Le tters to the
Editor
The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.
EN= 5t
Erdos Number Updates*
Marek A. Abramowicz
We give more accurate estimates of
Institute of Theoretical Physics
Erdos numbers (ENs) for three physi
Chalmers University
cists mentioned in the article "Famous
41 2-96 Goteborg
Trails to Paul Erdos" by Rodrigo de
Sweden
Castro and Jerrold W. Grossman (Math
e-mail:
[email protected] ematical lnteUigencer, vol. 21 (1999), no.
3, 51-63).
Future of Mathematical
3 be
John A. Wheeler has EN=
cause of the paper "On the question of a neutrino analog to electric charge,"
Rev.
Mod. Phys.
which
he
29, p. 516 (1957),
published
with John
R.
Klauder (known to have EN = 2). Roger Penrose has EN
4 at
Literature As an amateur Platonist I am well aware of problems with mathematical and philosophical archives. Much of what you are searching for is simply no longer available. It would be fine
most:
to have it all securely fixed on the
John R. Klauder has published to
Internet for free use by everybody.
gether
with
C.I.
=
Isham
an
article
However,
it
is
getting
increasingly
"Affine fields on operator representa
clear that the problem is not only in the
tions for the nonlinear sigma-model,"
relatively short lifetime of magnetic
JMP 31, p. 699 (1990), and C.I. Isham,
recording, but also in the possibility
3, published
(likely increasing) of vandalization of
with R. Penrose and D.W. Sciama a
electronic media by hackers. Whereas
book, Quantum Gravity, an Oxford
it took a fanatic mob to bum Alexan
Symposium, Clarendon Press, Oxford,
dria Library, it may now take just a few
who therefore has EN
=
1975, in which the Preface is jointly
keyboard
written by the three of them.
electronic Herostratus to erase irre
Stephen W. Hawking, who pub
strokes
by
some
shrewd
placeable records.
lished a book and several papers with
Unfortunately, the back-up problem
Roger Penrose, has therefore EN= 5
is far from trivial. A symbiosis of elec
at most.
tronic and paper storage systems (advo cated by D.L. Roth and R. Michaelson's
Jerzy Lewandowski
EN= 3
letter, Mathematical InteUigencer 22
(2000), no. 1, 5) is unlikely to provide
lnstytut Fizyki Teoretycznej Uniwersytet Warszawski
an adequate long-term solution be
ul. Hoza 69 00-681 Warsaw, Poland
cause of the sheer amount of paper it requires. Instead, more durable read
e-mail:
[email protected] only techniques are likely to enter the scene. Examples are atomic force mi-
Pawel Nurowski lnstytut Fizyki Teoretycznej
EN = 4
croscopy recording, with a potential capacity of 1012 bits per sq em, or iso
Uniwersytet Warszawski
topic information storage (information
ul. Hoza 69
is coded in the order of stable Si iso
00-681 Warsaw, Poland
topes on a crystal surface).
e-mail:
[email protected] Standardization and simplification
•This letter must serve as a representative of the hundreds of updates which could be written. The reader is reminded that the Web site http://www.oakland.edu/-grossman/erdoshp.html is perpetually updated. There you may find links joining several chemistry Nobel Prize winners. The updated Erdos numbers of all Fields Medallists are 4 or less, excepting Paul Cohen and Alexander Grothendieck at 5. - Editor's Note. tEN
=
4 after publication of this letter.
© 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 3, 2000
3
of such technologies (including read ing devices) will allow for inexpensive individual ownership of entire acade mic libraries stored in a briefcase-size box. Potentially, almost every house hold could have one (at a cost, per haps, comparable to a present set of the Britannica); they would be duly updatable. We could then feel there was no longer real danger of our records being permanently lost. Alexander A Berezin Department of Engineering Physics
be translated as, T " he sowerArepo takes pains to hold the wheels." But surely the magical form is more important than any artificially imposed meaning. It is transformed into the Pater Noster in the form of a cross:
s
A
Conceptual Magic Square
I would like to add the standard form of the magic square discussed by A Domenicano and I. Hargittai in The Intelligencer 22, no. 1, 52-53. (I am fol lowing [1], 142.) Usually it reads SAT 0R ARE P 0 T E N ET 0 P ERA R 0 TAS If
.
D-37073 Gottingen
Germany e-mail:
[email protected] p
A 0 T E R PAT ER NOS T ER 0
Hamilton, Ontario L8S 4L8 e-mail:
[email protected] Mathematisches lnstitut Bu n senstr 3-5
A
McMaster University Canada
Benno Artmann
T E R
0
Here exactly the same letters are used as in the "magic square." The addi tional copies of letters A and 0, transliterations of alpha and omega, stand as a metaphor for Christ as in the NewTestament,Apocal. Joh. 22,13. It is not clear to me why the authors claim that all the reading forwards, backwards, etc., has anything to do with the regular polyhedra.
Newton Scooped
In my article "Exactly how did Newton deal with his planets" in vol. 18 (1996), no. 2, 6-11, I credited Newton with the following discovery: LetS be a focus of an ellipse and P a point on the ellipse. The lineSP meets the line through the center that is parallel to the tangent at P at a point E.Then the length of PE is independent of P. (Principia, Book II, Proposition XII, Problem VII.) However, Bjorn Thiel of Bielefeld University has pointed out to me that Apollonius had obtained the result some 1900 years earlier. (Conics, Book III, Proposition 50.) Sherman K. Stein Mathematics Department University of California at Davis
1. G. Mazzola, ed.: Katalog: Symmetrie in
one takes the secondary meaning of opera as toil, labor, pains, then it might
Davis, CA 95616-8633
Kunst, Natur und Wissenschaft, Vol. 3.
USA
Darmstadt, Roether, 1 986.
e-mail:
[email protected] MOVING? We need your new address so that you do not miss any issues of THE MATHEMATICAL INTELLIGENCER. Please send your old address (or label) and new address to: Springer-Verlag New York, Inc., Journal Fulfillment Services P.O. Box 2485, Secaucus, NJ 07096-2485
U.S.A. Please give us six weeks notice.
4
THE MATHEMATICAL INTELLIGENCER
Opinion
Chaotic Chaos Denys A Hill
T
he word "chaos" is used in two dif
with the prevalent literary usage of
ferent contexts.
chaos as confusion!
It is more than two millennia since
Turbulence, understood as absence
"chaos" meaning confusion and empti
of underlying order, is sometimes used
ness was established in use in literary
as an analogy for chaos; so is random
contexts.
ness.
Vergil
and
Ovid
thought
"chaos" (the same word in Latin as in English)
this
Does chaos have a separate identