus, K = 1 corresponds to an underlying Poisson random process complementary to the exponential distribution. When K < 1 the random variable X represents spacings between events that are more clustered than for a Poisson process and when K > 1 the spacings X are more uniformly distributed than for Poisson. The case when iL = n is a positive integer and K = 2 gives the ChiSquared distribution with n - 1 degrees of freedom; this is the distribution of ~ for variances 8 2 of samples of size n taken from a Gaussian population (Je
fo
with variance 17~. The gamma distribution has a conveniently tractable information geometry [1,2], and the Riemannian metric in the 2-dimensional manifold of gamma distributions (1) is
(2)
76
C. T. 1. Dodson
So the coordinates (p" K,) yield an orthogonal basis of tangent vectors, which is useful in calculations because then the arc length function is simply
The system of geodesic equations is difficult to solve analytically but numerical solutions using the Mathematica programs of Gray [7] were obtained in [2]. Figure 1 shows a spray of some maximally extended geodesics emanating from the point (p" K,) = (1,1). Geodesic curves have tangent vectors that are parallel along them and yield minimal information arc length distances in the gamma 2-manifold . 2.00 1.75
K, 1.50 1.25 1.00 0.75 0.50 0.25
0.5
1.0
1.5
2.0
2.5
3.0
/1
Fig. 1. Some examples of maximally extended geodesics passing through (/1, K,) = (1,1) in the gamma 2-manifold.
We note the following important uniqueness property: Theorem 1 (Hwang and Hu (8)). For independent positive random variables with a common probability density function f, having independence of the sample mean and the sample coefficient of variation is equivalent to f being the gamma distribution.
This property is one of the main reasons for the large number of applications of gamma distributions: many near-random natural processes have standard deviation approximately proportional to the mean [2].
On the Entropy Flows to Disorder 77
2
Gamma entropy flows
Papoulis [13] Chapter 15 gives an account of the role of the Shannon entropy function in probability theory, stochastic processes and coding theory. The entropy of (1) is shown in Figure 2 using Sf
(tt, r;,)
=
-1
00
f logf
dx : lR 2+
-+
lR
~ r;, -- log (~) + log(r(r;,)) -
(r;,-1)1,b(r;,)
(~3)
with gradient
( ~, J.i
_ (r;, -1) (r;,1,b/(r;,) -1) ) . r;,
(4)
r;:
where 1,b = is the digamma function. At fixed r;" the entropy increases like log J.i. At fixed mean J.i, the maximum entropy is given by r;, = 1, the exponential distribution case of maximal disorder or chaos.
Fig. 2. Shannon entmpy function Sf for the gamma family as a sm:face with respect to mean fJ, and", (left) and as a conto'u r plot with entropy gradient flow and integral curves (right), The asymptote is", = 1, the exponential case of maximum disorder.
Figure 2 on the right shows entropy as a contour plot with superimposed also some examples of integral curves of the entropy gradient flow field, namely curves c satisfying c:
[0,(0)
-+ ]R2:
c(t) =
V'Sf l(c(t)) .
(5)
78
C. T. J. Dodson
By inspection, we can see that the entropy gradient components are each in one variable only and in particular the first component has solution
so the mean increases exponentially with time. Such curves represent typical trajectories for processes subordinate to gamma distributions; the processes become increasingly disordered as K, -+ 1. The entropy gradient curves correspond to systems with ext ernal input- the mean increases as disorder increases. The asymptote is K, = 1, the exponential case of maximum disorder. Conversely, the reverse direction of the curves corresponds to evolution from total disorder to other states (clustered for K, < 1, and smoothed out, ' more crystal-like', for K, > 1) while the mean is allowed to reduce-somewhat like the situation after the Big Bang, see Dodson [4].
3
Constrained degeneration of order
Lucarini [ll] effectively illustrated the degeneration of order in his perturbations of the simple 3D cubic crystal lattices (SC, BCC, FCC) by an increasing spatial Gaussian noise. Physically, the perturbing spatial noise intensity corresponds somewhat to a lattice temperature in the structural symmetry degeneration. With rather moderate levels of noise, quite quickly the three tessellations became indistinguishable. In the presence of intense noise they all converged to the 3D Poisson-Voronoi tessellations, for which exact analytic results are known [6]. Moreover, in all cases the gamma distribution was an excellent model for the observed probability density functions of all metric and topological properties. See also Jarai-Szabo and Neda [9] for some analytic approximations using gamma distributions for two and three dimensional Poisson-Voronoi cell size statistics. Lucarini provided plots showing the evolution of the mean and standard deviation of these properties as they converge asymptotically towards the Poisson-Voronoi case, illustrating the degeneration of crystallinity from K, ~ 00 to lower values. Of course, the constraint of remaining tessellations , albeit highly disordered ones, precludes convergence down to the maximum entropy limit K, = 1. In fact the limiting values are K, ;::,;; 16 for number of vertices and the same for number of edges and K, ;::,;; 22 for the number of faces; actually these are discrete random variables and the gamma is not appropriate. However, for the positive real random variables, polyhedron volume in the limit has K, ;::,;; 5.6 and polygon face area K. ;::,;; 16. Lucarini [10] had reported similar findings for the 2D case of perturbations of the three regular tessellations of the plane: square, hexagonal and triangular. There also the gamma distribution gave a good fit for the distributions during the degeneration of the regular tessellations to the 2D Poisson-Voronoi case; the limiting values were K, ;::,;; 16 for the perimeter of polygons and K, ;::,;; 3.7 for areas.
On the Entropy Flows to Disorder
79
We can give another perspective on the level of constraint persistent in the limits for these disordered tessellations: for infinite random matrices the best fitting gamma distributions for eigenvalue spacings have", = 2.420, 4.247, 9.606 respectively for orthogonal, unitary and symplectic Gaussian ensembles [2]. These values in a sense indicate the increasing statistical constraints of algebraic relations as the ensembles change through orthogonality, unitarity and symplectivity. In the context of analytic number theory, gamma distributions give approximate distributions for the the reported data on spacings between zeros of the Riemann zeta function [12]. The best fit gamma distributions to spacings between the first two million zeros of the Riemann zeta function has "':::::; 5.6 [2].
For the SARS disease epidemic outbreak [3,14] the gamma distribution gave a good model for the infection process with", :::::; 3, cf. [5].
4
Bivariate gamma processes
Next we consider the bivariate case of a pair of coupled gamma processes. The McKay family can be thought of as giving the probability density for the two random variables X and Y = X + Z where X and Z both have gamma distributions. This smooth bivariate family M of density functions is defined in the positive octant of random variables 0 < x < y < (Xl with parameters CYi, C, CY2 E ~+ and probability density functions cCCXl+CX2)XCXl-i(y _
m(x, y) =
x)cx 2- i e -c y
(6)
r(CYdr(CY2)
The marginal density functions, of X and Yare:
(7)
x>o c(CX1 +CX2)y(CX 1+cx 2 )- le- c y
my(y) =
r(CYi
y>
+ CY2)
o.
Note that we cannot have both marginal distributions exponential. covariance and correlation coefficient of X and Yare: 0"12
=
CYi and p(X, Y) 2"" C
(8) The
= { ; ! £ ; i. CYi
+ CY2
Unlike other bivariate gamma families, the McKay information geometry is surprisingly tractable and there are a number of applications discussed in Arwini and Dodson [2]. In fact the parameters CYi, C, CY2 E ~+ are natural coordinates for the 3-dimensional manifold M of this family. The Riemannian
80
C. T. 1. Dodson
information metric is
(9)
It is difficult to present graphics of curves in the McKay 3-manifold M, but it has an interesting 2-dimensional submanifold M1 C M: 001 = 1. The density functions are of form: (10) defined on 0 < x < y < = with parameters c,002 E jR+. The correlation coefficient and marginal functions of X and Yare given by:
(11) (12) (13) In fact 002 = l~t, which in applications would give a measure of the variability not due to the correlation. The matrix of metric components [gij 1 on M1 is
(14) Some geodesics emanating from (002, c) = (1,1) E M1 are shown on the left of Figure 3. The density functions can be presented also in terms of the positive parameters (OO1,CJ12,OO2) where CJ12 is the covariance of X and Y
(15)
x>O
(16) y > O.
(17)
On the Entropy Flows to Disorder
c
81
c
Fig. 3. Geodesics passing through (c, (2) = (1,1) (left) and a contour plot of the entropy wdh some integral gradient CU'f'1JC8 (right), in the McKay submanifold 1111 which has a1 = 1.
T he entropy function is
On the right of F igure 3 is a contour plot of t he entropy showing its gradient field and some integral curves. It may be helpful to express the entropy in terms of a2 and p, which gives
T he McKay entropy in M1 with respect to a2, P is shown in F igure 4 (left) with the gradient flow on a contour plot (right) together with the approximate locus curve of maximum entropy. The two thinner curves show t he loci of a 1 = 1 (upper curve) and a1 + a2 = 1 (lower curve), which correspond, respectively, to Poisson random processes for the X and Y variables.
82
C. T. 1. Dodson
a2
Fig. 4. Surface representation of the Shannon entropy function 8 m f01' the s'U.bmani-
fold !viI of the M cKay family 'with respect to p(L7'ameter' a2 and correlation coefficient p (left), and contour plot (right). Superimposed also are gradient flow arrows and the thick curve is the loc'us of maximum entmpy. The two thinner C1lr'Ves show the loci of al = 1 (upper curve) and al + a2 = 1 (lower curve), which correspond, respectively, to Poisson random processes for the X and Y variables.
Qualitatively, what we may see in Figure 4 is that for correlated random variables X and Y subordinate to the bivariate gamma density (10), the maximum entropy locus is roughly hyperbolic. The maximum entropy curve is rather insensitive to the correlation coefficient p when (X2 > 1 and the difference Y - X is dispersed more evenly than Poisson. When Y - X is actually Poisson random, with 0;2 = 1, the critical value is at p ~ 0.355. However, as 0;2 reduces fnrther- -corresponding to clustering of Y - X values- ~so the locus turns rapidly to increasing correlation. If we take the situation of a bivariate gamma process with constant marginal mean values, then the McKay probability density has constant correlation coefficient p; in this case the gradient flow lies along lines of constant 0;2· Dodson [5] gives an application to an epidemic model in which the latency and infectivity for individuals in a population are jointly distributed properties controlled by a bivariate gamma distribution.
5
Weibull processes
Like the gamma family, the Wei bull family of distributions contains the exponential distribution as a special case; it has wide application in models for reliability and lifetime statistics for random variable t > O. The probability density function can be presented in terms of positive parameters ~,f3 w : jR+ ---; jR : t
h>
f3 ( ~
f)/3-1.e~ (t/O{3 .
(21)
On the Entropy Flows to Disorder
83
Fig. 5. Surface repl-csentaJion of the Sha.nnon cntmpy ftmction Sw for the Weibnll family (left). and conto'Wr plot (T'ight) with gmdient flow and integml carves.
In applications of (21), reliability R(t) is the probability of survival to time t and it is related to the failure rate Z (t) at time t through
i· X
_'t!i.:(J
R(t)=w(t)dt=e (f~) >
t
1)
r.' 1JJ (t) " ( and Z(t) = _ . =[3 -. . R( t) . ~
(I
j
(22)
The INeihull mean, Htandard deviation and entropy are /1",
1
= m. As a result overembedding is used in an iterative process t.o obtain the highest degree of noise reduction.
3
Analysis of Measured Data
0.15
~,1(l
.
+-c---r-.,......,.~.,.~.--~ .....,.......-,~ !J
5001000
1500
2000
2500
;>Q!lC
3WQ
400U
.tSQO
soon
t(sj
Fig. 3. PG 1351+489 relative light intensity curve of 10 s binned data.
Fig. 4. Power spectrum o[PG 1351 +489 of 30 s integratpd data.
The measured data is the light curve of DB variable white dwarf PG 1351 +489 that was observed by the vVhole Earth Telescope. A section of the light. intensity curve is shown in Fig. 3. The light curve data are available a.t two integration t.imes: 10 sand ::\0 s. The lOs binned light curve is over 40 periods long. The PG 1351 +489 power spectrum consists of the fundament.al frequency fo = 2.041 mHz, (period of 489 s) , its harmonics, subharmonies and other nonlinearly generated lines (Fig. 4)
]04
3.1
N. Jevtic and J. S. Sch1,>'eitzer
Phase-space analysis of PG 1351+489 light curve
The :3-D time-delay phase space portrait of the 10 H integrated data is shown in Fig. 5.
Fig. 5. ;j-D time-delay phase space port.rait of the PG 1:551+4S9 lOs integrated data.
As part of a comprehensive nonlinear analysis of PG 1351+489 data Poincare sections were obtained at the average of the data for two direct.ions. The Poinca.re section 'space' aspects in the two directions are shown in Fig. 6. The effect of noise is evident in the form of 'upward' point.s appearing in the 'downward' bundle of trajectories and vice versa. c· ...
IJpw.ind d ('<W fil,!'l,l "ff d
-1:!lD·
, HUl
«I)
Fig. 6 . 'Space' Poincare section of the PC 1:351+489 lOs binned data at average of data.
The Poincare Section Return Times (PSRT) are shown in Fig. 7 (left) 311dFig. 7 (right) for the 30 sand 10 s binned data, respectively. nIH period points at 48D s and some noise at short P SRT are present in Fig. 7 (left). In Fig. 7 (right) however, in a.ddition to the full period and, at short PSRT, noise, there is an excess of points at about half the fundamental time. In theF'ourier power spectrum, the fundamental and the first harmonic are in a, 10:1 ratio. Here the ratio is about 1:1.
ldenti/ving Chaotic and Quasiperiodic Time-Series Candidates
J05
b.
=
~-'-'-'T-"'--""'---''''''''''''-I II :;rl .0(;' CI ;n
Cifl)"urt/>t$rlb7.
9 9.1
Discussion Poincare plane orientation
Theoretically a Poincare section is taken in the direction perpendicular to the trajectory. However, since "resetting" has been defined for stochastic fluctuations that are perpendicular to the trajectory, no PSHT resetting would be observed for these Poincare plane orientations. For the identification of candidates for efficient noise reduction the orientation has to be chosen in a.n
Identifying Chaotic and Quasiperiodic Time-Series Candidates
109
arbitrary direction in respect to the trajectory. This 'fuzziness' can, however, work for us. If in addition to the position of the Poincare surface, its orientation is varied through 27r, then for resetting which is normal to the trajectory, the 'theoretical' direction would yield a minimum in the the number of fractional PSRT. As a result, this orientation can be used to select a reference level that has minimal resetting, with additive noise being responsible for the observed noise. This has the potential to allow us to quantify the relative contributions of resetting and additive noise. Further work will address the question of the optimal orientation of Poincare surfaces for the purpose of noise reduction and quantification in other types of systems.
9.2
Noise quantification
Many questions remain to be answered before a noise quantification measure can be defined. The first that come to mind are: If a 'clean' traj ectory can be defined for comparison, a measure of resetting are both the amplitude of the displacement off the trajectory and the length of trajectory missing before reset. Would a product or a ratio of the two be a 'good' measure of the effect of resetting stochastic fluctuations? This in turn raises the question of the definition of a 'clean' trajectory. In an iterative process what is the standard number of iterations? And finally, whether we decide that the product or the ratio of the excursions off the trajectory and the length of the trajectory involved in the excursion are a 'good' measure, should an average of this measure or sum along the trajectory be used? Or should it be normalized on the whole trajectory?
10
Closing Comments
At this time, many issues remain open as can be seen in the previous section. In addition to the procedural issues addressed above also of importance is the more general question of the relationship of the traditional Fourier power spectrum and the 3-D PSRF spectrum proposed here. In the Fourier spectrum of PC 1351+489 the ratio between the power in the fundament al and the first harmonic is 10:1. In the 3-D PSRF spectrum at the average of the data this ratio is 1:1. Could this also be used as a measure of noise? Moreover , digital data acquisition, with the possibility of variable binning as in the example of PC 1351+489, can yield an estimate of the time scale of the turbulence. The answer to this question can only be given after the effect of the delay is identified and provision is made to eliminate this dependence. And last but not least, could this lead to an estimate of time-local parameters of the medium such as thermal response TC timescales (Montgomery [3])?
11
Acknowledgements Dr. Mausumi Dikpati of HAO, NCAR for the dynamo model data.
110
N. Jevtic and J. S. Schweitzer
The WET for the variable white dwarf data: the xCov12 campaign team, particularly Antonio Kanaan and Steve Kawaler. The University of Connecticut Research Foundation and the Bloomsburg University Faculty Travel Fund The Max Planck Institute for Complex Systems for the TISEAN software: R. Hegger, H. Kantz, and T. Schreiber, Practical implementation of nonlinear time series methods: The TISEAN package , CHAOS, 9,413 (1999)
References l.J. Stark et al. Embeddings for forced systems ii: Stochastic forcing. JNonlinSci , 13:519, 2003. 2.N. Jevtic et al. Nonlinear time series analysis of pg1351 +489 light intensity curves. ApJ, 635:527- 539, 2005. 3.M. Montgomery. A renorma lization group with periodic behavior. CoAst, 154:38, 2008. 4.J. Stark. Embeddings for forced systems i: Deterministic forcing. JNonlinSci, 9:255- 332, 1999. 5.F. Takens. Detecting strange attractors in turbulence. In D. A. Rand and L.S. Young, editors, Dynamical Systems and Turbulence, page 366381, London, 1981. Springer-Verlag.
III
Chaos from Observer's Mathematics Point of View Dmitriy Khots 1 and Boris Khots 2 1
2
3710 S. 202nd Avenue Omaha, Nebraska, USA (e-mail:
[email protected] ) Compressor Controls Corp 4725 121st Street Des Moines, Iowa, USA (e-mail:
[email protected])
Abstract: This work considers Chaos aspects in a setting of arithmetic provided by Observer's Mathematics (see www.mathrelativity.com). We prove that the physical speed is a random variable, cannot exceed some constant, and this constant does not depend on an inertial coordinate system. Certain results and communications pertaining to these theorems are provided. Keywords: observer, chaos, arithmetic, derivative, Lorentz.
1
Introduction
The following discussion is based on the work introduced in Khots et al.[l]. Further information can also be found in Khots et al. [2] and Khots et al. [3]. We consider a finite well-ordered system of observers, where each observer sees the real numbers as the set of all infinite decimal fractions. The observers are ordered by their level of "depth", i.e. each observer has a depth number (hence, we have the regular integer ordering), such that an observer with depth k sees that an observer with depth n < k sees and deals (to be defined below) not with an infinite set of infinite decimal fractions, but, actually, with a finite set of finite decimal fractions. We call this set W n , i.e. it is the set of all decimal fractions, such that there are at most than n digits in the integer part and n digits in the decimal part of the fraction. Visually, an element in Wn looks like _ ... _ . _ ... _. Moreover, an observer with a given '-v-" '-v-" n
n
depth is unaware (or can only assume the existence) of observers with larger depth values and for his purposes, he deals with "infinity". These observers are called naive, with the observer with the lowest depth number - the most naive. However, if there is an observer with a higher depth number, he sees that a given observer actually deals with a finite set of finite decimal fractions, and so on. Therefore, if we fix an observer, then this observer sees the sets W n1 , . . . , W nk with nl < .. , < nk indicating the depth level, and realizes that the corresponding observers see and deal with infinity. When we talk
112
D. Khots and B. Khots
about observers, we shall always have some fixed observer (called 'us') who oversees all others and realizes that they are naive. The "Wn-observer" is the abbreviation for somebody who deals with Wn while thinking that he deals with infinity. The following sections describe application of the idea of relativity in mathematics to various mathematical fields.
2
Arithmetic
We begin by defining sets Wn which consist of all finite decimal fractions such that there are at most n digits in the integer part and at most n digits in the decimal part. That is, the set Wn contains all elements of the form a = aO.al ... a n where the integer part can be written as ao = bn-1 ... bo, where bn - l , ... , bo, aI, .... , an E {O, 1, ... , 9}. If n < m, then Wn naturally embeds into Wm by placing O's in the n + pt through mth decimal places. We call the embedding ipn,m : Wn --> W m . Here are some examples: let 2.34 E W 2 and then ip2,4 (2.34) = 2.3400 E W 4 . Similarly, Wm projects onto Wn by cutting off the superfluous digits on the right of the decimal point. Let cPm.n : Wm --> Wn be the projection, then, for example, if 45.4301 E W 4 , then cP4,2 (45.4301) = 45.43 E W 2. If the integer part of a fraction contains more than n digits, then cPm,n is not defined. Now, given C = CO,Cl"'C n , d = do.dl ... d n E Wn we endow Wn with the following arithmetic (+n, -n, X n , -;-n):
Definition 1. Addition and subtraction C
±n d = { C ± d, if C ± dE Wn not defined, if C ± d ¢:. Wn N
and we write (( ...
(Cl
2.= nCi
+n C2) ... ) +n CN) =
for
Cl, ... ,
CN iff the contents
i=l
of any parenthesis are in W n
.
Definition 2. Multiplication n
cXnd=
2.=
n-k n
k=O
where c, d > 0, -
Co'
2.=
nO.O ... Ock·O.O ... Odm
m=O
'-v-"
'-v-"
k-l
m-l
do E W n , 0.0 ... 0 Ck' 0.0 ... 0 d m is the standard product, and '-v-" '-v-" k-l
k = m = 0 means that
O.~Ck k-l
=
m-l Co
and
O.~dm
= do. If either C < 0 or
m-l
d < 0, then we compute iel Xn Idl and define C Xn d = ± Ici Xn Idl, where the sign ± is defined as usual. Note, if the content of at least one parentheses (in previous formula) is not in W n , then C Xn d is not defined.
Chaos from the Observer's Mathematics Point of View
113
Definition 3. Division
-'-- d_{"if3!IEWn Ixnd=c not defined , if no such 1 exists or it is not unique
C • n
Let n = 2, so we are in W 2 . Here are some examples of elements of W 2 : 3.14, -99,0.1 E W 2 and 0.115, 123.9, - 100000 1- W 2 . Now, the examples of arithmetic: 2.08 +2 11.9 = 13.98; (-2.08) +2 11.9 = 9.82; 80 +2 24 = not defined; 21.36- 20.87 = 20.49; 1.36- 2 16.95 = -15.59; 1.36 -2( -99 .95) = not defined; 11 X2 8 = 88; (-5) X2 19 = -95; 11 X2 12 = not defined; 3.41 X2 2.64 = 8.98; 3.41 X2 (-2.64) = -8.98; 3.41 X2 42.64 = not defined; 99.41 X2 1.64 = not defined; 0.85 X2 0.02 = 0; 80724 = 20 ; 1 7n 0.5 = not defined (since we get 10 different I'S); 1 7n 3 = not defined (since no 1 exists) .
3
Derivatives
From the point of view of Wn-observer (we will call such observers " naive", since they "think" that they " live" in Wand deal with W) a real function Y of a real variable x, y = y(x), is called differentiable at x = Xo (see Khots et al. [4]) if there is a derivative '() y Xo
=
l' 1m
x->xo.x#xo
y(x) - y(xo) x - Xo
What does the above statement mean from point of view of lVm-observer with m > n? It means that I(y(x) - n y(xo)) - n (y'(xo) Xn (x - n xo))1 ::; 0.0 ... 01 whenever Iy(x) -n y(xo)1 = 0.0 . .. OYI YI +1 ... Yn and I(x _n xo)1 = ~ I
'-v--' n
0. 0 ... OXk Xk+1 ... Xn for 1 ::; k, I ::; n, and Xk - non-zero digit . '-.;---' k
We now state the main theorems.
Theorem 1. From the point of view of a W m -observer a derivative calculated by a Wn -observer (m > n) is not defined uniquely. Proof. Put y'(xo) = ±ao·a1 ... a p a p +l . ,. an with aO ·al ... a p a p +l ... an ;:::: Then 0.0 ... OYI Yl+l .. . Yn =
o and p ::; n.
~
I
aO.a1 ... a p ap +1 ... an
Xn
0.0 ... OXk Xk+1 ... Xn = '-.;---'
k
aO.a1 ... a p bp +1 .. , bn
Xn
0.0 ... OXk Xk+1 ... Xn '-.;---'
k
for any digits bp + 1 , ... ,bn and p = n - k. Hence
and
y'(xo) E V
= {±ao.a1 ., . ap a p +1 ... an la p +1,"" an E {O, 1, ... , 9}}
IVI = 10 k .
QED.
114
D. Khots and B. Khots
Theorem 2. From the point of view of a Wm -observer with m > n, ly'(xo)1 ::; C~k, where C~k E W n is a constant defined only by n, I, k and not dependent on y(x).
Proof. We have ± O. 0 . .. OYI Yl+1 . .. Yn = (±aO·a1 ... an) Xn (±O.O ... OXk Xk+1 . .. xn) '--v--'
'--v--"'
I
k
with Xk - non-zero digit and aO.a1 ... ap a p +1 .. . an 2' O. Now , if I > k then ao = 0; if 1 = k then ao::; 9 and if 1 < k then ao < 9 x 10k-1. Hence
Cl,k n =
I, if I > k { 10 , if I = k 9 X 10k- 1, if I < k
QED. Theorem 3. From the point of view of a W m -observer, when a Wn -observer (with m > n 2' 3) calculates the second derivative:
Y"(Xo) =
Y(X3)-Y(XJ) (x3-xd
lim
Y(X2)-Y(XO) X2 - XO
Xl -+XO , X l¥=- XO , X2---+XO , X2¥:XO,X3---+X l ,X:3# X l
we get the following inequality:
n
provided that y" (xo)
i- O.
Proof. For the Wm-observer existence of y"(xo) means that 1((y(X3) -n y(xd) Xn (X2 -n xo) -n ((Y(X2) -n y(:ro)) Xn (X2 -n xo))) - n y"(xo) Xn ((I X2 -n xol Xn IX3 -n :];11) Xn IX1 -n xol)1 ::; O.~, whenever I(X2 -n n
n
n
~
~
'-v-"
'-v-"
k
I
xo)1 ::; 0.0 ... Op * ... * and I(X3 -n xdl ::; 0.0 . . . Oq * ... * and I( X1 -n xo)1 ::; n
~
0.0 .. . Or * ... * where p, q, r are non-zero digits, asterisks are any digits and '-v-" s
3 ::; k + I + s ::; n. Then given y"(xo) Xli) Xn IX1-nxol2' O.~. QED. n
i-
0 we have (IX2 - n xol Xn IX3
- n
Chaos from the Observer's Mathematics Point of View
4
lIS
Chaos theory interpretation
The following hypotheses illustrate possible Chaos theory interpretation of previous theorems. • Hypotheses 1. Theorem 1 could offer an explanation of why physical speed (or acceleration) is not uniquely defined and, from the point of view of a measurement system (observer), it is possible to consider speed (or acceleration) as a random variable with distribution dependend on the measurement system. Let v be the speed with v = Vo.1il ... Vn - k + ~;;-,:k where ~;;-,:k E {a. 0 . . . Vn-k+1 ... Vn } - random va riable, m > n, and the
°
'-v--"
n-k
distribution function is F::.,k(x) = p(~;;-,:k < x). • Hypotheses 2. Theorem 2 could offer an explanation of why the speed of any physical body cannot exceed some constant, (the speed of light, for example). Independence of this constant on explicit expression of spacetime function could offer an explanation of why the speed of light does not depend on an inertial coordinate system. • Hypot heses 3. Theorem 3 could offer an explantion of the various uncertainty principles, when a product of a finite number of physical variables has to be not less than a certain constant. This can be seen not just from consideration of second derivatives, but of any derivative. • Hypotheses 4. Theorems 1, 2, and 3 combined may provide an insight into the connection between classical and quantum mechanics.
5
Newton equation with chaos theory interpretation
Let F(x , t)
= m Xn X. Then we have the following
Theorem 4. If the body with mass m = mO.m1 ... mkmk+1 ... m n , with mE W n , moves with acceleration X, Ixl = XO.X1 ... XI XI+1 .. , Xn, with X E W n , and mo = TTL1 = ... = mk = 0, mk+1 01 0, k < n, Xo = Xl = ... = Xl = 0, l < n, k + l + 2 E W n, n < k + I + 2 ::::: q, then F (x, t) = 0.
Proof. In t his case m Xn
Ixi
=
Corollary 1. If 1= n - 1 and k Theorem 5. If 1= n - 1 and xn
Proof. IF(x , t)1 = m Xn
0 . QED.
= 0, i.e., m < 1, then F(x, t).
01
°
then IF(x , t)1 < 9.
Ixl k]}-I}-A[sin(Bk+l)+I]=O 2g 2Bk cOJ (3)
where
Dynamics of a Bouncing Ball
167
fJk =2 ' C
and the velocity map
- c2
a[ C
V k+l
J+
V'k+l-Uk+l
2,
U k+l
- V k+l U k + 1
= -----[~-------'------,--
1- au
V'k+l-Uk+l 2 , C - V k+1 U k +1
k+l
J
(4) where Uk+l
= AU) cos (Bk + 1 )
is the table's velocity just after the (k+l)th impact, and ,
V k+l
C [2B g ( Ok+l = --cos - -k -
2Bk
-
Ok
) + 0] k
CO)
is the ball ' s velocity just before the (k+ 1)th impact. Our derivation of the general relativistic map [Eqs. (3) and (4)] for the bouncing ball, which is similar to the derivation [10] of the Newtonian map [Eqs. (1) and (2)], utilized Ladipus' result [9] for the general relativistic motion of a body in a uniform gravitational field. The impact phase maps, Eq. (1) and Eq. (3), which are implicit algebraic equations for
0k+l'
must be solved numerically by finding the
zero of the function on the left side of the equation given
Ok
and v k
.
We use the izera function in MATLAB for this purpose. Numerical accuracy of the solutions was carefully checked by varying the tolerances.
168
S.-N. Liang and B. L. Lan
3.
Results
In the example given here, the parameters of the bouncing ball system are: g = 981 cmls2, C = 2.998x103 cm/s (we have to use an artificially smaller c value for accurate numerical calculation of the
= 60 Hz, table's coefficient of restitution a = 0.5 . The
general relativistic map), table's frequency (OJ/2;r)
amplitude A = 0.012 cm, and initial conditions are 8.17001 cmls for the ball's velocity and 0.12001 for the normalized impact phase (i.e., impact phase divided by 2n.
The Newtonian and general relativistic trajectories are plotted in Figure 1. The figure shows that the two trajectories are close to each other for a while but they are completely different after 21 impacts although the ball's speed and table's speed remained low (about 1O-3c) and the gravitational field is weak (gravitational potential gy is about 1O-6c2). We have carefully checked that the breakdown of agreement between the two trajectories is not due to numerical errors.
4.
Conclusion
For a slow-moving dynamical system where gravity is weak, we have shown, contrary to expectation (see, for example, [1,9]), that the trajectories predicted by Newtonian mechanics and general relativistic mechanics from the same parameters and initial conditions could rapidly disagree completely. The bouncing ball system could be realized experimentally [10-12] to test which of the two completely different trajectory predictions is physically correct.
References 1.
2. 3. 4.
5. 6.
7.
A. Einstein, Relativity: the special and the general theory, Random House, New York, 1961. J. Ford and G. Mantica, Am. 1. Phys. 60, 1086-1098 (1992). W. D. McComb, Dynamics and relativity, Oxford University Press, Oxford, 1999. J. B. Hartle, Gravity: An Introduction to Einstein's General Relativity, Addison-Wesley, San Francisco, 2003. B. L. Lan, Chaos 16,033107 (2006). B. L. Lan, in Topics on chaotic systems: selected papers from Chaos2008 international conference, eds. C. H. Skiadas, I. Dimotikalis, and C. Skiadas, World Scientific, 2009, pp. 199-203. B. L. Lan, Chaos, Solitons and Fractals 42, 534-537 (2009).
Dynamics of a Bouncing Ball
169
8.
P. Davies, in The new physics, ed. P. Davies, Cambridge University Press, 1992, pp. 1-6. 9. I. R. Ladipus, Am. 1. Phys. 40, 1509-15lO (1972). lO. N. B. Tufillaro, T. Abbott, J. Reilly, An experimental approach to nonlinear dynamics and chaos, Addison-Wesley, California, 1992. 11. N. B. Tufillaro, T. M. Mello, Y. M. Choi and A. M. Albano, 1. Physique 47,1477-1482 (1986). 12. N. B. Tufillaro and A. M. Albano, Am. 1. Phys. 54,939-944, 1986.
I
.
'.
.'
0.8
'
II.)
'" 0.6 ~
...
Po
<J
~ 0.4
.§
impact
.~
~~
..' .
10
A '.
'.
'.
'
'
8
'.
<J
o
Q) >
7
", I
I' I I
.' I I
III
,
11
Il
t:.
.' t
I I
I
11
4 L-______________- 4_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
o
10
20
30
40
~
50
impact Figure 1. Comparison of the Newtonian and general relativistic trajectories: normalized impact phases (top) and velocities (bottom). Newtonian (general relativistic) values are plotted with triangles (diamonds).
170
Symmetry-Break in a Minimal Lorenz-Like System Valerio Lucarini I and Klaus Fraedrich2 1
2
Department of Mathematics, University of Reading Whiteknights, PO Box 220, Reading, RG6 6AX, UK Department of Meteorology, University of Reading Earley Gate, PO Box 243, Reading, RG6 6BB, UK Department of Physics, University of Bologna Viale Berti-Pichat 6/2 40127 Bologna, Italy (e-mail:
[email protected]) Meteorologisches Institut, KlimaCampus, University of Hamburg Grindelberg 5,20144 Hamburg, Germany
Abstract: Starting from the classical Saltzman 2D convection equations, we derive via spectral truncation a minimal 10 ODE system which includes the thermal effect of viscous dissipation. Neglecting this process leads to a dynamical system which includes a decoupled, generalized Lorenz system. The consideration of this process breaks an important symmetry, couples the dynamics of fast and slow variables, and modifies the structural properties of the attractor. When the relevant nondimensional number (Eckert number Ec) is different from zero, an additional time scale of 0(Ec·· 1) is introduced in the system. Moreover, the system is ergodic and hyperbolic, the slow variables feature long term memory with 11/ 312 power spectra, and the fast variables feature amplitude modulation. Increasing the strength of the thermal-viscous feedback has a stabilizing effect, as both the metric entropy and the Kaplan-Yorke attractor dimension decrease monotonically with Ec. The analyzed system features very rich dynamics: it overcomes some of the limitations of the Lorenz system and might have prototypical value in relevant processes in complex systems dynamics.
1. Introduction The Lorenz system [1] has a central role in modern science as it has provided the first example of low-dimensional chaos [2], and has literally paved the way for new scientific paradigms. The Lorenz system can be derived with a minimal truncation of the Fourier-modes projection of the 2D Boussinesq convection equations introduced by Saltzman [3], where a specific selection of the spatial symmetry of the fields is considered. Extensions of the Lorenz system taking into account higher-order spectral truncations in the 2D case have been presented, see e.g. [4,5,6], whereas in [7] the standard procedure has been extended to the 3D case. The mathematical properties of the Lorenz system have been the subject of an intense analysis; For several classic results, see [8]. Recently, at more theoretical level, the investigation of Lorenz-like systems has stimulated the introduction of the family of singular hyperbolic systems as extension of the family of hyperbolic systems [9]. Moreover, moving from the theory developed for non-equilibrium systems [10-12], in a recent paper a careful verification on the Lorenz system of the Kramers-Kronig dispersion relations and sum rules has been performed [13]. In spite of its immense value, the Lorenz system does not provide an efficient representation of several crucial phenomena typically associated to
Symmetry-Break in a Minimal Lorenz-Like System
171
complex, chaotic systems. When looking at finite time predictability properties, the Lorenz system features an unrealistic return-of-skill in the forecast, i.e. there are regions in the attractor within which all infinitesimal uncertainties decrease with time [14]. Additionally, the Lorenz system does not feature an interplay between fast and slow variables, so that it cannot mimic the coupling between systems with different internal time scales. Palmer [15] introduced artificially ad hoc components to the Lorenz system in order to derive a toy-model able to generate low-frequency variability. Moreover, as discussed in [16], the Lorenz system does not allow for a closed-form computation of the entropy production. In this work, we wish to propose a minimal dynamical system, possibly with some paradigmatic value, able to overcome some of the limitation of the Lorenz system, and endowed with a much richer dynamics. Starting from the 2D convection equations, we derive with a truncation a la Lorenz a minimal 10 d.oJ. ODEs system which includes the description of the thermal effect of viscous dissipation. We discuss how, while neglecting this process lead us to a dynamical system which includes a decoupled (generalized) Lorenz system, its consideration breaks an important symmetry of the systems, couples the dynamics of fast and slow variable, with the ensuing modifications to the structural properties of the attractor and in the spectral features of the system.
2. Rayleigh-Benard convective system We consider a thermodynamic fluid having kinematic viscosity V, thermal conductivity k, thermal capacity at constant volume Cl" and linearized equation
P = Po (1- cff). The 2D (x, z) Rayleigh-Benard top-heavy convective system confined within z E [0, H] can be completely described, when adopting
of state
the Boussinesq approximation, by the following system of PDEs:
d,V2V+J(V,V2V)= gad/J + vV 4V d/J+J(V,e)= I1T dxV+kV2e+~diiVdijV H
where
g
is
(la-lb)
C, . .
the
gravity
acceleration,
T = To - zl1T/ H + e , with H uniform depth of the fluid and I1T imposed temperature difference. The suitable boundary conditions in e(x, z = 0) = e(x, z = H) = 0 and the case of free-slip system are
V(x, z = 0) = V(x, z = H) = V 2V(x, Z = 0) = V 2V(x, Z = H) = 0 for all values of x. These PDEs can be non-dimensionalized by applying the linear transformations
x=Hx, z=HZ, t=H 2 /kt, v=k¥/,and e=kv/(gaH 3 )B:
V. Lucarini and K. Fraedrich
172
ai V2v/ + '(v/, V2v/)= ciJ/)+ cfi4v/
(2a-2b)
ai)+ '(Iii, V2B+ aEca,v/a,v/' t r B) = Ra.v/+ x IJ
I)
where fJ=v/k is the Prandl number, R=gaH 3f:,.T/(kv) is the Rayleigh number,
and
Ec =
e /(C/y..TH
2)
is
the
Eckert
number,
and
the
B(x, z= 0) = B(x, z= 1) = 0 and v/(x, z = 0) = v/(x, z= 1) = V2v/(x, z= 0) = V2v/(x, z= 1) = O. The Eckert number
suitable
boundary
conditions
are
in Eq. 2b quantifies the impact of viscous dissipation on the thermal balance of the system. As this number is usually rather small in actual fluids, the corresponding term in the 2previous set of PDEs is typically discarded. We perform a truncated Fourier expansion of
x
z
v/
and
B,
assuming that they are
periodic along the and with periodicity of 2/ a and 2, respectively. When boundary conditions are considered, we then derive a set of ODEs describing the temporal evolution of the corresponding (complex valued) modes 'Pm . and 1I
Om which are associated to the wave vector R = (J/Zln,mn). See [3] for a detailed derivation in the case Ec = 0, considering that the sign of the term proportional to R is wrong in both Eqs. 34 and 35. In the case Ec = 0, the seminal Lorenz system can be derived by severely truncating the system, considering the evolution equation for the real part of 'Pl.l and for the imaginary II '
parts of
°
0 ,2
°
1,1
and
°
0 ,2'
and performing suitable rescaling (see below). While
has no real part because of the boundary conditions [3], neglecting the
imaginary (real) part of 'Pl.l (° 1,1) amounts to an arbitrary selection of the phase of the waves in the system. An entire hierarchy of generalized Lorenz models, all obeying to this constraint, can be derived with lengthy but straightforward calculations. See, e.g., [4-6] for a detailed discussion.
3. Symmetries of the extended Lorenz system In this work we include the modes 'PI,I' 'P2,2' 01.1' 02,2 in our truncation, and retain both the real and the imaginary parts, whereas the considered horizontally symmetric modes 00,2 and 00.4 are, as mentioned above, imaginary. Finally, we assume, in general, a non vanishing value for Ec. If we define 'PI,I = a(X I+iXJ, O1,1 = f3(~ +iyJ, 'P2,2 = a(AI +iAJ, 0 2.2 = f3(B I+iBJ,
0 0,2=iJi.lI'
0 0 ,4=iJi.l2'
and
f3 = JZ'3 (a 2 + 1) j {2.J2a 2 ),
r=
JZ'3
following system of real ODEs:
with
a={a 2 +1)/{2.J2a),
(a 2 + 1JI{2a 2 )' set a = I/.J2
[1], we derive the
Symmetry-Break in a Minimal Lorenz-Like System
173
XI = O"Y2-(JX I
X2 =
~
-(J~
-(JX 2
= X 2 Z 1 -rX 2 -~ +s/(.J2;r)(JEc(AI X I +~X2)
1'2 = - XIZI + rX I - Y2 + s/(.J2;r )oEc(A2X I - AI X 2)
AI = (J/2 B2 -
4(JAI
(3a-3j)
,12 = -(J/2BI -4(JA2
BI = 4A2Z 2 - 2rA2 -4BI -1/(2.J2;r )oEC(XI2 - xi) B2 = -4AIZ 2 + 2rAI - 4B2 -1/(.J2;r )(JEcX I X 2
X2~
ZI = X I Y2 -
-bZ I
Z2 = 4AIB2 -4A2BI -4bZ2
where the dot indicates the derivative with respect to r=;r2(a 2 +1}=3/2;r2 t
r = R/ Rc
= R/(27;r4 /4)
,
is the relative Rayleigh number, and b is a geometric
factor. Note that if we exclude the faster varying modes
tp2,2
and O 2,2 in our
truncation (as in the Lorenz case) the Eckert number is immaterial in the equations of motion, the reason being that viscous dissipation acts on small spatial scales. Therefore, the ODEs (3a-31) provide the minimal system which includes the feedback due to the thermal effect of viscosity. If we retain all the variables in the system (3a-3j) and Ec is set to 0, the following symmetry is obeyed. Let
Set, So) = (X (t), y(t), A(t), B(t}, z(t)Y - with - (t) = (-I (t}'-2 (t)) - be a
solution of the system with initial conditions So = (X(O),Y(O),A(O),B(O),Z(O)Y. We then have that: y-I (m,¢ )S(t, T(m, ¢ )So) =
for
all
(m,¢)E 9\2,
(y-I (m, ¢) =
TT (m,¢))
(4)
T(m,¢)
where
is
a
real,
linear,
orthogonal
transformation:
0
0
0
0
X
0
0
y
0
RaJ 0
0
0
0
A
0
0
R¢ 0
0
B
0
0
0
RI/J 0
I
Z
RaJ 0
T(m,¢)S =
S(t, So)
(S)
174
V. Lucarini and K. Fraedrich
with I being the 2X2 identity matrix and R;; being the 2X2 rotation matrix of angle ~. The T(OJ,¢) -matrix shifts the phases of both the \f'1,1 and 0 1,1 waves by of the same angle OJ and, independently, the phases of both the \f'2,2 and 02.2
waves by the same angle ¢. The symmetry (4-5) gives a solid framework
to and generalizes the results given in [17].
4. Statistical properties of the system - symmetric case In the Ec = 0 case, we expect the presence of degeneracies in the dynamics, leading to non-ergodicity of the system. Since, from (4), we have that S(t,T(OJ,¢)SeJ=T(OJ,¢)S(t,So), we readily obtain that the statistical properties of the flow depend on the initial conditions, and that by suitably choosing the matrix T(OJ, ¢) , we can, e.g., exchange the statistical properties of XI and r; with those of X 2 and Y2 by setting OJ = ;r/2 (and similarly for the A and B variables by setting ¢ = ;r/2). Whereas, in actual integration, numerical noise coupled with sensitive dependence on initial conditions (see below) breaks the instantaneous identity S(t,T(OJ, ¢ )So) = T(OJ,¢ )S(t, So) after sufficiently long time, the statistical properties of the observables transform according to the symmetry defined in Eq. (5). The dynamics of the variables (Xi' X 2' r;, Y2 , ZJ defines an extended Lorenz system [17]. The classical three-component system results from a specific selection of the phase of the waves of the system, which is obtained by setting vanishing initial conditions for X 2 and r;. With such as choice, we have that X 2 =
r; = 0
at all times, whereas the x and y variables of the Lorenz system
are Xl and Y2 , respectively. As a consequence of the symmetry given in Eqs. (4-5), the statistical properties of ZI do not depend on the initial conditions, and agree with those of the z variable of the classical Lorenz system. Moreover, the statistical properties of the quadratic quantities X2 == X I2 + xi, y2 == r;2 + y22, and XY == X I Y2 - X 2r; do not depend on the initial conditions (whereas those of each term in the previous sums do!), and agree (for all values of r, a, b !) with those of x 2 and l, and xy of the classical Lorenz system, respectively. Analogously, since the symmetry defined in Eq. (5) is obeyed, the statistical properties of AI' A2, BI , B2 depend on the initial conditions, whereas the statistical properties of the quadratic quantities A 2 == AI2 + Ai
B2 == Bl2 + Bi '
AB == AIB2 - A2BI ' and of Z2 are well defined. The symmetry (4-5) implies that the attractor of the system (3a-3j) can be expressed as a Cartesian product of a 2torus times a "fundamental" attractor. The initial conditions will define where on the torus the system is located.
Symmetry-Break in a Minimal Lorenz-Like System
175
From a physical point of view, the symmetry (4-5) is related to the fact that the system does not preferentially select streamfunction and temperature waves of a specific phase, and, does not mix phases. Therefore, in a statistical sense, the relative strength of waves with the same periodicity but different phase can be arbitrarily chosen by suitably selecting the initial conditions. Instead, the statistical properties of the space-averaged convective heat transport, which is determined by a linear combination of Z, and Z2' as well as those of the total kinetic energy (determined by a linear combination of X 2 and A 2) and available potential energy (determined by a linear combination of y2 and B2) of the fluid [3] must not depend on the initial conditions - but only on the system's parameters - as they are robust properties of the flow. In general, since the dynamics of the variables (X"X2'~'Y2'Z,) is entirely decoupled from that of the variables {A" A2 , B" B 2 , ZJ, the attractor of the system (3a-31) will be strange at least for the same values of the parameters r, a, b providing the classical Lorenz system with a chaotic dynamics. Additionally, strange attractors could result from choices of the values of r, a, b determining a chaotic dynamics for the (A" A2, B" B 2, Z2) variables and a trivial or periodic behavior for the
(X" X 2' y, ' Y2 ,Z,) variables.
With the "classical" Lorenz parameter values r
= 28,
a
= lO,
b
= 8/3 ,
the variables (X"X2'~'Y2'Z,) obviously have an erratic behavior, whereas the variables (Ai' A2, B" B2, Z2)' which describe the faster spatially varying wave components, do not feature any time variability when asymptotic dynamics is considered, as they converge to fixed values above, the some
r
(A,=, A;, B,=,B;)
manipulations
on
(A,=, A; ,B,=, B;, Z; ) . As discussed
values depend on the initial conditions, while, after Eqs
(3a-3j),
we
obtain
(A2
r
=
b/8{r/2-8) ,
(B2 = 8b{r/2-8) ,
Z; = r/2-8, and (ABt = b{r/2-8). In general, since the fundamental properties of the system do not depend on where in the 2-torus the system converges, despite the fact that ergodicity is not obeyed, it makes sense to compute the Lyapunov exponents [2,18] of the system (3a-3j). In fact, if we compute the Lyapunov exponents (,11' ... '..110 ) using the algorithm by Benettin et al. [19] with the classical Lorenz parameter values, we obtain the same results independently of the initial conditions. The sum of the Lyapunov exponents is given by the trace of the jacobian J of the system (3a-3j) and yelds Tr(J) = LAj =-5{2+2a+b)"" -123.333. In particular, we obtain one positive exponent (AI "" 0.905 ), which gives the metric entropy h of the system [2] and matches exactly the same value as the positive Lyapunov exponent in the classical Lorenz system, and three vanishing exponents Az = ~ = ,14 = o. The non-hyperbolicity of the system, as described by the
176
V. Lucarini and K. Fraedrich
presence of two additional vanishing exponents, is related to the existence of two neutral directions due to the symmetry property (4-5). Moreover, we have 6 negative Lyapunov exponents, one of which agrees with the negative exponent of the Lorenz system (As "" -14.572). Given the very structure (and derivation) of the model discussed above, the fact that we recover the classical results of the 3-component Lorenz system is quite reinsuring. The Kaplan-Yorke dimension [20] of the system is: (6)
where k is such that the sum of the first k (4, in our case) Lyapunov exponents is positive and the sum of the first k+ 1 Lyapunov exponents is negative.
5. Symmetry-break and phase mixing Considering a non-vanishing value for Ec amounts to including an additional coupling between the temperature and the streamfunction waves. Such a coupling breaks the symmetry (4-5), so that a dramatic impact on the attractor properties is expected. Using standard multi scale procedure, we can emphasize the emergence of the additional time scale Ec- 1 • We introduce a slow time variable r = £t with c = Ec . Then we take the expansion Vi(t)=Voi(t,r)+c~j(t,r)+ .. +cl7v,/(t,r)+ ... where
Vi,j=l, ... ,lO is a generic
variable of the ODEs system considered, we redefine the time derivative d/ dt as d/ dt == d/dt + liJ/dr , we plug these new definitions in Eqs. (3a-3j), and group together terms with the same power of £. At zero order - the r variable is frozen - we obtain that the variables V0' obey the Eqs. (3a-3j) with Ec = 0 . Therefore, for time scales shorter than 0(c- 1 ), the symmetries (4-5) described in the previous section are approximately obeyed, which confirms that our multiscale expansion is well-defined. We hereby consider the specific case where we select the classical values for the other 3 parameters of the system, so that r = 28, (j = 10, b = 8/3 . When large values of Ec are considered (Ec > 0.045), the system loses its chaotic nature as no positive Lyapunov exponents are detected (not shown), whereas periodic motion is realized. The analysis of this transition and of this regime is beyond the scopes of this paper, as we confine ourselves to studying the properties of the system when chaotic motion is realized, thus focusing on the Ec ~ 0 limit. We consider Ec E [0,0.02]. In physical terms, the coupling allows for a mixing of the phases of the thermal and streamfunction waves: after a 0(c- 1 ) time the system "realizes" that the symmetry (4-5) is broken, so that the previously described degeneracies
Symmetry-Break in a Minimal Lorenz-Like System
177
are destroyed and ergodicity is established in the system. Note that, since also in the case of Ec > 0 the convection does not preferentially act on waves of a specific phase, we have that, pairwise, the statistical properties of XI and X 2 ' and of ~ and -Y2 are identical and do not depend on the initial conditions. The same applies for the A and B pairs of variables, respectively. If, in particular, we consider the long term averages of the observables
(A2), (B2),
and
(AB),
same contribution, e.g.
(X 2), (y2), (Xy) ,
we obtain that in all cases the two addends give the
(Xn = (xi) = 1/2(X
2 ).
In this case, long is considered
with respect to [,-I = Ec- I : these results can be obtained by first and second order £ -terms in the multi scale expansion envisioned above. As an example, the
(Xn = (Xi) = 1/2 ( X2)
identity is obtained by taking the long term average of
the first order £ -term of Eq. (3g). Whereas having Ec > 0 is crucial for mixing the phases of waves, the impact of the symmetry break on the long term averages of the physically sensitive observables 0 = X2 , y2, A2, B 2, ZI' and
Z2 is relatively weak in the considered Ec -values range.
6. Symmetry-break and hyperbolicity We hereby want to take a different point of view on the process of symmetry break of the dynamical system and on its sensitivity with respect to Ec, by analyzing the Ec -dependence of the spectrum of the Lyapunov exponents. Again, the classical values for r, (J, b are considered. The first, crucial result system is that the system is hyperbolic for any finite positive value of Ec: two of the vanishing Lyapunov exponents branch off from 0 with a distinct linear dependence on Ec, so that ~ '" 5.IEc and ,14 '" -6.0Ec for Ec < 0.008, whereas
~
=0
corresponds to the direction of the flow. The largest Lyapunov
exponent also decreases linearly as ~ '" 0.905-9.0Ec. Time scale separations emerges again, as the two globally unstable directions define two characteristic times II ~ and 1/ ~ , which are 0(1) and O(£c- I ), respectively. All the other Lyapunov exponents feature a negligible dependence on Ec, except As '" -14.572 + 1O.0Ec, which ensures that the sum of Lyapunov exponents is independent of Ec. Therefore, we derive that Ec < 0.008 the metric entropy decreases with Ec as h(Ec) = ~ + ~ '" 0.905 - 4.0Ec , whereas the Kaplan-Yorke dimension can be approximately written as dKy{Ec) = 4+(~ +~ +~ +A,4)J1AsI '" 4.l83-2.01Ec. For larger values of Ec, linearity is not obeyed (except for ~ and ,14)' whereas a faster, monotonic decrease of both the metric entropy and the Kaplan-Yorke dimensions are found.
178
V. Lucarini and K. Fraedrich
See Figs. la)-lc) for results for values of Ec up to 0.02. We then conclude that, in spite of introducing a second unstable direction, which is responsible for mixing the phases of the waves, the inclusion of the impact of the viscous dissipation on the thermal energy balance acts with continuity on the dynamical indicators, by reducing the overall instability, increasing the predictability, and by confining the asymptotic dynamics to a more limited (in terms of dimensionality) set. "'-~--~--~--~------~--~--~--~--~
0.9
1 - -_____
0,8 0,7 0,6 0 ,5
0,4 0,3 0,2
O,'~===========~
-O.l~ ~ -0_20
0.002
0.004
0.006
0.008 '
0.01
0.01 2 - 0.014
0.016
0.018
0.02
Be
(a) 0.95 0.9 0.85 0.8
------------------------------------------
~075 r
07r 0.65 0.6,
0.551
°
(b)
0.002
0.004
0.006
0.008
~12
0.014
0.016
0.018
0.02
Eo
4.2 1-
~ --~--,
-~--~
4.19
41B f 4. 17
"
"t::
4.15 416 4.14
1
413 4.1 2
1
4-11 0-
(c)
0.002
0.004
0,006
0.008
O~----O_014
0.016
0.0 18
0.02
Ec
Figure I: Four largest Lyapunov exponents (a), metric entropy h (b) and Kaplan-Yorke d KY dimension (cl as a function of Ec. Note the continuity for Ec = 0 of all parameters and the distinct linear behavior - see specifically the dashed lines in (b) and (cl - for Ec < 0.008. The linear behavior of the second and fourth Lyapunov exponents branching off zero in (a) extends throughout Ec '" 0.018. Details in the text.
Symmetry-Break in a Minimal Lorenz-Like System
179
7. Symmetry-break and low-frequency variability We now focus on spectral properties of the system. We first observe that the variables AI' A2 , B I , B 2 , Z2 feature mostly ultra-low frequency variability. In order to provide some qualitative highlights on these variables, in Fig. 2 we show, from top to bottom, the projection on Al of three typical trajectories for Ec = 10-3 , Ec = 10-4 , and Ec = 10-5 , respectively. Going from the top via the middle to the bottom panel, the time scale increases by a factor of 10 and 100, respectively. The striking geometric similarity confirms that the time scales of the dominating variability for the slow variables can be estimated as O(Ec- l ) As we take the Ec ~ 0 limit, such a time scale goes to infinity, which agrees with the fact that for Ec = 0 these variables converge asymptotically to fixed values. Moreover, as (A 2 )
= (A12 + An ~ b/8{r/2 - 8) = 2
when &.-1 -long time
averages are considered, Fig. I suggests that the system switches (on short time scales) back and forth between Al - and A2 -dominated dynamics. Same applies for the BI and B2 pair of variables. On the other hand, the dynamics of XI' X 2 ,
~, Y2'
ZI is basically
controlled by the 0(1) time scale 1/ ~ «1/ ~ , so that a clear-cut separation between fast and slow variables can be justified. In Fig. 3 we depict, from the top to bottom panel, some trajectories of the fast variable XI for Ec = 10- 3 , Ec = 10-4 , and Ec = 10-5 , respectively. Similarly to Fig. 2, the time scale of the x-axis is scaled according to Ec- I • Since XI is a fast variable, the dominating high-frequency variability component is only barely affected by changing Ec, because Al has a weak dependence on Ec, as discussed in the previous section (see also Fig. la). Nevertheless, in Fig. 3 we observe an amplitude modulation occurring on a much slower 0(&·-1) time scale of the order of
1/ ~ . Note that,
as in Fig. 2, such a slow dynamics of the "coarse grained" XI variable for various values of Ec is statistically similar when the time is scaled according to Ec- I • As in the previous case, the slow amplitude modulation determines the changeovers between extended periods of XI - and X 2 -dominated dynamics, thus ensuring the phase mixing and ergodicity of the system. In the Ec ~ 0 limit, as discussed above, the relative strength of the two phase components of the (l, I) waves is determined by the initial conditions, as mixing requires ~ Ec- I time units. The same considerations apply for the ~ and Y2 pair of variables. Further insight can be obtained by looking at the spectral properties of the fast and slow variables; some relevant examples of power spectra for Ec = 10-3 ,
V. Lucarini and K. Fraedrich
180
Ec = 10-4 , and Ec = 10-5 are depicted in Fig. 4, where we have considered the low frequency range by concentrating on time scales larger than 10/ A, . The
white noise nature of the fast variable X 1 is apparent and so is the overall lack of sensitivity of its spectral properties with respect to Ec. Note that any signature of the amplitude modulation, which is responsible for phase mixing observed in Fig. 3, is virtually absent. This shows how crucial physical processes are masked, due to their weakness, when using a specific metric, like that provided by the power spectrum. Instead, the slow variable Al has a distinct red power spectrum, which features a dominating
1-3/ 2
scaling in the
low frequency range (black line) between the time scales ~ 0.5 Ec- I and ~ 10 Ec- I • Such a scaling regime dominates the ultra-low frequency variability described in Fig. 2 and is responsible for the long-term memory of the signal. For higher frequencies, the spectral density p(p) decreases - see the sharp
I"" O.SEc
corner in the for DC
1-
2
- as 1-4 , and then a classical red noise spectrum
is realized for very low values of the spectral density. Note also that, in
agreement with our visual perception of Fig. 2, the spectral density
p(p) scales
according to Ec - I in a large range of time scales. 7
2,---,----,----~--_,--_,----,_--_,----,_--,_--_,
~:~
"'"
0
1
2
3
4
5
6
7
8
9
10
X 10'
1
2 , - - -, _- - - ,- -- - , -- -, -- -- ,- - - -, -- -- ,- -- , - -- -, - - -_,
~:~
"'"
0
1
2
3
4
5
6
7
8
9
10
X 10 5
~:~ "'" -20
1
2
3
4
5
t
6
7
8
9
10
X 10 6
Figure 2: Impact of the viscous-thermal feedback on the time scales of the system. From top to bottom: typical evolution of the variable A I for different values of the Eckert number (Ec = 10-3 • Ec = 10-4 , Ec = 10-5 , respectively). Note that the time scale is magnified by a factor of I, 10 and 100 from top to bottom. Details in the text.
Symmetrv-Break in l.l Minimal Loreflz-Liki' S:vstem
181
"
S 'i
~ ~-2m---'---·c_----~-
----:~-----~----~---.-.-~---~~-----~-----.-. . ~
Figure 3: Impact of the viscous-thennal feedback on the time scales of the system. From top to bottom: typical evolution of the variable Xl for different values of the Eckert number (Ee '" 1O-J, Be '" 10-.1, Ee = 10-', respectively). Note that the ljme scale is magnified by a factor of 1, 10 and 100 from top to bottom. Details in the text.
Figure 4: Power spectrum of 1'1 1 and Xl (multiplied times 10-8) for Ee = 10', Be = 10-", and Ee ",10-5 in units of power per unit frequency. The black straight lines corre,pond to 312 scalings. Details in th(, texL
r
8. Summary and Conciusions III this work we have derived and thoroughly analyzed a IO-variable system of Lorenz-like ODEs obtained by a severe spectral truncation of the 2D x-z convective equations for the streamfunction ljf and temperature {} in Boussinesq approximation and subsequent nondimensionaIization. In the tmncation we retain the terms corresponding to the real and imaginary part of the modes 'P"w (streamfunction) and em.n (temperature) associated to the x-z
182
V. Lucarini and K. Fraedrich
wave vector
K = (nan,nm)
with
(m,n)= (1,1)
imaginary part of the x-symmetric modes
and
(m,n) = (2,2)'
gO,2
and
gO,4'
and the
As modes
characterized by a faster varying spatial structure and different parity are added to the Lorenz spectral truncation, which describes the dynamics of the real part of P 1,] and of the imaginary part of g1,1 and gO,2 only, the system we consider in this work represents the thermal impact of viscous dissipation, controlled by the Eckert number Ec. The presence of a forth parameter - together with the usual Lorenz parameters r (relative Rayleigh number), (j (Prandl number), and b (geometric factor) - marks a crucial difference between this ODEs system and other extensions of the Lorenz system proposed in the literature - see, e.g., [46]. Moreover, as can be deduced following the argumentations of Nicolis [16], this system specifically allows for the closed-form computation of the entropy production, as the Eckert number enters into its evaluation. Up to our knowledge, this is the first ODE system obtained as a truncation of the convective equations presented in the scientific literature featuring all these additional properties. The following results are worth mentioning: 1. When the Eckert number is set to 0, as common in most applications, the dynamical system is invariant with respect to the action of a specific symmetry group, which basically shifts the phases of the streamfunction and thermal waves with m > 0 . At a physical level this implies that the initial conditions set the relative strength of waves with the same spatial structure but n12 -shifted phases. In particular, it is found that the Lorenz system is included in the ODEs system analyzed here when specific initial conditions, which set the parity of the slower spatially varying modes, are selected. The absence of phase-mixing implies a lack of ergodicity, and the degeneracy of the dynamics is reflected also in the non-hyperbolicity of the system. When the classical parameters' values r = 28, (j = 10, b = 8/3 are selected, three of the Lyapunov exponents vanish, one corresponding to the direction of the flow, the other two accounting for the toroidal symmetry. The value of the only positive exponent coincides with that of the Lorenz system, and the value of one of the six negative exponents agrees with that of the negative Lyapunov exponent of the Lorenz system. Therefore, the Lorenz system contains already all the interesting unstable dynamics described by this extended ODEs system, and features exactly the same value for the metric entropy. Correspondingly, while the five variables (fast) describing the modes Pl,l , gl,l , and
2.
gO,2
have an erratic behavior, the other five variables (slow)
converge to fixed values. When Ec 0, the symmetry of the system is broken, and coupling occurs
'*
between the fast and slow variable over a time scale O(Ec -1). This is clarified by adopting multi scale formalism. If we select, as in the original Lorenz system, r = 28, (j = 10, b = 8/3, the system is chaotic for
Symmetry-Break in a Minimal Lorenz-Like System
183
o< Ec ~ 0.045 , whereas for higher values of Ec a quasi-periodic regime is realized. In the chaotic regime, the symmetry-break is accompanied by the establishment of a hyperbolic dynamics: two Lyapunov exponents branch off from zero (one positive, one negative) linearly with Ec. Therefore, a second unstable direction with a second
3.
o (Ec-
I)
time scale
11 ~ » 11 AI is established. The impact of the thermal-viscous feedback is stabilizing, as indicated by the metric entropy and the Kaplan-Yorke attractor dimension monotonically decreasing with Ec, with a marked linear behavior for Ec ~ 0.008. The coupling establishes dynamics on time scales of the order of Ec- I responsible for the changeovers between extended periods of dominance of waves of specific phase, both for the slow and for the fast variables. Such processes, which result from small terms in the evolution equations, correspond to the phase mixing of the waves and ensures the ergodicity of the system. In particular, the slow variables have a non-trivial time-evolution and are characterized by a dominating
1-312
scaling in the
low frequency range for time scales between 0.5 Ec- I and 10 Ec- I . Instead, in the case of fast variables, the phase mixing appears as a slow amplitude modulation occurring on time scales of Ec - I which superimposes on the fast dynamics controlled by the 0(1) time scale 11 AI . The system introduced in this paper features very rich dynamics and, therefore, may have prototypical value for phenomena generic to complex systems, such as the interaction between slow and fast variables and the presence of long term memory. Moreover, analysis shows how, neglecting the coupling of slow and fast variables only on the basis of scale analysis - as usually done when discarding the Eckert number - can be catastrophic. In fact, this leads to spurious invariances that affect essential dynamical properties (ergodicity, hyperbolicity) and that cause the model losing its ability to describe intrinsically multiscale processes. We have shown that a standard multiscale approach allows for understanding the role of the small parameter controlling the coupling. This may suggest that a careful re-examination of the scaling procedures commonly adopted for defining simplified models, especially in the climate science community, may be fruitful in the development of more efficient modeling strategies. Note that in a recent study [21] we have tested that refeeding the kinetic energy lost to dissipation as positive thermal forcing to the fluid (which corresponds to considering Ec > 0) brings the long-term global energy budget of the system closer to zero by an order of magnitude. While the numerical values presented in this paper refer to a specific choice for the parameters r, (J, b, analogous properties - like the symmetry break and the ensuing emergence of ergodicity and multiscale processes, the linearity of the Lyapunov exponents with respect to Ec- I , the low frequency variability are expected for all values of r, (J , b leading to a chaotic dynamics.
184
V. Lucarini and K. Fraedrich
Bibliography [I] E.N. Lorenz, J. Atmos. Sci. 20,130 (1963) [2] D. Ruelle, Chaotic Evolution and Strange Attractors, Cambridge University Press, Cambridge, 1989 [3] B. Saltzman, J. Atmos. Sci. 19, 329 (1962) [4] J.H. Curry, Commun. Math. Phys. 60, 193 (1978) [5] Z. Wenyi and Y. Peicai, Adv. Atmos. Sci. 3, 289 (1984) [6] D. Roy and Z.E. Musielak, Chaos, Solitons and Fractals 31 (2007) 747 [7] P. Reiterer, C. Lainscsek, F. Schtirrer, C. Letellier, and J. Maquet, J. Phys. A 31,7121 (1998) [8] C. Sparrow, The Lorenz Equations. B!furcations, Chaos. and Strange Attractors, Springer, New York, 1984 [9] c. Bonatti, L.J. Diaz, and M. Viana, Dynamics Beyond Un!form Hyperbolicity: A Global Geometric and Probabilistic Perspective, Springer, New York, 2005 [10] V. Lucarini, l. Stat. Phys. 131,543 (2008) [II] D. Ruelle, Phys. Lett. A 245, 220 (1998) [12] D. Ruelle, Nonlinearity 11, 5 (1998) [13] V. Lucarini, J. Stat. Phys. 134,381 (2009) [14] L.A. Smith, C. Ziehmann, and K. Fraedrich, Q. J. R. Metero!. Soc. 125,2855 (1999) [15J T.N. Palmer, Bul!. Am. Met. Soc. 74, 49 (1993) [16] C. Nicolis, Quart. J. Roy. Meteor. Soc. 125, 1859 (1999) [17] Z.-M. Chen and W.G. Price, Chaos, Solitons and Fractals 28 571 (2006) [18] V. Oseledec, Moscow Math. Soc.19, 197 (1968) [19] G. Benettin, L. Galgani, A. Giorgilli and J .M. Stre1cyn, Meccanica 15, 9 (1980) [20] J.L. Kaplan and l.A. Yorke, Chaotic Behavior of Multidimensional Difference Equations, in Functional Differential Equations and Approximations of Fixed Points, edited by H.-O. Peitgen and H.-O. Walter, (Springer, Berlin, 1979) [21] V. Lucarini, K. Fraedrich, F. Lunkeit, Thermodynamic Analysis of Snowball Earth Hysteresis Experiment: Efficiency, Entropy Production, and Irreversibility, submitted to Q. l. Roy. Met. Soc. (2009); also at arXiv:0905.3669vl [physics.ao-ph]
185
Sub-Critical 'Transitions in Coupled Map Lattices P. Manneville Hydrodynamics Laboratory Ecole Poly technique 91128 Palaiseau, France (e-mail:
[email protected]) Abstract: Sub-critical transitions between two collective regimes with different periods have been observed in a 4-dimensional hypercubic lattice of coupled logistic maps. Statistical findings can be interpreted in terms of macroscopic quantities (measuring global properties) escaping from potential wells in the presence of noise induced by chaos in the microscopic dynamics (local variables). The relevance of these results to a "thermodynamic" understanding of the dynamics of distributed complex systems is briefly discussed. Keywords: coupled map lattices, sub-critical transitions, escape from attractors.
1
Introduction
Large assemblies of interacting nonlinear units give examples of complex systems in which collective behaviour emerges, characterised by the ordered evolution of averaged quantities in spite of strong chaos persisting at the scale of the individual units. The acronym NTCB, meaning non-trivial collective behaviour, has been coined to label this dynamics, emphasising the fact that conventional thermodynamic arguments seem to rule them out. In the present study, we consider coupled map systems [1] in the form: XHI -
.i
-
_1_ " 2d + 1
~ j'EVj
f r (xt) J'
,
where t denotes (discrete) time and Vj the neighbourhoods of sites j at nodes of ad-dimensional hypercubic simple lattice. Here, Vj is limited to nearest neighbours and the local evolution is governed by the logistic map fr(X) = rX(l-X). Such models have already been studied in various space dimensions [2]. They are known to display NTCB for r between the accumulation point of the period-doubling cascade of the isolated map, roo ~ 3.57, and its crisis point rmax = 4. Thc nontrivial features arise from the fact that the mechanisms at the origin of the different macroscopic states and the transitions from one to another are not known. In particular, the mean-field approximation obtained by replacing variables by mean values and neglecting correlations is unable to give hints on the emergence of a collective behaviour apparently unrelated to the properties of the isolated map.
186
P. Manneville
Bifurcations between different NTCB has been observed upon varying r. Supercritical period-doubling cascades for d = 2 and 3 and the Hopf bifurcation toward quasi-periodic behaviour for d = 5 are well documented [3,2]. The size of the system as measured by the number of sites N = n d , where n is number of sites along a space direction, is an important parameter and, remarkably, NTCB is better and better defined as N increases (the so-called thermodynamic limit). Accordingly, it is believed that the asymptotic evolution in the infinite-time limit is accounted for by a genuine attractor in the infinite-size limit. Whereas size effects on continuous transitions, especially period-doubling bifurcations, can be understood within appropriately adapted standard approaches [4], the role of the noise that they introduce in discontinuous transitions needs further study. We focus here on the subcritical bifurcation between two NTBC periodic regimes T2 and T4, with respective periods 2 and 4, coexisting in the range 3.88 < r ::; 4 [2]. Only preliminary results are presented; a more complete analysis is in progress.
2
Results
NTCB is identified from the evolution of the mean value of the Xj over the lattice at time t, X t = N- 1 ~j Xj. Size effects alluded to above are easily observable at r = 4 as seen in Fig. 1 which displays time series of X t for systems with n = 16, 19, 23, 27, and helical boundary conditions,! the numbers of sites increasing by factors of 2 from 6.510 4 to 5.310 5 . The T4 regime corresponds to 4 bands and the T2 regime to 2 bands only with the 't = 4k + 3' and' 4( k + 1)' series partly masking the' 4k + l' and' 4k + 3' series. For n = 16, switchings between T4 and T2 are frequent and it may happen that a phase change occurs upon return to the T4 regime, e.g. after the T2 episodes at t :::::; 1.3105 , :::::; 1.42105 and:::::; 1.82105 . For n = 19, switchings are less frequent, even less for n = 23, and for n = 27 regime T2 seems sustained (notice the change of time scales for n = 23, 27). This situation no longer changes as n is increased further. These observations helps us understanding that bifurcation diagrams strongly depend on the experimental protocol. In [2], regimes T2 and T4 were found to exist in the range r E [3.90,3.97] with n = 27 in sweepup/sweep-down adiabatic simulations with 2,000 iterations and detection of the state obtained after elimination of 1,500 iterations. Considering instead 10 4 iterations, after elimination of 8 10 3 transients, regime T2 is obtained in sweep-up simulations up to r :::::; 3.94 where a transition to T4 is observed; regime T4 is then kept up to rmax = 4. On the other hand, in sweep-down 1
The lattice is arranged as a one-dimensional array of size N = n 4 with periodic boundary conditions. The coupling then reads ~ (Yj + Yj+l + Yj -1 + Yj+n + Yj-n + YJ+n2 +YJ-n2 +YJ+n3 +YJ- n3) for j = 1, ... ,n4, where Yj = Jr(Xj). Other kinds of boundary conditions, e.g. periodic, give the same results provided that the system is large enough.
Sub-Critical Transitions in Coupled Map Lattices 0.9
187
~----~--~-------,
o'r:"-
Xo for any t fixed ;
xl!...rICioJ f(y)pt( x,dy) for a ll continuous functions
=./
f(y)pt( xo, dy)
f on Sj;
(4) Pt(x, U(x)) ---. 1 as t 1 0, for any neighborhood U(x) Of x; (5) the Chapman-Kolmogorov equation holds:
Ps+t(x,A) = }SPt(x, dy)ps(y, A).
226
G. V. Orman
In a similar manner the transition operators can be defined. It is considered that C = C(S) is the space of all continuous function s (it is a separable Banach space with a supremum norm). The following operators, denoted by Pt ,
(ptf)(x) =
is
Pt(x, dy)f(y),
f EC
are called transition operators. Now the conditions for the t ransition probabilities can b e adapted to the transition operators. We do not insist here on this case (for more details see [4], [6], [2]). Let us observe that the conditions (1) - (5) above are satisfied for Brownian transition probabilities. One can define 1
pt(x , dy)
=
pt(oo , A)
=
_
=e tv 27r o=A.
(,, _ x)2
dy
2,2
inR
Definition 1. A Markov process is a system of stochastic processes
{Xt(w),t E T,w E (O,K,Pa)}aES. [For each a E S, {Xt}tES is a stochastic process defined on the probability space (0 , K , Pall. The transition probabilities of a Markov process are {pet , a, Let us d enote by {Ht} the transition semigroup and let R ", b e the resolvent operator of {Ht}. Now pet, a, B), Ht and R", can be expressed in terms of the process as follows.
Bn.
Theorem 1. Let f be a fun ction in C(S). Then 10. p(t,a,B) = Pa(Xt E B). 20. For EaO = .f~ ·Pa(dw) one has Htf(a) = Ea(f(Xt)). 30. R ", f (a) = Ea e-"'t f(Xt)dt).
Uo=
Proof. One can observe that 1° and 20 are immediately. Now, for 30, it is considered the following equality:
R",f(a) =
1=
e - ",t Hd(a)dt =
1=
e-",t Ea(f(Ht))dt .
But f (X t (w)) is right continuous in t for w fixed and measurable in w for t fixed and therefore it is m easurable in the pair (t, w). Thus, the Fubini's theorem can be used and one gets
that is just 30..
Definition 2. The operator
et
:
° ° --t
defined as follows
Uhw)(s) = w(s
+ t)
for every s E T, is called the "shift operator". Evidently, (the semigroup property). Notation. For e a a-field on 0 , the space of all bounded e-measurable functions will be denoted by B(O, e), or simple B(e).
A Survey of the Transition Probabilities and Markov Property
2
227
Markov property
We shall develop below some aspects concerning the Markov property and related ideas in a vision of Kiyosi Ito. Firstly the classical and the extended Markov properties are presented. Then the strong Markov property will be considered. The Markov property is expressed in the theorem below. Theorem 2. Let be given r E K. The following is true
that is to say Note 2. The following notation can be used
Proof. It will be suffice to show that (1)
for r E K a nd DE K t . One can distinguish three cases. [I.] Let us consider rand D as follows:
and with
o :S Sl < S2 < ... < Sn o :S t 1 < t2 < ... < tm :S t and B i , Aj E K(S). Now it will be observed that the both sides in (1) are expressed as integra ls on srn+n in terms of transition probabilities. Thus, one can see that they are equal. [II.] Let now be r as in the case [I.] and let us denote by D a general member of Kt. For r fixed the family 'D of all D's satisfying (1) is a Dynkin class. If M is the family of all M's in the case [I.] then , this family is multiplicative and Me 'D. In this way it follows 'D(M) c 'D = K(M) = K t and one can conclude that, for r in the case [I.] and for D general in K t , the equality (1) holds. [III.] (General case.) This case can be obtained in a same manner from [II.] by fixing an arbitrary D E K t . It will be obtained that Pa(r) is Borel mea surable in a for any r E K. Corollary 1.
Ea(G
0
et , D)
Ea(EXt (G), D)
for G E B(K), DE Kt,
ell ) etlKtl
Ea(F. EXt (G))
for G E B(K), FE B(Kt) ,
E a(F · (G 0 Ea(G
0
EXt (G)
(a.s.)(Pa ) for G E B (K) .
228
G. V. Orman
2.1
The extended Markov property
It is interesting to see that the Markov property can be extended. The following theorem will refer to the extended Markov property.
Theorem 3. for
r
E K.
Proof. Let us come back to the equality (1) before. Now it will be proved for D E K t +. To this end the following equality will be shown:
for fi E C(S), DE Kt+ and 0 0, so that by Corollary 1 it results
Ea(h (XSI (et+l,w)) ... fn(Xsn (et+hw)), D) = Ea(Ext+,,(h (X S1
) •..
fn(Xsn)), D). (3)
Now one can observe that
is continuous in a, if it is considered that
and Hs : C ---+ C. But Xt(w) being right continuous in t, one gets
1 o.
as h
Now , the equality (2) will result by taking the limit in (3) as h 1 o. In this way, for G i open in S, the following equality will result from (2)
Ea(XSi(etw) E G 1 ,'" ,XSn(etw) E G n , D)
= Ea(PXt(XSl
E G 1 ,'"
(4)
,Xsn E Gn) , D),
and now the Dynkin's theorem can be used. 1
2.2
The strong Markov property
As it is known, the intuitive meaning of the Markov process (for example X(t)) is the fact that such processes "forget" the past, provided that t n -1 is regarded as the present. Now, the intuitive meaning of the Markov property is that under the condition that the path is known up to time t, the future motion would be as if it started at the point
Xt(w) E S. But a question arises: what will happen if "t" will be replaced by a random time
a(w)? 1Theorem (Dynkin's formula). Let us suppose that a is a stopping time with Ea(a) < 00. Then, for u E :D(A) it follows:
Ea
(~OO AU(Xtl dt )
=
Ea(u(Xcr)) - u(a).
A Survey of the Transition Probabilities and Markov Property Definition 3. A random time to {Kt} if
0" :
it
-->
229
[0, (Xl] is called a "stopping time" with respect (5)
for every t. But such a condition is equivalent to the condition
(6) for every t.
Observation 1. If the condition (5) holds, then it results
But if the condition (6) holds, then we have {O" 0.
C,,+ y) 2 e - ~ 'P(x)dx
--+
'1'( -y) =
-00
°
Therefore,
p(x, t, y)
--+
o(x - y) as t 10 for all x> 0, Y
> 0,
where 0 is the Dirac 's o-function. 2. Now let us consider the Brownian motion on x > but with a reflection barrier at the origin. The forward Kolmogorov 's equation for a Brownian motion on x > with an reflection boundary at x = is given by ap 1 a 2p y>o at 2 ay2'
°
°
°
ap(x , t , y) / =0 ay y=o p(x,t,y)
--+
o(x - y)
as
t
1 0, x > 0, y > 0,
with 0 - the Dirac's o-function. In this case the following solution is found
p(x,t,y)
= -1tV21T
[ ( X_ y ) 2
e-
~
and the condition
ap(x, t, y) -0 / ax x= o holds too.
( x +y) 2 ]
+e- ~
A
Survey of the Transition Probabilities and Markov Property
231
Note 4. It is known that in some conditions of existence, the transition density satisfies the folowing two Kolmogorov's equations (they will be referred to as the backward Kolmogorov 's equation and respective the forward Kolmogorov 's equation). o - .f~(c:..:t,-=-:.x,--;T-,--,y~) ( ) of(t,x;T,y) - -1 b( t x ) 0 2 f(t,x;T , y) = -a t x (8) ot 'ox 2' ox 2 and of(t , x; T, y) 0 1 02 OT = - oy [a(T,y)f(t,x;T,y)] + 2 oy2 [b(T,y)f(t,x;T,y)], (9)
where aCt , x), bet , x), aCT, y) , beT, y) are functions satisfying some conditions to assure the existence and the uniqueness of the solution of the equations. The forward Kolmogorov's equation is also referred to as the Fokker-Planck equation. We emphasize that the expression "the backward Kolmogorov equation" m eans: in the "backward" variables t and x. Similarly, "the forward Kolmogorov equation" must be understood that it is considered in the "forward" variables T and y. One can observe that this Application implyes stochastic differential equations in connection with the K olmogorov 's equations for a Markov process; more details and related topics can be found in [lS}, flO}, (13).
References [1] BERTOIN, J. , Levy Processes , Cambridge U niv. Press , Cambridge, 1996. [2] BHARUCHA-REID, A.T. , Elements Of The Theory Of Markov Processes And Their Applications, Dover Publications, Inc., Mineola, New York, 1997. [3] G NEDENKO, B.V., The Theory of Probability, Mir Publisher, Moscow , 1976. [4] ITO , K. , Selected Papers. Springer, 1987. [5] ITO , K. , McKEAN , H.P. , Diffusion Processes and their Sample Paths. Springer-Verlag Berlin Heidelberg, 1996. [6] ITO, K., Stochastic Processes. Eds. Ole E . Barndorff-Nielsen, Ken-iti Sato, Springer, 2004. [7] KALLENBERG , 0. , Foundation of Modern Probability, Springer , New York, 1997. [8] LOEVE, M., Probability Theory, vol. I , Springer-Verlag, New York , 1977. [9] LOEVE, M. , Probability Theory, vol. II, Springer-Verlag, New York , Heidelberg Berlin, 1978. [10] 0KSENDAL , B. , Stochastic Differential Equations: An Introduction with Applications, Sixth Edition, Springer-Verlag, 2003. [ll] 0KSENDAL, B. , SULEM A., Applied Stochastic Control of Jump Diffusions, Springer, 2007. [12] ORMAN , G . V., Capitole de matematici aplicate, Ed. Albastra, Microlnformatica, Cluj- Napoca, 1999. [13] ORMAN G. V., Lectures on Stochastic Approximation Methods and Related Topics , Preprint. " Gerhard Mercator" University, Duisburg, Germany, 2001. [14] ORMAN , G.V., Handbook Of Limit Theorems And Sochastic Approximation, Transilvania University Press , Brasov, 2003. [15] SCHUSS Z., Theory and Application of Stochastic Differential Equations, J ohn Wiley & Sons, New York, 1980. [16] STROOCK , D.W. , Markov Processes from K. Ito Perspective, Princeton Univ. Press, Princeton, 2003.
232
Qualitative Dynamics of Interacting Classical Spins Leonidas Pantelidis Department of Physics and Astronomy, Hanover College Hanover, IN 47243, USA (e-mail:
[email protected]. edu)
Abstract: \Ve consider the classical Heisenberg model (Hl'vI) with z-axis anisotropy and external magnetic field. Its phase space is foliated into a family of invariant leaves diffeomorphic to (S2)A The flow on each leaf S is Hamiltonian. For the isotropic Hl'vI with zero field, the manifold :F of longitudinal fixed points (LFPs) intersects each leaf S orthogonally. In addition, we show that the ferromagnetic (FR) state and the antiferromagnetic (AF) state with non-zero total spin are both stable LFPs. This is a direct implication of a Lemma which extends Lyapunov stability from an invariant subspace to the whole leaf by exploiting the rotational symmetry. The lemma does not apply in the case of zero total spin, and indeed, the AF state on an equal-spins leaf is shown to be unstable. PACS numbers: 05.45.-a
L Introduction The Heisenberg model (HM) is a simplified model of magnetic ordering in materials [1, 2, 3, 4] which has drawn considerable attention for a long time. Theoretical and experimental studies on the quantum version of the HM are abundanq. The classical HM has also been of interest, both as a model of ferromagnetic and other phenomena and as a testground for methods of nonlinear dynamics. However, cla.'isical Heisenberg spin systems, especially chains, have been mostly studied in the continuum limit§. From the point of view of quantitative analytical dynamics, the discrete HM is barely tractable with even a simple Heisenberg ring (closed chain of spins with equal nearest-neighbor only interactions) being non-integrable. The only known exact solutions for the HM on a general periodic lattice arc the (propagating) ferromagnetic (FR) and antiferromagnetic (AF) nonlinear spin waves (NLSWs), and some special planar solutions [32, 33, 34]. These arc finite-amplitude generalizations of the smallamplitude linear spin waves, the latter being approximate periodic solutions of the nonlinear equations of motion in the vicinity of the FR and AF fixed points (FPs) II. :j: Some important work in the field can be found in [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 251 and references therein, although this list is by no means comprehensive. § For instance, see [26,27, 28, 29, 30, 311 and references therein.
II
Also termed as critical points, equilibrium points, or steady states.
Qualitative Dynamics a/Interacting Classical SpillS
233
Certain linear combinations of NLSWs havc also been found to satisfy the equations of Illotion [35], although these solutions do not, in general, reside on the leaf (shell) of equal spins ("off-shell" NLSW s). In all the aforementioned solutions the spins have constant and equal (in each sublattice, in the AF case) z-components, that is, all known solutions are of a rather special type. This paper focuses on the qualitative dynamics of the classical HM. In section 2, we introduce the general classical HM with z-axis anisotropy and magnetic field. In section 3, we discuss its phase space structure and in particular the foliation of the phase space into invariant leaves with Hamiltonian flow. In section 4, we discuss the usual FPs of the flow and point out that they form manifolds which intersect the leaves orthogonally. Section 5 addresses the question of stability of FPs. This question is answered in the special but interesting cases of the FR and AF states on the basis of a general lemma which extends Lyapunov stability from an invariant subspace to the whole leaf by exploiting the rotational symmetry of the model. 2. The classical HM The standard quantum HM Hamiltonian reads A
A
(2.1 )
H = - LgjlSJ' S[- LBJ' Sj, j dV = -jAI dr
. dl y' JI =rb --->-=--V dr jA
(8)
Where we have introduced the propagation constant defined by the relation Equivalently we may define the transmittances to be Z, = -jA
Y,
=-j~
Then the characteristic impedance are defined through the relations: -Z
(9)
Z and the propagation factor y of the equivalent transmission line
rz; ..,1, '~l =Vy:=Jy=V;:y--l
(10)
y=~Z,y, From standard transmission line theory we have that propagation is defined for IAI
2
Y
~
(11)
-yl(l+l)
While damping (evanescent modes) for ,
1.,1,1
r>O or r< ~ -yl(l+l)
(12)
An element of such a transmission line corresponds to a lumped-element model of the initial domain [0, ooj of the radius r into a set of mesh points ro' r, "'" r" with each element interpolating into the arbitrary intervals [r",r,,+,j,We then take each element to correspond to a segment of infinitesimal width
LY"
= ':,+1 - ,:, for which the propagation factor y and the characteristic impedance Z can be
244
C. D. Papageorgiou and T. E. Raptis
taken approximately constant (defined as Yn and Z,, ) . For such an element we may take the equivalent relations for a homogeneous transmission line as follows.
[ cosh(q,,) [ V(~,ll I (~,l = sinh(q,,) / Z"
(13)
where we have put q" = r" . /';.~, . Thus the value of the input impedance at each point of the line will be given by the recursive formula Z(r ) = V(r,,) = " I (r,,)
Z
Z(r,,+l) + Z" tanh(q,,) " Z(r,,+l) tanh(q,,) + Z"
(14)
The boundary conditions are defined through the respective limitsZ(r ~ O),Z(r ~ 00) based on the assumption that the asymptotic limit of a lossless transmission line corresponds to the fixed value of the characteristic impedance given by Z" according to (14) for r ~ 0 and r ~ 00.
4. Trapped modes and bound states The form of the propagation constant leads to the possibility of "trapped" waves and their respective eigen-modes. A necessary condition for trapping is to have at least one region r1 ::; r ::; r, where y2 (r) < oallowing propagation and two disparate regions r < r" r > r 2 with y2 (r) > 0 where only evanescent modes exist. This situation is reminiscent of the bound states of quantum mechanical systems with the function 112 (r) playing a role analogous to that of the potential in the radial Schroedinger equation. Evanescent modes are then analogous to the scattered wave functions above a threshold. On the other hand, a constant value of A allows only a simple root r3 = ~l(l + I) / Au beyond which propagation is allowed up to the infinity. We will next evaluate one such example of an eigenvorticity function supporting a trapped mode. According to a standard technique introduced in [11] and [12] , a method for locating the eigenvalues and compute the associated eigen-mode is based on a simple resonance theory for transmission lines. In this viewpoint, we may take the bound state to correspond to an area where ordinary resonance takes place through the condition Z L = Zc between the "capacitive" and "inductive" elements in the imaginary part of the respective complex impedance. This can effectively be restated as (15) Where Z + (~) and Z - (~.) are the input impedances of the equivalent transmission line as shown from the point rc (right and left of it). Evolution of the above quantity in the region [0,00] prescribes also the associated eigen-mode. In order to locate a certain mode for a given value of angular momentum we iterate the mapping (14) forward (0 ~ r, ) and backward (00 ~ 1',. ) and expand the interval until we find a root of (15). As an example let us make a simple choice of the propagation factor given below
r'= L2(r-~)(r-r2) r + fir2r 4
2'
h
were,
L=l(l
+ 1)
(16)
A Transmission Line Modelfor Spherical Beltrami Problem The propagating region is 'i
:0;
245
r :0; r, and the evanescent regions are r < ~, r > r,
· ,.2 ,'.It can be proved that: · th Usmg ereIatlOn r = 2L' - /L. r
,12 = L2 .(~ +r2)
(17)
r·(r' +~r,)
For given integer values of I and real positive values of rl using the equation (15) for rl and variable values of r2>rl i.e. (18) Quantities Z+(~) and Z·(~)are calculated through the recursive relation (14). Then the value of r2 for which the equation (18) holds can be defined and though it the proper function A(r) by (17) for which the Beltrami equation (1 a) is fulfilled. In Fig. 1 we show fori = I rl=1 and r2=8.12 the exact form of the propagation factor y' and the eigen·vorticity ,12 that is positive definite everywhere. 1=1 ;r1=1 ;r2=B.12 3,-----~-----,------._----_,------,_----~
.£ " t o > C
0.5 stretch it.
---l
1
r-- ---,
1r------
I
\
-1
-1
-1
Li. ____--'
-1
-1
(a) r = 0.1.
(c) r=0.8.
(b) r = 0.42.
1\\
\
L i
o -1
-1
(d) r=O.1.
/ \
/
·/ I
---~
0
(e) r=0.42.
I I
o ~~_.---.J -1
0
(f) r = 0.8.
Fig. 1. Three examples of deformed circular maps and their invariant densities. (a): Map for r = 0.1, (b): map for r = 0.42, (c): map for r = 0.8, (d): density for r = 0.1, (e): density for r = 0.42, (f): density for r = 0.8.
It can be confirmed that the function f(x), given by f(x) '= {-2(1 - r)x, -1::; x ::; 0 .' 2rx, 0 < x ::; 1 '
(5)
satisfies the Perron-Frobenius equation. Thus, f(x) can be seen as the invariant distribution of spreading iteratively produced by deformed circular maps. Figure 1 displays the plot of f(x) for three values of r. For r = 0.5, the so called circular map has the "V-shaped" invariant density f(x) = lxi, x E [-1,1], which results in zero mean. The meanadjusted quadratic ACF is known for any lag (see [6]):
(6) lag(l) = -0.5 according to (6) and therefore the circular map has smaller first lag than the tent and Bernoulli maps, which have lag(l) = 0.25.
252
T. Papamarkou
The lo,g(l) value of the mean-adjusted quadratic ACF has been calculated as a function of deforming para~eter r, see Figure 2. With the help of numerical minimization, it has been found that the minimallo,g(l) is approximately equal to -0.722 and is achieved for r c:::' 0.42. So, the optimal deformed circular map, with respect to the third optimality criterion, is the one with deforming parameter r c:::' 0.42.
Fig. 2. Dashed line: Plot of lag(l) of mean-adjusted quadratic ACF of deformed circular family versus its deforming parameter r . Solid line: Fnkhet lower bound of lag(l) of mean-adjusted quadratic ACF for the class of spreading sequences with invariant density given by (5).
Jag(l ) 0.4 0.2 - 0.2 -0.4 - 0.6
- 0.8 -1.0
Also, the Fnkhet lower bound of lo,g(l) has been computed, see [7]. Such a lower bound gives the minimal attainable lo,g(l) mean-adjusted quadratic autocorrelation among all stationary processes with invariant distribution described by (5), see Figure 2. Despite having not reached the lo,g(l) lower bound, spreading produced by the deformed circular maps outperforms that generated by the logistic, Bernoulli, tent and other conventionally used maps. BER simulations confirm the superiority of deformed circular family, see Figure 3.
10-2 +------------ , Tent Logistic Del Circ, r=0.42
10-3
10-3 ~---L-B~------~----~ o 5 10 -5
10-4
(a) N
=
2.
Tent Logistic Del Circ, r=O.42 LB
-5
0
5
10
(b) N=5.
Fig. 3. Simulated BER of tent, logistic, circular and deformed circular (r = 0.42) spreading for two values of spreading factor N.
Two Aspects of Optimum CSK Communication
4
253
Paired Bernoulli circular spreading (PBCS)
When r = 0.5, the Frechet lower bound of the first lag equals - 1. This implies that there exists a stationary process with the "V-shaped" invariant density f( x ) = lxi, x E [- 1, 1] and lag(l) mean-adjusted autocorrelation equal to -1. Starting from that knowledge of the distribution of spreading which fully meets the third optimality criterion, the Bernoulli circular spreading (BCS) has been defined, see [7]. Although BER simulations showed that BCS is a relatively efficient type of spreading, it has a main drawback; BCS can take only four values. As an improvement over BCS , paired B ernoulli circular spreading (PBCS) has been suggested. To introduce PBCS, let X := (Xo, Xl,'" , XN - d denote the N - length spreading sequence, where N is any even natural number. Also , consider the function f
k lR. j '( x) '= { Ix - kl, x E [k - 1, k + 1] . 0, x E lR. \ [k - 1, k + 1]' E ,
(7)
and the random vector (AI , A3 , . .. , AN-d given by A 2i -
l
:=
(_1)1-O
(5)
Numerical Exploration of Dynamical Behaviour of q-Deformed Nonlinear Maps
x 1- (1- q)(x -1)
259
(6)
such that
lim[x]q q--.l
= x.
form xn+l
=f
The q-deformation scheme for ID nonlinear map [2] of the
=f
(X n) is given by xn+l
([Xn]q) .
In this paper we are aimed to numerically analyze the dynamical behaviour of the q-deformed versions of the widely studied 1D nonlinear map-the Gaussian map and another famous 2D nonlinear map-the Henon map. In the next subsections, we briefly introduce the Gaussian and Henon maps and their q-deformed versions. 2.1. The q-dejormed Gaussian map The Gaussian map [6] is based on the Gaussian exponential function. It is characterized by two parameters band c as follows:
x n +J
= e-bx,~
+C
(Gaussian map)
(7)
The q-deformed version (i.e. q-Gaussian map) of the Gaussian map by following the above definition of the q-deformation of ID nonlinear map is given as:
xn+1
=e -b {{x"lql
2
+c
(q-Gaussian map)
(8)
or
b{ xn+1 = e
}2
x" J-(J -q)(x -1)
"
+c (q-Gaussian map)
(9)
Under the limit q -7 1 , the q-Gaussian map becomes the original Gaussian map (Eq. (7»). In this paper, throughout the discussion on q-Gaussian map, we prefer to use deformation parameter [; instead of q (which are related by the relation 1 - q = [; ). Hence the explicit form of q-Gaussian map, which we use, is given as:
xn+1 = e
-b{ 1-"'(;"-J)}' +c, (q-Gaussian map)
(10)
which under the limit [; -70, becomes the original Gaussian map (Eq. (7». 2.2. The q-dejormed Henon map One of the simplest mathematical models, which exhibits strange attractor is the quadratic map introduced by Henon in 1976 [7]. The mathematical form of the Henan map is given by
260
V. Patidar. G. Purohit, and K. K. Sud
X,,+l =f(x",y,,)=I-ax,~+ y"
(Henan map)
(11)
Y,,+l =g(X 'YIl)=j3x" Il
If we generalize the above definition of q-deformation of ID nonlinear map to the 2D case then we may introduce different deformation parameters for different state variables. Such generalized form of a 2D q-deformed map is given by
X,, +l = f([xJ qx,[Y,,]q , ) (2D q-defarmed map),
(12)
Y,, +l = g([xJ qx,[yJ q , ) X
"
1- (1- qx )( x" -1) an Clearly under the limits
~ land
qx
reduces to original map.
d[
y"
]
q ,-
1- (1- q y)(y" -1)
qy ~ 1the q-deformed 2D map
The explicit form of the q-Henon map (by
introducing 1- qx = Cxand 1- qx =
cx )' which we use throughout this
study, is given by
x
,,+1
=1-a(
x" l-c ,(x -1)
J2
ll
-
y".,
=
pC -£,~:"
which under the limits
cx
~
-I)
0 and
+[
y" l-c y (y,,-1)
J (13)
J
cy
~
0, becomes the canonical
Henan map (Eq. (II )). In the next section, we present results of our numerical exploration of the dynamical behaviour of above described qdeformed nonlinear maps.
3. Results and Discussion A recent study [2] on q-deformation of nonlinear dynamical systems has revealed that the q-deformed logistic map exhibits a variety of dynamical behaviours: fixed point, periodic, chaotic and more interestingly the coexistence of attractors, which is a rare phenomenon in the 1D nonlinear maps. In the first part of our study we are interested in analyzing another ID mapthe Gaussian map, which is known to exhibit period doubling route to chaos and co-existing attractors [3,6] under the same q-deformation scheme. Recently, Patidar [6] has done a detailed study on the regions of the parameter space of Gaussian map where co-existing attractors exist. Here we embark to analyze the effect of q-deformation on the regions of the parameter space of Gaussian map where co-existing attractor exist. In Figure 1, we have shown the results of the analysis on the Gaussian map [3]. Particularly in Figure 1(a), we have identified different regions of the parameter space.
Numerical Exploration o/Dynamical Behaviour ()f q-De{ormed Nonlinear Maps
261
I)
+I).,w
&. ~ ~~".,..,..,,~,...j..,~=,-~-+~""""_-i,",=-~_ _
(a)
4::14
1-·-~·--~"""'T~·"""·--'-"!""",-·'~""",,,-~~,,,,'~....,.----+·1.1(i I)
11,. P,"';lM__~
:; 0.00
un
HI
. , . ' ' - - _ - - ' -_ _~_ _ _L __
1~
~\J
20
'15
__'__ __'__ _~_,__+
0 .00
(b) .0.29
·0.29
ill iil .0.50
.1l.58
~g'
,.~
IL
.lUll
·1.1&
-0.81
-"-~--
...---~--.--.-~--,--~--+
·1.16
P Ilr ameter (b)
Figurc 1. Paramcter space (b, c) of the Gaussian map (a) showing the regions, where chaotic (black shade) and regular (white shade) motions appeaL (b) showing the regions where a period-] attractor co-exists with some other periodic attractor (grey shade) and the regions where a petiod-1 atlmctor co-exists with a chaotic attractor (black shade),
(b ,c) where the dynamics of Gaussian map is chaotic and regular (black and
white shades correspond to chaotic and periodic attractors respectively). The results shown in Figure lea) are based on the extensive numerical calculation of Lyapunov exponent by iterating the Gaussian map 1000 times (after neglecting initial transient behaviour up to 400 iterations) at 2 X 106 different points of the pammeter space defined by 0::;: b S 20 and -1.0 S c sO,
262
V Patidar, G. PUlVhit. and K. K. Sud
Chaotic situation has been recorded, whenever the Lyapunov exponent becomes greater than 1 X 10-4 • The calculation has been repeated for several sets of initial conditions to include all the co-existing attractors. In Figure 1 (b), we have shown the complete region of parameter space where Gaussian map exhibits co-existing attractors. -Q..&l O.Q
-i"-----,.,...·--""----·--'----'-------+t,.J(.I (a)
·0.50
-0.25
.\).!i411
,1).2:/j
O,;jfofrf,lIith';,r.
fI.O
Pi\'it'~ITWi'Iw !Ii:~
o.~
.~U
r -.......- . . . . . L -.......--iL....~........-..L--.-........-~-{M. b:5.0
(b)
Figure 2. Parameter space (E, c) of the q-Gaussian map for b=5.0 (a) showing the regions, where chaotic (black shade) and regular (white shade) motions appear, (b) showing the regions where a period· 1 attractor coexists with some other periodic attractor (grey shade) and the regions where a period-] attractor co-exists with a chaotic attractor (black shade).
Numerical Exploration of Dynamical Behaviour ofq-Deformed Nonlinear Maps
263
Particularly, the grey shade shows the regions of the parameter, where a period-l attractor co-exists with some other periodic attractor (i.e., period-I, period-2, period-4, ... , period-3, etc.), however the black shade shows the regions of the parameter space where a period-l solution co-exists with a chaotic solution . • (1.00 OO·i-~~--~~~--~~~--~--~--~~-~+~'.O
(a)
.... liO (1,(1
+------'--'---......- .......- ........- ......--~+Q.1i' b=7.5
(b)
Figure 3. Parameter space ( c , c) of the q-Gaussian map for b==7.5 (a) showing the regions, where chaotic (black shade) and regular (white shade) motions appear. (b) showing the regions where a period-l aUmctor co-exists with some other periodic attractor (grey shade) and the regions where a period-l attractor co-exists with a chaotic attJactor (black shade).
264
V. Patidar, G. Purohit. and K. K. Sud
As explained in Section 2 that the q-deformation of any function/map is to introduce an additional parameter (£ ) in the definition of that function/map in such a way that under the limit £ --7 0, the original function/map is recovered. The q-deformation of the Gaussian map (Eq. (10)) leads to a three parameter one dimensional nonlinear map. Now it becomes more complicated to analyze the dynamical behaviour of the q-Gaussian map in a three dimensional parameter space (b, £ , c). The effect of parameter b on the dynamical behaviour of Gaussian map has been analyzed [4], so we prefer to work in the two dimensional parameter space (£ , c) for some fixed values of parameter b. The results of our analysis for the co-existing attractors in qdeformed Gaussian map (the analysis similar to the Gaussian map reported in Fig. I) have been depicted in Figures 2 and 3 for b=5.0 and b=7.5 respectively. It can be easily observed from Figure 2 that for b=0.5, the nondeformed Gaussian map ( £ = 0 ) there is a range of c for which co-existing attractors exist. For all the values of c which belong to this range, both the co-existent attractors are periodic i.e., chaotic solution does not co-exist with a period-l attractor for any value of c. Now if we deform the Gaussian map by changing the value of deformation parameter £, then for all positive values of £ both the co-existent attractors are periodic and the range, for which co-existing attractors exist, is decreasing with the increase in the value of £ in positive direction. However, if we increase the value of deformation parameter £ in the negative direction, initially the range of c, for which coexisting attractors exist, increases and then after a particular value of £ , it starts decreasing. We also notice an important feature that for some negative values of deformation parameter £, one of the co-existent attractors is chaotic. A similar feature we observe for b=7.5, the only difference is that the whole pattern is shifting towards the higher values of deformation parameters. In conclusion to the study on q-deformation of Gaussian map, we may infer that the q-deformation of the Gaussian does not lead to a drastic change in the dynamical behaviour (i.e., the qualitative behaviour is similar), however it introduce an additional parameter in the definition of the map, which sometime may be useful for making a choice of the desired dynamical behaviour required for some specific purposes (in case if we do not have direct access to change the parameters of the Gaussian map i.e. b and c). In the second part of our study, we analyze the dynamical behaviour of Henon map under the same q-deformation scheme. Since the Henon map is a 2D map and to consider the most general case, we introduce the two different deformation parameters corresponding to the deformation of two different state variables as explained inSection 2.2. Moreover in the canonical Henon map two system parameters are present hence after introducing the qdeformation, it becomes a four parameter system. To analyze the dynamical behaviour of this four parameter system, we choose fixed values for the parameter of canonical henon map i.e. a and f3 and then analyze the the dynamical behaviour of the q-deformed system in the space of deformation
Numerical Erploration of Dynamical Behaviour of !I-Deformed Nonlinear Maps
-2
~
Henon Map Parameter Space a 1
2
3
1.0
1.0
0.5
0.5
is 0.0
0.0
...
-... Q)
~0.5
·0.5
(!J
(tf
0.
-1.ll
-1.0
-1.5
-1.5
-2.0
-2.0 -2
-,
0
1
2
3
Parameter (0:)
Figure 4, Pm'ameter space of the Henon map showing the regions correspond to variety of dynamical behaviours: fixed point solution (Black shade), pCl10dic solutions (blue shade). chaotic solutions (red shade) and unbounded ~olUl.ions (white shade).
·1.0 1.0
q-Hel1on Map Parameter Space for II ::: 1.4 & j3 ::: 0.3 -0.5 0.0 0.5
1.0 1.0
£05 ... .
0.5
(U
Gi E I!:! \'II
0.0
0. 0.0
c:
10
:; E ... o
-0.5
1il -0.5
o
-1.0 -'1.0
-1.0
-o.S
0.0
0.5
i .0
Deformation Parameter (E.)
Figure 5. Parameter space of the q-Henol1 map showing the regions correspond to variety of dynamical behaviours: fixed point solution (Black shade), periodic solutions (blue shade), chaotic solutions (red sbade) and unbounded solutions (white shade).
265
266
V. Patidar, G. Purohit, and K. K. Sud
parameters C x and
cy .
In this way, we will be able to see the effect of q-
deformation on the dynamical behaviour of henon map for a particular choice of parameters a and f3. In Figures 4 and 5, we have shown the results of one such analysis. Particularly in Figure 4, we have shown the regions of parameter space (a, f3) of non-deformed Henon map (i.e., c x = c y =0), where different different types of dynamics occur i.e., fixed point solutions, periodic solutions, chaotic solutions and unbounded solutions (diverges to infinity), etc. These results are based on an extensive calculation of Lyapunov exponent at 15 X 104 different points of the parameter space defined by - 2 a 3 and - 2 S f3 1. The black shade in Figure 4 represents the region of parameter space with fixed point solutions, blue and red shades represent periodic and chaotic solutions respectively and white shade represents the regions where dynamics diverges to infinity i.e., unbounded solutions. Now we choose a particular set of parameters a =1.4 and f3 =0.3 for which a chaotic strange attractor exists and see the effect of q-deformation on the chaotic dynamics of the Henon map. The results of this analysis are shown in Figure 5, which is again based on extensive Lyapunov calculation at 4 X 104 points of the deformation parameter space defined by -1 S C x S 1 and -1 S c y 1. The same shading scheme has been used
s s
s
s
as in Figure 4. It is clear from Figure 5 that the q-deformation leads to the suppression of chaos in the Henon map as in the large part of the deformation parameter space the dynamics becomes periodic. Now several questions arise - (i) How this conversion from chotic to periodic solution occurs in this case of q-deformation or in other words what's the route to chaos in qdeformed Henon map?, Whether the topological shapes of the strange attractor of canonical Henon and q-deformed Henon maps are same or different?, Is it a universal feature in the all 2D q-deformed nonlinear map?, etc. Work in this direction is in progress and will be reported elsewhere.
4. Conclusions In this paper we investigated the dynamical behaviour of q-deformed Gaussian and Henon maps. We numerically explored the dynamics of these maps in the complete parameter space including the deformation parameters using extensive computation of the Lyapunov exponent. We observed a variety of dynamical behaviours and we are able to produce the desired behaviour by slightly changing the deformation parameter without disturbing the canonical system parameters of the system, which are not accessible in some practical situations. A more detailed analysis on the q-deformed Henon map is in progress to clarify the routes to the suppression of chaos due to the deformation and some universal features (if exist!) in such q-deformation of nonlinear maps. We strongly believe that such studies of q-deformation of nonlinear maps can be used beneficially in modelling of several phenomena,
Numerical Exploration of Dynamical Behaviour of q-Deformed Nonlinear Maps
267
which are not modeled exactly with the standard maps but their q-deformed versions could serve the purpose.
References [I] C. Chaichian and A. Demichev.lntroduction to quantum groups. World Scientific, Singapore, 1996. [2] R. Jaganathan and S. Sinha. A q-deformed nonlinear map. Phys Lett A, 338:27787,2005. [3] Vinod Patidar.Co-existence of regular and chaotic motions in the Gaussian map, Electronic Journal of Theoretical physics, 3(13): 29-40, (2006). [4] Vinod Patidar and K. K. Sud. A comparative study on the co-existing attractors in the Gaussian map and its q-deformed version, Communications in Nonlinear Science and Numerical Simulation, 14: 827-838,2009. [5] E. P. Borges. On a q-generalization of circular and hyperbolic functions. J Phys A, 31 :5281-8, 1998. [6] R. C. Hillborn. Chaos and nonlinear dynamics. Oxford University Press, Oxford, 2000. [7] M. Henon. A two-dimensional mapping with a strange attractor. Comm. Math. Phys., 50: 69-77, 1976.
268
Spatiotemporal Chaos in Distributed Systems: Theory and Practice George P. Pavlos, A. C. Iliopoulos, V. G. Tsoutsouras, L. P. Karakatsanis, and E. G. Pavlos Department of Electrical and Computer Engineering Democritus University of Thrace, Xanthi, Greece (e-mail:
[email protected])
Abstract:
This paper presents theoretical and experimental results concerning the hypothesis of spatiotemporal chaos in distributed physical systems far from equilibrium. Modern tools of nonlinear time series analysis, such as the correlation dimension and the maximum Lyapunov exponent, were applied to various time series, corresponding to different physical systems such as space plasmas (solar flares, magnetic-electric field components) lithosphere-faults system (earthquakes) brain and cardiac dynamics during or without epileptic episodes. Futhermore, the method of surrogate data was used for the exclusion of 'pseudo chaos' caused by the nonlinear distortion of a purely stochastic process. The results of the nonlinear analysis presented in this study constitute experimental evidence for significant phenomena indicated by the theory of nonequilibrium dynamics such as nonequilibrium phase transItIOn, chaotic synchronization, chaotic intermittency, directed percolation, defect turbulence, spinodal nucleation and clustering. Keywords: nonlinear time series analysis, spatiotemporal chaos, Self Organized Criticality (SOC), distributed systems, nonequilibrium dynamics.
1. Introduction In this study we are interested in distributed nonlinear physical systems such as space plasmas, earthquakes, brain and cardiac dynamics which include practically infinite degrees of freedom, as well as the theoretical presuppositions for understanding their complex dynamics. At the meso scopic level, these systems are described by variables which are functions of space and time as coarse grained fields of microscopic variables. Also, the spontaneous formation of the spatiotemporal structures is the result of sudden qualitative changes known as bifurcations, which occur if an externally controllable driving passes through critical values. At bifurcation points the system can perceive its environment and adapt to it by forming preferred patterns of modes of behaviour. According to Wilson, 1979 [15], precisely at the critical point, the scale of the largest fluctuations becomes infinite, but the smallest fluctuations are in no way diminished. The theory that
Spatiotemporal Chaos ill Distributed Systems
269
describes a physical system near its critical point must take in account the entire spectrum of length scale. Recently, for the understanding of such distributed physical systems two theoretical concepts, contradicting for many scientists, have been introduced: the self organized criticality (SOC) and the low dimensional chaos. In a series of papers we have supported experimentally and theoretically the broad universality of SOC and low dimensional chaotic processes for distributed physical systems. In this study, we present new experimental results and theoretical concepts that reveal the deeper and universal character of the nonequilibrium phase transition and the critical point theory characteristics, applied to distributed physical systems. Particularly, in section 2 we introduce significant theoretical concepts concerning the far from equilibrium phase transition process. In section 3, we present new results obtained by modem nonlinear time series analysis applied to data concerning five different physical systems, such as the solar system, the earth's magnetosphere and crust, the human brain and cardiac dynamics. In section 4 we summarize crucial results obtained by the data analysis and in section 5, we give theoretical explanations of our results embedded in the modem theory of spatiotemporal chaos.
2. Far from Equilibrium Physical Processes The far from equilibrium nonlinear dynamics includes significant collective phenomena such as: power law distribution and critical scale invariance, nonequilibrium fluctuations causing spontaneous nucleation and evolution of turbulent motion from metastable states, defect mediated turbulence and localized defects changing chaotically in time and moving randomly in space, spatiotemporal intermittency, chaotic synchronization, directed percolation causing levy-flight spreading processes, turbulent patches and percolation structures, threshold dynamics and avalanches, chaotic itinerancy, stochastic motion of vortex like objects. In particular, in the phase space of a distributed system various finite - dimensional attractors can exist such as fixed points, limit cycles or more complicated finite dimensional strange attractors, which correspond to chaotic dynamics as well as states known as self organized critical states (SOC). Generally, spatiotemporal chaos includes early turbulence with low effective dimensionality and few coherent spatial patterns or states of fully developed turbulence. These states are out of equilibrium steady states related to the bifurcation
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G. P. Pavios et ai.
points of the nonlinear distributed dynamics, as well as to out of equilibrium second order phase transition processes. Also, spatial and temporal patterns as well as spatially localized structures are possible solutions of the nonlinear spatially extended dynamics. Solar flares, magnetospheric super-substroms, earthquakes and epileptic seizures could be some paradigms of physical systems which include dynamical coherent localized structures underlying these physical events.
2.1. The modern unified theory of the far from equilibrium physical processes A distributed system is governed by a generalized Langevin stochastic equation
a;; =.t:
(-
x
Fig. 2. a) Poincare Map of the ABC flow; with color we interpret the computed value of IT for every closed streamsurface. b) The streamsurface that attracts inertial particles obtained by direct numerical simulation of the inertial equation for t( 1) = -0.01.
3: -
For the computation of the normal vector we exploit the ergodic property of the flow on the invariant streamsurface and we express it as
nT(ip(i,xo)) =O:U(ip(t,xo)) x [rp (t,xo) - ip(s ,xo)] where 0: > a is a normalization constant such that is defined as 8=0;-1 [ inf
I.,.-tl>f)
lin (x)11 = 1 and 8
Ilip(t,XO)-ip(T, Xo)ll]
(12 ) E [0, T]
300
T. Sapsis and U. Haller
:I
x 10"
'1.5 ~~
f;~~11A
I:: :~1'1
1
'1 '0.9
0,8
x -4
-3
-2
,.-,
0,(
Fig. 3 . Invariant streamsllrface for inertial particles. The color shO\vs the proba-, bility density function of the position of an inertial particle.
with Ot (s) == 11'P(t,xo) - 'P(s,xo)11 and () the characteristic time for a particle to run over a typical diameter dso of 8 0 (Figure 1). In this ease the ergodic property of the flow and the Central Limit Theorem gua.rantee that IlnT'(x) - 11 (x) II~'O as T-., 00 with rate of convergence of order 0 (r-l) . To illustrate our results we consider the ease A 2 1, 8 2 = ~) C 2 = by Dobre at aL [5]. For this choice of parameters the phase space is devided into 'ordered' and 'chaotic' regions and the system is not integrable. vVe first compute the Poincare section defined by the plane z = 1\ and sub"equently calculate the integral IT' (1' = 50 ) over the ordered regions where closed streamsurfaces exist. ~ studied
In Figure 2 we present the F)oincare map for the flow, superimposed by a colornlap that represents the value of h· for every closed streamsurface. As we are able to observe there is a pair of dosed streamsurfaces in S-
0
-0.2
-0.2 ·0.4
-0.4
·0.6
-0.6 ._~_~_ .
I
-0 8 L-----~~~--~
-3
-2
-1
0
X-axis
Figure 8. Trajectorie in XY axes, for ill = 0.1, Go = 1500, Y = 3 dB, A = 0.02.
-2
-1
0
__, __ ~_ _i 1
2
X-axis
Figure 9. Trajectories in XZ axes, for ill = 0.1, Go = 1500, Y == 3 dB, A=0.02.
3
Chaos ill a Fractional-Order Jerk Model Using Tanh Nonlinearity
-O:~. 5
'2 -1 ~. S ~ - I -~S-I~~1.5-~ 2
2.5
X-axis
·2
Figure 10. Trajectorie in XY axes, for M = 0.1 , Go = 2000, Y = 3 dB, A=0.02.
-():~.5
-=2'''---::ts--:-1-:O.-s
-o~- -t5-~I--t5--X-axis
343
2
Figure 12. Trajectorie in XY axes, for M = 0.1 , Go = 3000, Y = 3 dB, A= 0.02.
-1.5
-1
-0.5
0 X-axis
0.5
1
1.5
2
2.5
Figure II. Tr~jectorie in XY axes, for m = 0.1, Go = 2000, Y = 3 dB , A=O.02.
2.5
Figure 13. Trajectorie in XY axes, for M = 0.1, Go =3000, y = 3 dB , A = 0.02.
5. Conclusions In this paper, chaos in a fractional-order jerk model using the hyperbolic tangent nonlinearity has been presented. A fractional integrator in the model has been approximated by linear transfer function approximation in the frequency domain. Resulting chaotic attractors have been observed with the system orders as low as 2.1.
Acknowledgements This work is supported by Telecommunications Research and Industrial Development Institute (TRJDI), National Telecommunications Conunission (NTC), Thailand.
References [1] G. Contopoulos, Order and. Chaos in Dynamical Astronomy, New York: Springer, 2004. [2] G. Contopoulos, and N. Voglis, Eds_ Galaxies and Chaos, Lecture Notes in Physics, Germany: Springer, Germany, 2003. l3] H, -J. Stockmann, Quantum Chaos : an introduction, UK: Cambridge University Press, 2000.
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[4] G. Chen and T. Ueta, Chaos in Circuits and Systems, Singapore: World Scientific, 2002. [5] F. C. M. Lau, and C. K. Tse, Chaos-Based Digital Communication Systems, Germany: Springer, 2003 . [6] S. MandaI, and S. Banerjee, Analysis and CMOS Implementation of a ChaosBased Communication System, IEEE Trans. Circuits and Systems,S 1(9): 17081722,2004. [7] G. Kolumban, M. P. Kennedy and L. O. Chua, The Role of Synchronization in Digital Communications Using Chaos - Part I: Fundamentals of Digital Communications, IEEE Trans. Circuits and Systems, 44(10):927-936, 1997. [8] G. Kolumban, M. P. Kennedy and L. O. Chua, The Role of Synchronization in Digital Communications Using Chaos - Part II: Chaotic Modulation and Chaotic Synchronization, IEEE Trans. Circuits and Systems, 45(11): 1129-1140, 1998. [9] P. Stavroulakis, Chaos Applications in Telecommunications, New York: Taylor & Francis, 2006. [10] H. Adeli, S. Ghosh-Dastidar, N. Dadmehr, A Wavelet-Chaos Methodology for Analysis of EEGs and EEG Subbands to Detect Seizure and Epilepsy, IEEE Trans Biomedical Engineering, 54(2):205-21 I, 2007. [11] S. Ghosh-Dastidar, H. Adeli, N. Dadmehr, Mixed-Band Wavelet-Chaos-Neural Network Methodology for Epilepsy and Epileptic Seizure Detection, IEEE Trans. Biomedical Engineering, 54(9):1545-1551 , 2007. [12] A. Dalgleish, The relevance of non-linear mathematics (chaos theory) to the treatment of cancer, the role of the immune response and the potential for vaccines, Q. 1. Med., 92:347-359,1999. [13] M. P. Kennedy, Robust Op Amp Realization of Chua's Circuit, Frequenz, 46(34):66-80, 1992. [14] J. C. Sprott, A New Class of Chaotic Circuit, Physics Letters A, 266:19-23, 2000. [IS] B. Srisuchinwong, and C. -H. Liou, High-Frequency Implementation of Sprott' s Chaotic Oscillators Using Current-Feedback Op Amps, Proceedings of International Symposium on Signals Circuits and Systems - ISSCS 2007, 1:9799,2007. [16] B. Srisuchinwong, C. -H. Liou, and T. Klongkumnuankan , Prediction of Dominant Frequencies of CFOA-Based Sprott' s Sinusoidal and Chaotic Oscillators, Proceedings of CHAOS 2008, Chaotic Modeling and Simulation International Conference, 3-6 June, Crete, Greece, 1-8, 2008. [17] B. Srisuchinwong, C. -H. Liou, and T. Klongkumnuankan, Prediction of Dominant Frequencies of CFOA-Based Sprott's Sinusoidal and Chaotic Oscillators, in Topics on Chaotic Systems: Selected Papers from CHAOS 2008 International Conference, c. H. Skiadas, Ed. Greece: World Scientific, (in press). [18] B. Srisuchinwong, and W. San-Urn, A Chua's Chaotic Oscillator Based on a Coarsely Cubic-Like CMOS Resistor, Proceedings of Asia-Pacific Conference on Communications -APCC 2007,2007,47-49,2007. [19] M. W. Hirsch, and S. Smale, Differential Equations: Dynamical Systems and Linear Algebra, New York: Academic Press, 1974, Chapter 1 I , 238-54. [20] J. C. Sprott, Simple chaotic systems and circuits, Am. 1. Phys. 68(8):758-763, 2000. [21] K. B. Oldham and J. Spanier, The Fractional Calculus, New York: Dover Publications, 1974.
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[22] A. Charef, H. H. Sun, Y. Y. Tsao, and B. Onaral, Fractal System as Represented by Singularity Function, IEEE Trans. Automatic Control, 37(9): 1465-1470, 1992. [23] T. T. Hartley, C. F. Lorenzo, and H. K. Qammer, Chaos in a Fractional Order Chua's System IEEE Trans. Circuits and Systems, 42(8):485-490,1995. [24] W. M. Ahmad, and 1. C. Sprott, Chaos in Fractional-Order Autonomous Nonlinear Systems, Chaos, Solitons & Fractals, 16:339-351 , 2003.
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Nonlinear Systems in Brunovsky Canonical Form: A Novel Neuro-Fuzzy Algorithm for Direct Adaptive Regulation with Robustness Analysis Dimitris C. Theodoridisl, Yiannis Boutalis 1 ,2, and Manolis A. Christodoulou3 1
2
3
Department of Electrical and Computer Engineering Democritus University of Thrace 67100 Xanthi, Greece (e-mail:
[email protected] ) Department of Electrical, Electronic and Communication Engineering Chair of Automatic Control University of Erlangen-Nuremberg 91058 Erlangen, Germany (e-mail:
[email protected]) Department of Electronic and Computer Engineering Technical University of Crete 73100 Chania, Crete, Greece (e-mail:
[email protected] )
Abstract: The direct adaptive regulation of unknown nonlinear dynamical systems in Brunovsky form with modeling error effects, is considered in this chapter. The method is based on a new Neuro-Fuzzy Dynamical System definition, which uses the concept of Fuzzy Adaptive Systems (FAS) operating in conjunction with High Order Neural Network Functions (HONNF's). Since the plant is considered unknown, we propose its approximation by a special form of a Brunovsky type fuzzy dynamical system (FDS) assuming also the existence of disturbance expressed as modeling error terms depending on both input and system states. The development is combined with a sensitivity analysis of the closed loop in the presence of modeling imperfections and provides a comprehensive and rigorous analysis of the stability properties of the closed loop system. Simulations illustrate the potency of the method and its applicability is tested on well known benchmarks, where it is shown that our approach is superior to the case of simple Recurrent High Order Neural Networks (RHONN's). Keywords: direct control, Neuro-Fuzzy Algorithm, Brunovsky form, robustness analysis.
1
Introduction
Adaptive control theory has been an active area of research over the past years [2,5-7,9,11,17,18,20,25,27]. The identification procedure is an essential part in any control procedure. In the Neuro or N euro - Fuzzy adaptive
Nonlinear Systems in Brunovsky Canonical Form: A Neuro-Fuzzy Algorithm
347
control two main approaches are followed. In the indirect adaptive control schemes, first the dynamics of the system are identified and then a control input is generated according to the certainty equivalence principle. In the direct adaptive control schemes [5,9,18] the controller is directly estimated and the control input is generated to guarantee stability without knowledge of the system dynamics. Also, many researchers focus on robust adaptive control that guarantees signal boundness in the presence of modeling errors and bounded disturbances [2,5-7,9,11,18,25,27]. It has been established that neural networks and fuzzy inference systems are universal approximators [8,19,26]'i.e., they can approximate any nonlinear function to any prescribed accuracy provided that sufficient hidden neurons and training data or fuzzy rules are available. Recently, the combination of these two different technologies has given rise to fuzzy neural or NeuroFuzzy approaches, that are intended to capture the advantages of both fuzzy logic and neural networks. Numerous works have shown the viability of this approach for system modeling [3,10,13,14,16].
The neural and fuzzy approaches are most of the time equivalent, differing between each other mainly in the structure of the approximator chosen. Indeed, in order to bridge the gap between the neural and fuzzy approaches several researchers introduce adaptive schemes using a class of parameterized functions that include both neural networks and fuzzy systems [3,10,14,16]. Recently [4,12], HONNF's have been proposed for the identification of nonlinear dynamical systems of the form i; = f(x, u), approximated by a Fuzzy Dynamical System. This approximation depends on the fact that fuzzy rules could be identified with the help of HONNF's. The same rationale has been employed in [1,23,24]' where a Neuro-Fuzzy approach for the indirect and direct control of square unknown systems has been introduced assuming only parameter uncertainty. In this chapter HONNF's are also used for the Neuro-Fuzzy direct control of nonlinear dynamical systems in Brunovsky canonical form with modeling errors. In the proposed approach the underlying fuzzy model is of Mamdani type [15]. The structure identification of the underlying fuzzy system is made off-line based either on human expertise or on gathered data. However, the required a-priori information obtained by linguistic information or data is very limited. The only required information is an estimate of the centers of the output fuzzy membership functions. Information on the input variable membership functions and on the underlying fuzzy rules is not necessary because this is automatically estimated by the HONNF's. This way the proposed method is less vulnerable to initial design assumptions. The parameter identification is then easily addressed by HONNF's, based on the linguistic information regarding the structural identification of the output part and from the numerical data obtained from the actual system to be modeled. We consider that the nonlinear system can be expressed in Brunovsky canonical form. We also consider that its unknown nonlinearities could be
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D. C. Theodoridis, Y. Boutalis, and M. A. Christodoulou
approximated with the help of two independent fuzzy subsystems. We also assume the existence of disturbance expressed as modeling error terms depending on both input and system states. Every fuzzy subsystem is approximated from a family of HONNF's, each one being related with a group of fuzzy rules. Weight updating laws are given and we prove that when the structural identification is appropriate and the modeling error terms are within a certain region depending on the input and state values, then the error reaches zero very fast. Also, an appropriate state feedback is constructed to achieve asymptotic regulation of the output, while keeping bounded all signals in the closed loop. The chapter is organized as follows. Section 2 presents the Fuzzy systems description using indicator functions and HONNF's. The direct adaptive Neuro-Fuzzy regulation of systems being in Brunovsky canonical form is given in section 3, where the problem formulation and the weight updating laws are derived. Section 4 presents simulation results and comparisons while section 5 concludes the work.
2
FUzzy system description using rule firing indicator functions and HONNF
Consider a nonlinear function f(x) E R, x E X c R n approximately described by a Mamdani-type Fuzzy System (FS). Let .fl;"h, ... ,j" be defined as the subset of x E X belonging to the (jl,j2, ... ,jn)th input fuzzy patch and pointing - through the function fe) - to the subset which belong to the [th output fuzzy patch. In other words, .fl;,,12, ... ,j" contains input values x that are associated through a fuzzy rule with output values f(x). Furthermore, the FS receiving as input the n-tuple of x = (Xl, X2, ... ,xn ) gives as output an approximate of f (x) using fuzzy rules and well known fuzzy inference procedure. According to the above notation the Rule Firing Indicator Function(RFIF) l . or simply Indicator Function (IF) connected to.flJl . is defined as fol,J2,··· ,In lows: l . I . [.. . ( x ()) t = { a(x(t)) if (x(t)) E.fl)1,)2,···,)" (1) )1,)2,.·.,)n 0 otherwise where a(x(t)) denotes the firing strength of the rule. According to standard fuzzy system description, this strength depends on the membership value of each Xi in the corresponding input membership functions /-LFji and more specifically [26], a(x(t)) = min[/-LFj1 (Xl (k)), ... /-LFj" (xn(k))]. Then, assuming a standard defuzzification procedure (e.g. weighted average), the functional representation of the fuzzy system can be written as
f(x(t)) =
L !J" ...
,jn X
(I1)~" ... ,jn (x(t))
(2)
Nonlinear Systems in Brunovsky Canonical Form: A Neuro-Fuzzy Algorithm
349
where the sumation is carried over all the available fuzzy rules. l)" ... ,j" is any constant vector consisting of the centers of fuzzy partitions of f determined by l and (I1)~" ... ,j" (x(t)) is the IF defined in (1) divided by the sum of all IF participating in the summation of 2. Based on the fact that functions of high order neurons are capable of approximating discontinuous functions [4] and [12] use high order neural network functions HONNF's in order to approximate an IF. A HONNF is defined as: L
N(x(t); w, L)
L
=
II
Whot
hot=l
p~j(hot)
(3)
jEhot
where hot = {h, h , ... , h} is a collection of L not-ordered subsets of {1,2, .. .,n}, dj(hot) are non-negative integers. Pj are the elements of the following vector,
(4) where
S
denotes the sigmoid function defined as: 1
s(x) -- a 1 + e-{3x -
(5)
rv I
with a, /3, 'I being positive real numbers and W := [Wl" weights. Eq. (3) can also be written as
'WL]T
are the HONNF
L
N(x(t); w, L)
=
L
WhotShot(X(t))
(6)
hot=l where Shot(X(t)) are high order terms of sigmoid functions of x. Following the above notation (II)~, ,.. ,j" in (2) can be approximated by
1 . (x) = N(x(t);wJJ ,···,jn;I,LJI,···,jn;l ) N J1,···,]n So, Eq. (2) can be rewritten as ~-l
f(x(t)) = L
h ,... ,jn
I
x N j"
.. ,jJx(t))
(7)
From the above definitions and Eq. (7) it is obvious that the accuracy of the approximation of f(x) depends on the approximation abilities of HONNFs and on an initial estimate of the centers of the output membership functions. These centers can be obtained by experts or by off-line techniques based on gathered data. Any other information related to the input membership functions is not necessary because it is replaced by the HONNFs.
3 3.1
Direct adaptive neuro-fuzzy regulation Problem formulation and neuro-fuzzy representation
Problem formulation We consider nonlinear dynamical systems of the Brunovski canonical form
350
D. C. Theodoridis, Y. Boutalis, and M. A. Christodoulou ::i;
= Aex + bc[j(x) + g(x) . u]
(8)
f
where the state x E Rn is assumed to be completely measured, the control input u E R, j and 9 are scalar nonlinear
functionS[O~~f..the tate
volved in the dynamic equation of x n . Also , Ae =
o o
ein:
onl~oo~'lll]-
0
0 0
and be = [0 The state regulation problem is known as our attempt to force the state to zero from an arbitrary initial value by applying appropriate feedback control to the plant input. However, the problem as it is stated above for the system (8), is very difficult or even impossible to be solved since the j, 9 are assumed to be completely unknown. To overcome this problem we assume that the unknown plant can be described by the following model arriving from a neurofuzzy representation described below, plus a modeling error term w(x, u)
(9) where x j , Xg are vectors containing the centers of the partitions of every fuzzy output variable of j(x) and g(x) respectively, determined by the designer and Wj, W; are unknown weight matrices. 8 j, 8 g are sigmoidal vectors containing high order sigmoidal terms terms with parameters which also determined by the designer. Therefore, the state regulation problem is analyzed for the system (9) instead of (8). Since, Wj and W; are unknown, our solution consists of designing a control law u(Wj, W g , x) and appropriate update laws for Wj and Wg to guarantee convergence of the state to zero and in some cases, which will be analyzed in the following sections, boundedness of x and of all signals in the closed loop. The following mild assumptions are also imposed on (8), to guarantee the existence and uniqueness of solution for any finite initial condition and u E Uc . Assumption 1. Given a class Ue C R of admissible inputs, then for any
u E Ue and any finite initial condition, the state trajectories are uniformly bounded for any finite T > O. Meaning that we do not allow systems processing trajectories which escape at infinite, in finite time T, T being arbitrarily small. Hence, Ix(T) I < 00. Assumption 2. The j, 9 are continuous with respect to their arguments and satisfy a local Lipschitz condition so that the solution x( t) of (8) is unique for any finite initial condition and u E Uc .
Nonlinear Systems ill Brullovsky Canonical Form: A Neuro-Fuzzy Algorithm
351
Neuro-fuzzy representation We are using a fuzzy approximation of the system in (8), which uses two fuzzy subsystem blocks for the description of f( x) and g( x) as follows
f(x) = ~ Xfl. ~
. xNfl.
}1,}2"",)n
. (x)
J1,)2,· ·,)n
(10)
(11) where the summation is carried out over the number of all available fuzzy l . . rules, N f , N g are appropriate HONNF's and the meaning of indices. )1 ,J2,··· ,)n has already been described in Section 2. In order to simplify the model structure, since some rules result to the same output partition, we could replace the NNs associated to the rules having the same output with one NN and therefore the summations in (10),(11) are carried out over the number of the corresponding output partitions. Therefore, the system of (8) is replaced by the following equivalent Brunovsky form Fuzzy - High Order Neural Network (F-HONN), which depends on the centers of the fuzzy output partitions xfi and xgi
(12) where Npf and Npg are the number of fuzzy partitions of f and g respectively. Or in a more compact form
(13) where, similar to (9), xf, Xg are vectors containing the centers of the partitions of every fuzzy output variable of f (x) and g( x) respectively, sf (x), Sg (x) are vectors containing high order combinations of sigmoid functions of the state x and Wf, Wg are matrices containing respective neural weights according to (6) and (12). The dimensions and the contents of all the above matrices are chosen so that both xfWfsf(x) and XgWgSg(x) are scalar. For notational simplicity we also assume that all output fuzzy variables are partitioned to the same number, m, of partitions. Under these specifications x f is a 1 x m row vector of the form
where xfp denotes the center or the fuzzy p-th partition of f. These centers can be determined manually or automatically with the help of a fuzzy c-means clustering algorithm as a part of the off-line fuzzy system structural identification procedure mentioned in the introduction. Also, sf(x) = [sj,(x) s!k(x)f, where each Sfi(X) with i = {1,2, ... ,k}, is a high
352
D. C. Theodoridis, Y. Boutalis, and M. A. Christodoulou
order combination of sigmoid functions of the state variables and Wf is a m x k matrix with neural weights. Wf can be also written as a collection of column vectors Wi, that is Wf = [W} Wi Wi]' where l = 1,2, ... , k. Similarly, Xg is a 1 x m row vector of the form
where xgk denotes the center or the k-th partition of g. Wg, Sg(x) have the same dimensions as Wf, sf (x) respectively. 3.2
Adaptive regulation with modeling error effects
In this subsection we present a solution to the adaptive regulation problem and investigate the modeling error effects when the dynamical equations have the Brunovsky canonical form. Assuming the presence of modeling error the unknown system can be written as (9). The regulation of the system can be achieved by selecting the control input to be (14) with
v = kx
(15)
where k is a vector of the form k = [k n ... k2 klJ E R n be such that all roots ofthe polynomial h(s) = sn + k1s n - 1 + ... + k n are in the open left half-plane. Define now, the regulation error as ~ =-x
(16)
After substituting Eq. (14) to the n - th state equation of Eq. straightforward manipulations we have that
(9) and
where ilc = Ac - bck is a matrix with its eigenvalues on the left half plane. Wf = Wf - Wj and Wg = Wg - W;. Wf and Wg are estimates of Wj and respectively and are obtained by update laws which are to be designed in the sequel. After substituting Eq. (16), (17) becomes
W;
~ = ilc~ + bc[x f Wfs f(x)
+ Xg WgSg(x)u -
w(x, u)J
(18)
v = ~e p~ + 2~1 tr { Wf T Wf } + 2~2 tr { WJWg}
(19)
To continue, consider the Lyapunov candidate function
Where P > 0 is chosen to satisfy the Lyapunov equation
Pile
+ il~ P = - I
Nonlinear Systems in Brunovsky Canonical Form: A Neuro-Fuzzy Algorithm
353
If we take the derivative of Eq. (19) with respect to time we obtain
. 1 2 V = -211~11
+~
T
-
PbcXfWfSf(X) +
T T ~ PbcXgWgSg(X)u-~ Pbcw(x,u)+
~1 tr { WfTWf } + ~2 tr { Wg TWg } Hence, if we choose (20) (21 )
V becomes
v ~ -~ 11~112 + 11~llllPllllw(x, u)11
(22)
It can be easily verified that Eqs. (20) and (21) after making the appropriate operations, result in the following weight updating laws (23)
(24) where ~ is the vector defined in (16), u is a scalar and 11, 12 are positive constants expressing the learning rates. The above equations can be also element wise written as: a) for the elements of Wf
(25) or equivalently wj = -11~nPn (xf)T sf, (x) for alll = I, 2, ... , k, j = 1, 2, ... , m, is the n - th element of the vector ~ and Pn is the last n - th row of P. b) for the elements of Wg
~n
(26)
WJ
or equivalently = -12~nPn (Xg)T Sg, (x)u, for all l = 1,2, ... , k and j = 1,2, ... ,m. Furthermore, we can make the following Assumption. Assumption 3. The modeling error term satisfies
I w(x, u)
I~ e~
Ixi + e~ lui
where e~ and e~ are known positive constants. Also, we can find an a priori known constant
eu
> 0, such that
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D. C. Theodoridis, Y. Boutalis, and M. A. Christodoulou
and Assumption 3 becomes equivalent to (27) where
£1 = £~
+ £~£u
(28)
is a positive constant. Employing Assumption 3, Eq. (22) becomes
v : :; -~ 11~112 + £1 11~llllPllllxll since
Ilxll
= II~II
(29)
then Eq. (29) becomes
v : :; -~ 11~112 + £1 IIPIIII~112 :::; -
(~ -
£1
IIPII)
11~112
(30)
Hence, if we chose IIPII < 2t then the Lapyunov candidate function becomes negative. We are now ready to prove the following theorem Theorem 1. The control law (14) and (15) together with the update laws (23) and (24) guarantee the following properties
• limt-+oo ~(t)
provided that
= 0,
limt-+oo x(t)
IIPII < 2}!
=0
and Assumption 3 is satisfied.
Proof. From Eq. (30) we have that V E Loo which implies ~,Wj, Wg E Loo. Furthermore Wj = Wj + Wi E Loo and Wg = Wg + Wg * E Loo. Since ~ E Loo this also implies x E Loo. Moreover, since V is a monotone decreasing function of time and bounded from below, the limt-+ oo V(t) = Voo exists so by integrating V from 0 to 00 we have
which implies that
~
=
I~I E
L 2 . We also have that
Ac~ + bc[xj WjSj(x)
+ Xg WgSg(x)u -
w(x, u)]
Nonlinear Systems in Brunovsky Canonical Form: A Neuro-Fuzzy Algorithm
355
Hence and since U E L oo , the sigmoidals and center matrices are bounded by definition, Wf , Wg E Loo and Assumption 3 hold, so since ~ E L2 n Loo and ~ E L oo , applying Barbalat's Lemma we conclude that limt->oo ~(t) = O. Now, using the bounded ness of u, sf(x), Sg(x), x and the convergence of ~(t) to zero, we have that Wf, Wg also converge to zero. Hence we have that limt---+oo x(t)
= -limt->oo ~(t) = 0
Thus, limt---+oo x(t) = O.
Remark 1. We can't conclude anything about the convergence of the synaptic weights Wf and Wg to their optimum values Wi and W; respectively from the above analysis. The existence of signal u is associated with conditions which guarantee that XgWgSg(x) # O. It can be shown that using appropriate projection method [9], the weight updating laws can be modified so that the existence of the control signal can be assured. However, this development is not presented in this chapter. The relevant results along with the ones given here are to be presented shortly in a forthcoming work.
4
Simulation results
To demonstrate the potency of the proposed scheme we present two simulation results. One of them is the well known benchmark "Inverted Pendulum" and the other one is the "Van der pol" oscillator. Both of them present comparisons of the proposed method with the well established approach on the use of RHONN's [21]. The comparison shows off the regulation superiority of our method under the presence of modeling errors. 4.1
Inverted pendulum
Let the well known problem of the control of an inverted pendulum. Its dynamical equations can assume the following Brunovsky canonical form [22]
rnlx~ cos Xl sin Xl
.
X2 =
'Tnc+ rn
gSlnxl
1(1- =C08 3
with
e
-
2
Xl)
rHC+rn
d = 5X2
~
+
=C+ m
I
U
+d
=c082 " " ) 4 ( '3rrtC+ m
+ 10sin(lOx2) + sin(2u)
(31)
where Xl = and X2 = iJ are the angle from the vertical position and the angular velocity respectively. Also, 9 = 9.8 m/ S2 is the acceleration due to
356
D. C. Theodoridis, Y. Boutalis, and M. A. Christodoulou
gravity, m e is the mass of the cart, m is the mass of the pole, and I is the half-length of the pole. We choose me = 1 kg, m = 0.1 kg, and I = 0.5 min the following simulation. It is our intention to compare the direct control abilities of the proposed Neuro-Fuzzy approach with RHONN's [21] (page 62-71). We also, make the appropriate changes to RHONN's in order to be equivalent with F-HONNF's (brunovsky form) for comparison purposes. For the proposed F-HONNF approach we use the adaptive laws which are described by Eqs. (23) and (24) and the control law described by Eqs. (14) and (15). Numerical training data were obtained by using Eqs. (31) with initial conditions [Xl (0) X2(0)] = [ - ;2 0] and sampling time 10- 3 sec. We are using the proposed approach with Eq. (13) to approximate Inverted Pendulum dynamics and send data to sigmoidal terms. Our NeuroFuzzy model was chosen to use 5 output partitions of f and 5 output partitions of g. The number of high order sigmoidal terms (HOST) used in HONNF's (for F-HONNF and RHONN) were chosen to be 5: (s(xd, S(X2), s(xd .
S(X2)' s2(xI), S2(X2)). In order the proposed model to be equivalent with RHONN's regarding to other parameters except the initial weights (Wf(O) = 0 and Wg(O) = 1 for F-HONNF and Wf(O) = 0 and Wg(O) = 0.016 for RHONN) we have chosen the updating learning rates 1'1 = 5 and 1'2 = 1 and the parameters of the sigmoidal terms such as al = 0.1 , a2 = 3, bl = b2 = 1 and Cl = C2 = O. Also, after careful selection, kl = 2 and k2 = 80 . Fig. (1) shows the regulation of states Xl and X2 (with blue line for F-HONNF and red line for RHONN's), while fig. (2) gives the evolution of control input u, the errors el, e2 and the modeling error d(X2' u) respectively. The mean squared error (MSE) for RHONN and F-HONNF approaches were measured and are shown in Table 1, demonstrating a significant (order of magnitude) increase in the regulation performance of F-HONNF against RHONN's. Example RHONN F-HONNF Inverted Pendulum MSExj 0.0010 6.0794 . 10 - 4 MSEx2 0.9252 0.3368
Table 1. Comparison of RHONN's and F-HONNF approaches for the Inverted Pendulum.
4.2
Van der pol
Van der Pol oscillator is usually used as a simple benchmark problem for testing control schemes. It's dynamical equations are given by
Nonlinear Systems in Brunovsky Canonical Form: A Neuro-Fuzzy Algorithm
357
0.2 ,\
* x
II
jll\A
0.1
~~~~~~~~-----------1
0
II
(J5
~
-0.1 -0.2
o
2
3
4
5
1\ . il
4 1
*
';i
2 0
II,.
mly:0,\P\iXX'7VV'''XI''NV~v'''A-~~-
,Ii V
(J5 -2
-4
\/ V
_6L---__L -_ _ _ _ _ _
o
~
_ _ _ _ _ _ ~_ _ _ _ _ _~ _ _ _ _ _ _ ~
2
3
4
5
Time (sec)
Fig. 1. Evolution of angle state variable Xl and angular velocity for RHONN's (red line) and F-HONNF approach (blue line)
X2 = X2 . with
d
(a - xi) . b -
Xl
X2
respectively,
+ U+ d
= 2X2 + 4sin(5x2) + 5sin(10u)
(32)
The procedure of the approximation was the same as that of Inverted Pendulum. The proposed Neuro-Fuzzy model was chosen to have initial conditions, [Xl (0) X2(0)] = [0.4 0.5] and the number of high order sigmoidal terms (HOST) used in HONNF's (for F-HONNF and RHONN) were chosen to be 2 (s(xd, S(X2)' In order our model to be equivalent with RHONN's regarding to other parameters except the initial weights (Wf(O) = 0 and Wg(O) = 0.1 for FHONNF and Wf(O) = 0 and Wg(O) = 0.05 for RHONN) we have chosen the updating learning rates {I = 0.1 and {2 = 10 and the parameters of the sigmoidal terms such as al = 0.1, a2 = 4, bI = b2 = 1 and CI = C2 = O. Also, kI = 2 and k2 = 80 after careful selection. Fig. (3) shows the regulation of states Xl and X2, while fig. (4) presents the evolution of control input u, the errors eI, e2 and the modeling error d. The mean squared error (MSE) for RHONN and F-HONNF approaches were measured and are shown in Table 2 demonstrating as before (Inverted
358
D. C. Theodoridis. Y. BOlltalis, and M. A. Christodoulou
5
~ _: Hhilu-v II
~
j'-\D_'
30
60
P
90
200
400
1: 600
Fig. 5. (a) Momentum probability distributions vV at the fixed time moment in the nonadiabatic (bold curve, Li. = 0.2), adiabatic (dashed curve. Li. = 1), and resonant (dotted curve, Li. = 0) c&ses. (b) Mean atomic momentum VB time in the nonadiabatie (bold lower curve, Li. = 0.2), adiabatic (bold upper curve, Li. = 1), and resonant (dotted curve, Li. = 0) eases.
At Ll = 0.2, the atomic ground state is the following superposition of the dressed states: 11) c:: 0.741 +).c, + 0.661- ).c,. The part of the initial wave packet, starting from the top of the potential E~+) , overcomes the barriers of that potential and moves in the positive direction of the axis :r proliferating at the nodes. The larger part of the initial packet, starting from the bottom of the potential E~-), will be trapped in the first well of that potential. Only its small part is capable to overcome the barrier. To illustrate that we show in Fig. 4 evolution of the initial Gaussian wave packet. As compared to Fig. 3a, we see an interference-like picture in the InomentulYl space and dear evidence of the Ilonadiabatic transitions between the nonresonant poten[,ials
Atomic de Broglie-Wave Chaos
375
in the position space. This results in a proliferation of the components of the atomic wave packet at the nodes of the standing wave. Figure 5a shows the momentum probability distributions W at the fixed time moment T = 30 in the nonadiabatic, adiabatic, and resonant cases. The distribution is narrower in the adiabatic case than in the other ones. The peaks of Ware more pronounced in the nonadiabatic case than in the resonant one. Figure 5b shows the dependence of the mean atomic momentum < P > on time for the three cases. This quantity oscillates in the adiabatic case in much nore narrow range than in the other cases where < p > is a wildly oscillating function.
3
Quantum-classical correspondence
In the semiclassical approximation, one treats the electronic degree of freedom quantum mechanically and the translational motion classically. The problem of motion of a point-like atom in a standing-wave laser field has been studied in detail in Refs. [6,1]. The five-dimensional Hamilton-Schrodinger equations of motion have been deduced. Along with two integrals of motion, they constitute a Hamiltonian autonomous system with two degrees of freedom moving in the phase space on a three-dimensional hypersurface with a given energy value. A number of nonlinear dynamical effects have been found in Refs. [6,1]: chaotic Rabi oscillations, Hamiltonian chaotic atomic transport and dynamical fractals, Levy flights and anomalous diffusion. One of the impressive effects is a random-like motion of an atom in a rigid optical lattice when the atom can fly through the optical potential in one direction, be trapped in its wells for a while and then continue to fly over its hills in the other direction [1]. This and the other effects are caused by local instability of the center-of-mass motion in a laser field. A set of atomic trajectories under certain conditions becomes exponentially sensitive to small variations in initial quantum internal and classical external states or/and in the control parameters. A detailed theory of Hamiltonian chaotic transport of atoms in a laser standing wave has been developed in [1]. The maximal Lyapunov exponent ,.\ characterizes the mean rate of the exponential divergence of initially close trajectories and serves as a quantitative measure of dynamical chaos. In Fig. 6a we show the dependence of ,.\ on the detuning Ll for an initially deexcited atom with the fixed value of the recoil frequency Wr = 10- 3 and initial conditions: Xo = 0 and Po = 55. As a measure of complexity of the wave packets, we take a number of local extrema N in the distribution function W(p) on a large time interval. Figure 6b plots the dependence of this quantity on the value of the detuning Ll. Comparing this quantum result with the semiclassical one in Fig. 6a, we see rather good correlation. It follows from the semiclassical simulation that the maximal Lyapunov exponent is zero nearby the resonance and when the detuning is large enough providing a regular point-like atomic motion.
376
V. O. Vitkovsky, L. E. Konkov, and S. V. Prants 0.04
A
,
"")\'
'(
O.Q3,
0.02
(
\
a) .~
\
""L", "' ~~"J
2---:O:-:.4---:0:-:.6---:0'"'.S-L\---'1 "O':----C0:-::.
Fig. 6. (a) Maximal Lyapunov exponent A with the classical atomic motion and (b) number of local extrema in the momentum distribution with the quantized atomic motion vs the detuning .:1.
Positive values of A mean chaotic atomic motion. Quantum evolution is rather regular in the same ranges of the detuning and it is complicated in that range of the detuning where the corresponding classical atomic motion is chaotic. We have shown that, varying the Rabi frequency or the detuning, it is possible to some extent to control propagation of de Broglie waves creating regular and chaotic wave packets. This work was supported by the RFBR (project no. 09-02-00358) and by the Program "Fundamental Problems of Nonlinear Dynamics" .
References l.V.Yu. Argonov and S.V. Prants. Theory of chaotic atomic transport in an optical lattice. Physical Review A, 75:art.no. 063428 , 2007. 2.R. Graham, M. Schlautmann, and P. Zoller. Dynamical localization of atomicbeam deflection by a modulated standing light wave. Physical Review A, 45:R19, 1992. 3.W.K. Hensinger, N.R. Heckenberg, G.J. Milburn, and H. Rubinsztein-Dunlop. Experimental tests of quantum nonlinear dynamics in atom optics. Journal of Optics B, 5:R83, 2003. 4.A. P. Kazantsev, G. 1. Surdutovich, and V. P. Yakovlev. World Scientific, Singapore, 1990. 5.D. Makarov, S. Prants, A. Virovlyansky, and G. Zaslavsky. World Scientific, Singapore, 2009 . 6.S. V. Prants. Chaos, fractals and flights of atoms in cavities. JETP Letters, 75:651, 2002.
377
Chaotic Behavior of Plasma Surface Interaction. A Table of Plasma Treatment Parameters Useful to the Restoration of Metallic Archaeological Objects C. L. Xaplanteris,·2 and E. Filippaki2 , Hellenic Military Academy, Vari Attika, Greece 2 Plasma Physics Lab, IMS, NCSR "Demokritos", Athens, Greece (e-mail:
[email protected]) Abstract: In Plasma Physics laboratory of N.C.S.R. "Demokritos" the plasma chemistry method has been used for the restoration and conservation of metallic archaeological objects during the last decades. The obtained experience had led us to conclude that plasma parameters and different status of treated objects are so specific, so as to become unique. In the present paper the theoretical and experimental results of our laboratory are summarized. A treatment table of plasma parameters is given, which claims to be useful for the conservators. It is obvious that this treatment table needs to be completed and extended, so that it meets the uniqueness of each artifact. A theoretical study and the treatment of a variety of iron objects are presented. Keywords: plasma sheath potential, plasma restoration, corrosion, external potential, plasma parameters.
1. Introduction Three decades have passed since Daniels and coworkers used for first time the mechanical and chemical action of the plasma [1] to reduce the tarnish of silver on daguerreotypes. During the 80's, Veprek and coworkers systematized the plasma cleaning and restoring method [2-4], by creating the homonymous plasma reactor, where a low-pressure hydrogen plasma acts upon the artifacts. These procedures were very soon adopted by the Plasma laboratory of NCSR "Demokritos" due to the many archaeological artifacts, which are found at the Hellenic area. The plasma treatment processes are usually preferred, for the following reasons: the plasma temperature is low and consequently, the destruction risk of artifacts is avoided; the chemical reactions produce steady compounds on the objects (e.g. magnetite for iron objects), which consist the restorating behavior of plasma; finally, the plasma reactor volume allows the simultaneously treatment of many objects. Many experiments and studies have been carried out [5, 6], at the two plasma reactors of "Demokritos" since the early 90's. Recently, the utilization of an external d.c. electric current on the treated objects have showed very good results [7, 8]. The whole previous meditation leads to the conclusion that plasma cleaning proceedings are chaotic due to multi-parametric behavior of plasma and the complicated corrosion of metallic objects. Thus, the factors which affect the cleaning and restoring can be listed as below.
378
C. L. Xapianteris and E. Fiiippaki
•
The plasma production way, the nature of plasma gas, accompanied with its parameters (plasma density, temperature, pressure ect). • The kind of metal, and its metallurgy. • The type of corossion and the corossion rate which are depended on the burial environment of the objects and on the time of their burial. The purpose of the present paper is to classify the previous results, to add new ones from the last experimental results and to draw up a table of treatment methods useful to the restorators. The paper is organized as following: the new experimental studies are presented in Sec.2; the precedent results, many of which were studied in our laboratory [5-8], are briefly described in Sec. 3 and in SecA treatment tables are given, the main results are summarised and concluded.
2. New experimental results Beyond the previous work which has been done during the last five years, the plasma treatment was extended by the enforcement of the d.c potential on restorated objects. In this way, a d.c current passes both through objects and plasma, and proves that the Langmuir probe theory which basically is valid, must be modificated. It is known that the rf plasma production method gives to the electrons much more acceleration than the ions because of the very small electron mass. Furthermore, the electron energy loss by the collisions with ions or neutrals, is negligible due to the small value of the ratio me /
1m; . In this
way, the electron
temperature increases considerably (10.000-100.000 K O), while the ion and neutral temperature remains at low levels (about 500 K Ofor ions and 300 K Ofor neutrals). Consequently, plasma is not in thermodynamic equilibrium, while the high electron temperature makes plasma more effective for the oxide reduction. When none external potential is imposed on the treated objects, the plasma sheath theory is valid as it was proved in an extented previous work [7,8]. However, if an external d.c. potential is enforced on the objects, as in our experiment, the Langmuir probe theory is valid with the presupposition that the following thoughts are taken into consideration. Firstly, when a voltage is applied on the treated object with reference to the earth, the simple probe with very large surface is formed . From plasma sheath theory [7, 8] is known that the ion current density to the object surface, is ji = no ·e.c s (1) with no the plasma number density and C s is the ion sound speed. It is possible to approach, 2_
c,
Te + T;_Te_
2
= - - - = - = Vo
mi
mi
where Vo is the ion velocity at the sheath edge.
Chaotic Behavior of Plasma Surface Interaction
Similarly,
the
electron
current
density
to
the
object
surface,
379 IS
e¢(x)
je =
-~4 . n e . e . ce'
Taking into consideration that
ne = no' e
T,
and
the electron current density is JlJn e
je = -
L
no . e . ( STe
4
)
Yz 2.
e¢(x)
e
T,
(2)
JlJn e
The current I to the object (probe) is I = (Ji + j e)' S where S is the projected surface area into the plasma. By substitution of ion and electron current density from (1) and (2), the current I which is biased to voltage V is (3)
Vf is the floating potential, which is equal to the external enforcemed potential,
so that the current I vanishes. Since the object (probe) dimensions are large compared to the ion and electron gyro-radii and the Debye length, the area S is determined as following: S
f is .
= ~ . B.
When the probe is biased negatively enough, all ellectrons are repulsed while what remains is the ion current. This current is not affected by the voltage and is known as ion saturation current. The ion current is saturated because the bias potential gathers all ions across the surface sheath and the ion flux is determined by the flux of ions which enter into the sheath at the ion sound velocity. When the probe potential becomes larger than the plasma potential, there is no sheath and the electrons strike on the object surface with the thermal distribution of velocities. In this case the electron current is saturated by value I es =
~ noece . S, with Ce the electron thermal velocity. 4
Figure 1 shows the typical probe characteristic (I versus V), which presents three regions: the ion current saturation, the exponential electron distribution and the electron current saturation.
380
C. L. Xaplanteris and E. Filippaki
sma
ion saturation io V
v
f
v p
p
e ponential region y-recorder
eo
electron saturation
Figure 1. The Langmuir probe and its characteristic.
If plasma is non-magnetized (as the plasma we use to treat corroded objects), it follows the above simple theory; the characteristic I - V draws the two saturation regions with the ratio of electron to ion saturation current is equal to ratio of electron thermal velocity to ion sound speed. However, in our experiment the probe's metallic surface into the plasma has been replaced with the treated objects, which have much more area than the probe's one. In spite of this big difference the Langmuir probe theory is valid in a very satisfactory way, and the measurements give currents proportional to the object area. Figure 2 shows the d.c. current, which passes through the objects and the plasma. As Figure 2 shows, a big ohmic resistance R is necessarely lined at the circuit to reduce the current I , which is measured on the value of some Amperes. The negatively charged object (cathode) attracts plasma ions, which strike on the surface with increased velocity; then, while the ions take free electrons from the cathode, reductive reactions take place as following:
3Fe203 +2H+ =? 2Fe304 +H 20
or
Fe203 +2H+ =? 2FeO+H 20
On the contrary, the positively charged object (anode) attracts the plasma electrons, which strike on the surface and reduce the oxides as the reactions show: 6Fe203 + 4e- =? 4Fe304 + 02
or
Fe203 + 2e- =? 2FeO +
h 02
Chaotic Behavior of Plasma SU/jace Interactioll
381
From the above chemical reactions it is obvious that the extemal d.c. potential makes the plasma much more active and reduce the object oxides to more stable compounds (magnetite) as the XRD examination can confirm.
plasma
Figure 2. The external dc potential circuit during plasma treatment.
3. Previous results Many inferences have been taked out from the early experimental work in our laboratory, as the plasma cleaning method is in operation during the last
two decades. An excavated nail fron the Holy Monastery of Simonopetra of Mount Athos before and after plasma treatment is shown in Photo 1.
Photo 1. An excavated nail before and after plasma treatment.
The factors and parameters, which affect the plasma cleaning, are too many and as a result, it is impossible to mentjon them without the risk of forgetting one. The present work has the ambition to start the drawing of some treatment tables in order to achieve the classification of the chaotic state. Firstly,
382
C. L. Xaplanteris and E. Filippaki
the kind of the material used e.g. iron, copper, silver e.t.c. are presented; secondly, the corrosion depth, which is separated in three sizes for practical reason: lightly corroded, moderately corroded and heavy corroded; the corrosion depth depends on the time and the environment where the object is being corroded. Another parameter, which strongly affects the cleaning, is the plasma gas; gases such as H 2' N 2' CH 4 as combinations of various gases were used, except for N 2 and CH 4 due to identification of CN- on the pyrex surface. Along with the plasma gas, the other plasma parameters, as plasma pressure, the plasma density, the rf absorbed power and the ion temperature, are listed. Consequently, the treatment time is a remarkable factor; practically, three period of treatment times were used, periods of lOhours, 1O-20hours and a period that exceed 20 hours of treatment.
4. Summation-elaboration tables The factors and parameters, which were mentioned in Sec.3 have been tabulated on cleaning Tables I, II and III.
Table I. Plasma treatment table for iron objects
Light corroded
Iron obJects Medium corroded
Heavy corroded
Gas Pressure (Torr) Power (kW) Time Temperature
H2 0.8 0.7 8h-IOh 200°C to 300°C
H2 0.9 1 to 1.2 lOh-20h 250°C to 350°C
H2 0.7 0.7 more than 20h 200°C
Gas Pressure (Torr) Power (kW) Time Temperature
H2, N2 0.4,0.2 0.8 8h-10h 200°C to 300°C
H2,N2 0.8,0.4 1.1 to 1.2 lOh-20h 250°C to 350°C
H2, N2 0.4,0.2 0.7 more than 20h 200°C
Gas Pressure (Torr) Power (kW) Time Temperature
N2 0.7 0.8 more than 20h 200°C to 300°C
N2 0.9 1.0 10h-20h 250°C to 350°C
N2 0.5 0.5 more than 20h 200T
Chaotic Behavior of Plasma Surface Interaction
383
Table II. Plasma treatment table for bronze objects
Gas Pressure (Torr) Power (kW) Time Temperature
Light corroded
Bronze objects Medium corroded
H2 0.8 0.65 4h-6h 170' C-190'C
H2 1.2 1.0 2h-4h 230'C-240'C
Heavy corroded H2
Table III. Plasma treatment table for silver objects
.___________________________________________________________________~!!~~!:_~~J~~_~~______________________________________ _ Gas Pressure (Torr) Power (kW) Time Temperature
Light corroded
Medium corroded
Heavy corroded
H2 0.8 0.7 2h-4h 21O'C
H2 0.8 0.7 6h-8h 21O'C
H2 0.8 0.7 lOh-12h 21O' C
It is obvious, that the above tables need to be expanded not only for other kinds
of material s but also by taking into account other parameters which were mentioned in section 1, so that a useful plasma treatment manual for restorators will be completed. From the table I, which concerns the treatment of the iron objects we can see that the state of the corrosion of the object (light to medium or heavy corroded objects) is critical for the duration and for the temperature of the plasma treatment. So, as far as the light corroded objects concerns, when treated either with hydrogen or a combination of hydrogen and nitrogen, the temperature varies between 200-300 degrees of C and the duration 8-lOhours, for the medium corroded objects the temperature is between 250-300 degrees of C and the duration varies between 1O-20h. For the heavy corroded objects the temperature does not exceed 200'C and the duration is more than 20h. The lower temperature is essential in order to preserve the integrity of the object as well as the metallographic characteristics of the metal itself. If the treatment gas is nitrogen then the treatment duration is much longer for all the three categories of the corrosion state. The second table concerns bronze objects treated with hydrogen plasma. The duration of treatment varies between 2-4 hours and the temperature between 170 to 240' C. As far as silver objects concerns, the temperature is always the same 21O' C, while the duration varies between 2-12 hours depending on the corrosion state of the object.
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C. L. Xaplanteris and E. Filippaki
Evaluating the above results we assume that 1. Gas, temperature and time are the key parameters of the plasma process. 2. The extend of the corrosion layer is essential on the plasma duration and the temperature of plasma treatment. 3. The plasma treatment at all conditions is successful in decreasing the density of the corrosion products and therefore facilitates subsequent mechanical cleaning. The gradual elimination of the chlorine containing corrosion products in favor of the formation of more stable species and sometimes even the complete reduction back to metal is proportional to the duration of the plasma treatment. It should be taken into account that in case of heavy corroded objects a longer treatment in lower temperature is preferable to a shorter treatment in higher temperature as it does not make the object too brittle.
References [I] Daniels V., Plasma Reduction of Silver Tarnish on Daguerreotypes, Stud. In Conserv, 26,1981.
[2] Veprek, S., Patscheider, J. and Elmer, J., Restoration and Conservation of ancient artifacts: A new area of application of plasma chemistry, Plasma Chern. Plasma process, Vol. 5(2), 1985. [3] Patscheider, J. and Veprek, S. Application of low pressure Hydrogen plasma to the conservation of ancient iron artifacts, Stud. In Consern., Vol. 31,1986. [4] Veprek, S., Eckmann, Ch. ,and Elmer, 1. Th., Recent progress in the restoration of archaeological metallic artifacts by means of low-pressure plasma treatment Plasma Chern. Plasma process. Vol. 8 (4), 1988. [5] Kotzamanidi I., Sarris Em., Vassiliou P.,Kollia c., Kanias G.D., Varoufakis G.l. Filippakis S.E., Corrosion behavior of oxidized steel treated in reducing plasma environment-Chloride ions removal, British Corrosion Journal, 34(4), 285-29 I, ( 1999). [6] Kotzamanidi, I. Vassiliou, P. Sarris, Em. Anastasiadis, A., Filippaki, E., Filippakis, S.: Anti-corrosion methods and materials, Vol 49, No 4, 2002. [7] Xaplanteris, c., Filippaki, E. Plasma -surfaces interaction and improvement of plasma conservation system by application of a d.c. electrical potential. Topics on Chaotic Systems, Selected Papers from CHAOS 2008 International Conference, pp. 406-415, 2009. [8] Xaplanteris, C., Filippaki, E. Mechanical and chemical results in the plasmasurface contact. An extended study of sheath parameters. (to be published).
385
Simulation of Geochemical Self-Organization: Acid Infiltration and Mineral Deposition in a Porous Ferruginous Limestone Rock Farah Zaknoun, Houssam EI-Rassy, Mazen AI-Ghoul, Samia AI-Joubeily, Tharwat Mokalled, and Rabih Sultan Department of Chemistry, American University of Beirut P.O. Box 11-0236, 11072020 Riad EI-Solh, Beirut, Lebanon (e-mail:
[email protected]) Abstract: The phenomenon of Liesegang banding finds its most striking natural similarity in the band scenery displayed in rock systems. Whereas theoretical modeling studies are extensive in the literature to simulate the reaction-transport dynamics of the medium, experimental simulations in-situ are very scarce. We carry out an empirical observation of the evolution of a deposition pattern embedded in a rock. A sulfuric acid solution is infiltrated into a porous ferruginous limestone rock by means of an infusion pump. This acidization causes the dissolution of the calcite (CaC0 3) mineral and deposition of calcium sulfate in the form of anhydrite (CaS04) and mainly gypsum (CaS04'2H20), behind the advancing dissolution front. We observe the growth of such a deposition pattern and investigate the extent of its resemblance with the well-known Liesegang patterns obtained in the laboratory. The composition of the seemingly banded zones is analyzed by powder X-ray diffraction, both qualitatively and quantitatively. The irregular shape of the band zone boundaries seems to be of fractal nature. Keywords: Liesegang, acidization, acid-infiltration, banding in rocks, ferruginous limestone, fractals.
1. Introduction In this paper, we shed light on the similarities between the band formation in rocks and the well-known Liesegang banding phenomenon [1,2]. Liesegang precipitation bands are obtained when co-precipitate ions interdiffuse in a gel medium due to a cycle of supersaturation, nucleation and depletion [3], coupled to the on-going diffusion process. Beautiful bands of precipitate appear as a result of this complex physico-chemical scenario. Representative specimens of colorful banded Liesegang patterns are displayed in Fig. 1, for a variety of precipitate systems. Liesegang systems share a great deal of similarity with natural systems displaying striped patterns and ripple morphology, notably in Biology and Geology. Biological rhythmic patterns are manifest in the stripes on tigers and zebras, belts of bacteria colonies [4,5], gallstones [6] and cysts. In Geology, agates and banded rocks present a typical example of rhythmic patterns [7-13], and are perhaps the closest widespread natural phenomenon to the Liesegang stratification, because they directly involve precipitation and
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mineral deposition. Pictorial scenes of periodic banding in real natural geological systems are displayed in Fig. 2.
Figure 1. Liesegmlg precipitation patterns grown in gel for a variety of precipitate systems.
Between the rapid growth of bacterial colonies and the tremendously slow formation of rock patterns, we realize that self-organization can take place on a huge range of time scales (minutes to millions of years).
a.
b.
c.
Figure 2. Patterns of bands in natural stone systems (geology). a. Liesegang stone, a highly silicated banded rhyolite; b. an agate of malachite, hydrous copper carbonate; c. donut agate.
In the present study, we investigate the phenomenon of band formation in rocks by simulating one possible scenario that would take place in a real, rock medium. Instead of focusing on the study of existing, "ready made" banded patterns in rock specimens, we force and steer the band deposition inside a rock bed, by simulating the physico-chemical processes which lead to such deposition, thus exactly mimicking the real natural scenario. This is done by infiltrating a ferruginous limestone rock with sulfuric acid, causing the dissolution of the bare rock calcite (CaC0 3) medium, followed by the deposition of anhydrite (CaS04) or gypsum (CaS04'2H20), behind the dissolution front: Cae0 3 (s) + 2W (aq)
~ Ca 2+ (aq) + CO 2 (g) + H20 (l)
(1)
Simulation of Geochemical Self-Organization
Ca 2+ (aq) +
sol- (aq)
~ CaS04 (s)
387
(2)
Thus, we carry out our experiments in-situ, spanning long times as required by the appearance of the desired patterns. To our knowledge, no investigator tackled this approach to geochemical self-organization before. The study would rather routinely be based on the analysis of existing rock specimens displaying parallel bands, or essentially, on theoretical modeling and numerical simulations. The mechanism involves diffusion and percolation of the infiltrating solution, coupled to chemical reactions driving dissolution, porosity changes and precipitation. The mineral deposition requires a collective scenario of nucleation, as well as competitive growth of both nuclei and larger particles of precipitate. A representative schematic picture of the dynamical events is illustrated (in reduced form) in the following scheme, and portrayed in Fig. 3:
A
~---PAO
space Figure 3. Moving dissolution front of mineral A, infiltrated by aqueous species X, and producing the banded deposition of mineral B.
A + 2X ~ Y Y +Z k2) B
(3) (4)
where A respresents a solid mineral (here calcite, CaC0 3 ), Y the dissolved salt material (Ca2+), from the reaction with X (acidic water, H+), Z the counter-ion from the infiltrating solution (here SOl-) which reacts with Y to form precipitate B (CaS04). Thus this scheme (also illustrated in Fig. 3) is a simplified form of chemical reactions 1 and 2. kJ and k2 are rate coefficients used in the numerical solution of the mathematical model adopted in Ref. 14.
2. Experimental Section 2.1. Acid-Infiltration Experiments A red brown, porous, ferruginous limestone rock is used in the experiment. A relatively high porosity is an advantage in our experiments as less time is required for the acid percolation inside the rock. The acid was infiltrated into the rock by means of an infusion pump (M361 Multi-rate Infusion Pump).
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The pump syringe was filled with 50 mL of 4.3 M H2S04, which were delivered at the rate of 1.0 mL per hour, through a piece of teflon tubing fitted into the tiny orifice. As this type of rock is weak and brittle, it was neccessm'y that a small orifice, 3 mm diameter and 1 cm depth, be cm'efully drilled in a way to allow the insertion of the il~jection tube. With this setup, a well-controlled flow is attained. The pump syringe is refilled as soon as the acid is ·consumed. Figure 4 shows pictures of the treated rock at various stages of the infiltration-acidization evolution.
Figure 4. Ferruginous limestone rock acidized with H2S04 through the teflon tube shown. Time evolution of the rock treatment showing the gradual formation of the deposits. The numbers indicate the time in days.
2.2. Preliminary Tests A number of simple preliminary chemical tests are carried out to guide our analysis, while the final rock composition in various band domains is confirmed by X-ray analysis. 1.
A zone rich in CaC03 is indicated by an effervescence upon treatment with 6M HCI, due to the reaction: CaC03 (.I') + 2HCI (aq) ~ Ca2+ (aq) + 2Cl- (aq) + CO2 (g) + H20 (l)
2. 3.
A zone predominantly consisting of CaS04 would show no fizzing upon treatment with HCl. An inter-band zone void of CaS04 showing no reaction upon addition of HCl is an indication of the depletion of the bare rock medium of any carbonate, due to prior consumption by the originally infiltrating acid (H2S04), This would leave only the other minerals supporting the bare rock medium.
Simulation of Geochemical Se(f-O';r;anization
4.
5.
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A small sample of white salt from a CaS04 ring dissolved in HCI, yields a yellow solution, attributed to the presence of ferric ions originating from the rock medium (ferruginous limestone). The presence of Fe3+ is confirmed by testing with KSCN solution. Upon addition of a few drops of SCN-, the solution turned blood-red indicating the formation of the FeSCN 2+ complex. Flame atomic absorption analysis of the precipitate (done on a Thermoelemental Solaar atomic absorption spectrophotometer), in CaS04-rich regions (after dissolution in HCI) was carried out to determine the elemental mass ratio of calcium to iron [14].
2.3. Composition Analysis of the Treated Rock
The treatment was fulfilled in a series of experiments, the latest of which was extended over a period of about one year and eight months. In the latter (most recent) expeliment, the H2S04 infusion started on October 5, 2005 and was stopped on May 23, 2007. The final pattern was examined to delineate the various regions wherein CaS04 bands may have formed. The pattern span area was divided into eight main textural regions, as highlighted in Fig. 5.
Figure 5. Final rock pattern at t:= 692 days. The rock was partitioned into eight "banded" zones for differentiation by X-ray analysis.
Powder samples from each region were taken and placed · in small tubes, which were stoppered thereafter. Each sample was analyzed by powder X-ray diffraction. Each diffractogram was mapped onto a Library of standard diffractograms for the calcite, gypsum and anhydrite minerals. Mter the peaks for each mineral were identified, the relative mass proportions of each mineral in the sample mixtures were determined by integrating the area under the peaks of each mineral [15]. The average composition of every region in each of the three minerals (over a representative number of samples) could then be calculated.
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3. Results and Discussion 3.1. Stage I At a first stage, the analysis of a treated ferruginous limestone rock was performed in a preliminary experiment wherein the H2S04 infiltration was stopped at time t = 161 days.
Figure 6. CaS04 deposition in the limetsone rock sample, infiltrated by H 2 S0 4 . The central region is a dense CaS04 precipitate zone. Then, beyond a brown depleted zone, a distinct belt of CaS04 suggests the formation of a first Liesegang ring. TIle photo is taken on day 161. A 1 em scale is placed on the picture.
The analysis of the white deposits in the central region and the ring shown in Fig. 6 confirmed the dominant presence of the gypsum mineral [14J. Photomicrographs of the deposit areas (notably in the ring region), before and after infiltration, depicted in Fig. 7, show the white crystals of the gypsum mineral embedded in the rock medium.
Figure 7.a. Rock texture before the acid injection. Magnification (lOx). The scale is indicated on the picture.
Fig. 7.b. Rock texture after acid injection, displaying the white crystals of CaS04 (labeled W) emhedded within the rock. Magnification (lOx).
Figure 7.c. Rock texture in the central region wherein the rock is entirely "stuffed" with CaS04.
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3.2 Stage II The results of the experiment described in Sect. 2.1 (analysis procedure described in Sect. 2.3) are now presented. Samples from each of the regions shown in Fig. 5 were taken by drilling small holes in the desired area, collecting the resulting powder in small vials and storing them for later X-ray analysis. The diffractograms confirmed the formation of gypsum (CaS04'2H20), which was quite dominant over anhydrite (CaS04, anhydrous) as the main deposit product in reaction 2. The central region (domains 1, 2 and 3) consists almost entirely of gypsum. Then depletion starts in region 4 (almost exactly like in a real Liesegang pattern wherein bands start separating from a uniform zone that extends over a relatively wide spatial domain) . The % composition in gypsum (calculated for the relative mass proportions of gypsum and calcite exclusively) drops to 28.4 % (see Table 1, and notably Fig. 8). 120
Region
% gyps.
1
100
..,::I
2
97.5
Q)
3
98.9
E 100
~
80
~
60
4
28.4
40
5
85.8
20
6
17.6
7
5.4
0 0
2
3
4
5
6
7
8
Region
Table 1. Variation of % gypsum over the rock regions.
Figure 8. Variation of % gypsum over the different rock regions.
Although some gypsum is present, it has dropped significantly in favor of unreacted calcite, suggesting the residence of an inter-band zone, relatively depleted in the diffusing sol-, just as predicted by the Ostwald cycle [3]. Then comes a new gypsum band (region 5; 85.8% gypsum) and so on. Hence, the % composition in gypsum essentially alternates. The complete depletion of gypsum between the bands (no 0% zones) is rendered impossible by the complex random percolation of the infiltrating acid (not a simple diffusion process), given the broad distribution of the pore sizes. The unexpectedly low composition in the last point (instead of an increase; see Fig. 8) is due to the incomplete formation of the gypsum band at that stage.
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4. Fractal Reaction Front Contours We now focus on the shape of the front contours in the different regions. The shape of the acidization front is to a large extent governed by porosity variations [16,17]. As a result, irregularities emerge, and thus may give rise to fractal structures. We analyze the front contours using Matlab and the Image Processing Toolbox. After choosing a specific region (from domains 1-8), the RGB image is cropped, and the region of interest is retained. The interface between the two corresponds to the contour of the chosen region. We used color segmentation using the L*a*b color space whereby we select the color of the contour to a tolerance of 10%. The plate was then transformed into a binary image. Morphological operations were afterwards applied to fill any open holes. Contours of regions 8 (the outer front) and 4 delimited by this method are shown in Fig. 9.
Figure 9.a. Contour of region 8, of fractal dimension 1.87.
Figure 9.b. Contour of region 4, of fractal dimension 1.44.
The fractal dimension using the box count algorithm was computed and found to be 1.87 for region 8 (frame. 9.a) and 1.44 for region 4 (frame. 9.b). This study is under continuing investigation, wherein all the contours will be meticulously traced and analyzed.
5. Conculsions In this paper, we simulated a Liesegang banding expeirment in-situ, i.e. inside a real rock bed. After a long treatment with acid infiltrated from a central source, band formation was induced in a way similar to what we see in a naturally banded rock. X-ray analysis revealed the alternation of minerals, thus resembling a 2D Liesegang pattern. The contours of the acidization fronts were found to be fractal.
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Acknowledgements: This work was supported by a grant from the Lebanese National Council for Scientific Research (LNCSR). The authors thank Dr. Abdel-Fattah Abdel-Rahman, for his valuable and very helpful advice. All the X-ray and AA measurements were performed in the Central Research Science Lab (CRSL) of the American University of Beirut.
References [1] R. E. Liesegang, Chemische Fernwirkung, Lieseg. Photograph. Arch. 37: 305, 1896; continued in 37: 331,1896. [2] H. K. Henisch, Crystals in Gels and Liesegang Rings, Cambridge University Press, Cambridge, 1988. [3] Wi. Ostwald, Lehrbuch der Allgemeinen Chemie, 2. Aufl., Band 2, 2. Teil: Verwandtschaftslehre, Engelmann: Leipzig, 1896-1902, p. 778. [4] J. A. Shapiro and M. Dworkin, Eds., Bacteria as Multicellular Organisms, Oxford University Press, New York, 1997. [5] V. V. Kravchenko, A. B. Medvinskii, A. N. Reshetilov and G. R. Ivanitskii, Dokl. Akad. Nauk, 364: 114,1999; ibid, Dokl. Akad. Nauk" 364: 687,1999. [6] D. Xie, J. Wu, G. Xu, Q. Ouyang, R. D. Soloway, and T. Hu, 1. Phys.Chem. B, 103: 8602, 1999. [7] R. E. Liesegang, Geologische DijJusionen, Steinkopff, Dresden, 1913; R. E. Liesegang, Die Achate, Steinkopf, Dresden-Leipsig, 1915. [8] T. Moxon, Educ. Chem. July issue, 105,2001. [9] T. Moxon, Agate: Microstructure and Possible Origin, Terra Publications, Doncaster, 1996. [10] J. H. Kruhl, Ed., Fractals and Dynamic Systems in Geoscience, Springer-Verlag, Berlin, 1994. [II] P. Ortoleva, Geochemical Self-Organization, Oxford University Press, New York, 1994. [12] M. Ohkawa, Y. Yamashita, M. Inoue, R. Itagawa and S. Takeno, Mineralogy and Petrology, 70: 15,2000. [13] B. Jamtveit and P. Meakin (Eds.), Growth, Dissolution and Pattern Formation in Geosystems, Kluwer, Dordrecht, 1999. [14] M. Msharrafieh, M. AI-Ghoul and R. Sultan, in Complexus Mundi: Emergent Patterns in Nature, Proceedings of the Fractals 2006 Conference, Vienna, Austria; M. M. Novak, Editor; World Scientific, Singapore, 2006, pp. 225-236.
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[15] C. Suryanarayana and M. Grant Norton, X-Ray Diffraction, A Practical Approach, Plenum Press, New York, 1998; Experimental Module 7, Quantitative Analysis of Powder Mixtures, pp. 223-236. [16] K. Lund, H. S. Fogler and C. C. McCune, Chern. Eng. Sci. 28: 691-700,1973. [17] K. Lund and H. S. Fogler, Chern. Eng. Sci. 31: 373-380, 1976.
395
Author Index Akkaya E. E., 93
Hacinliyan A. S., 93
AI-Ghoul Mazen, 385
Haller George, 294
AI-Joubeily Samia, 385 Alomari Majdi M., 1 Aniszewska Dorota, 9
Iliopoulos A. c., 268
AwrejcewiczJan, 17, 139
Jevtic N., 101
Aybar O.
b., 93 Karakatsanis L. P., 268
Baehr Christophe, 27
Khots Boris, 111
Becerra Alonso David, 35
Khots Drnitriy, 111
Boutalis Yiannis, 346
Kolesnikov Alexander A., 119
Brianzoni Serena, 43
Kolesnikov Anatoly A., 128, 134
Byrne G., 147
Konkov L. E., 369
Chin Chou Hsin, 215
Kudra Grzegorz, 17
Christodoulou Manolis, 346
Kut;;beyzi
Krysko V. A., 139
Commendatore Pasquale, 59
t, 93
Kuzmenko Andrew A., 134 Kuznetsova E. S., 139
Daniels S., 147 De Campos Cristina, 51
Lan Boon Leong, 165
Dingwell Donald B., 51
Law V. J., 147
Dobrescu Loretti I., 67
Lawrance Anthony J., 155
Dodson C. T. J., 75
Liang Shiuan-Ni, 165
Dowling D. P., 147
Lucarini Valerio, 170
EI-Rassy Houssam, 385
Mammana Cristiana, 43
Ertel-Ingrisch Werner, 51
Manneville P., 185 Mello Pier A., 191
Filippaki E., 377
Mendez-Bermudez J. A., 191
Fraedrich Klaus, 170
Michetti Elisabetta, 43, 59 Mikishev Alexander B., 207
Gopar Victor A., 191
Mitskevich S., 139
Grigoras Carmen, 85
Mokalled Tharwat, 385
Grigoras Victor, 85
396
Author Index
Neamtu Mihaela, 67
Sapsis Themistoklis, 294
Nepomnyashchy Alexander A., 207
Schweitzer J. S., 101
Ni David C., 215
Skiadas Christos H., 302 Smorodin Boris L., 207
Oommen B. John, 284
Sokolov Valentin V., 309
Opri~
Soldatov V., 139
Dumitru, 67
Orman Gabriel V., 224
Sotiropouios Anastasios D., 320, 327 Sotiropoulos Vaggelis D., 320, 327
Pannekoucke Olivier, 27
Srisuchinwong Baniue, 338
Pantelidis Leonidas, 232
Sud K. K., 257
Papageorgiou C. D., 241 Papamarkou Theodore, 249
Temizer L, 93
Patidar Vinod, 257
Tereshko Valery, 35
Pavlos E. G., 268
Theodoridis Dimitris, 346
Pavlos George P., 268
Tsoutsouras V. G., 268
Perugini Diego, 51
Tynan 1., 147
Pinto Antonio, 59 Poli Giampero, 51
Umut Omiir, 362
Prants S. V., 369 Purohit G., 257
Vitkovsky V. 0., 369
Qin Ke, 284
Xapianteris C. L., 377
Rabih Sultan, 385
Zaknoun Farah, 385
Raptis T. E., 241
Zhigalov M., 139
Rybaczuk Marek, 9
Zhu Jian Gue, 1
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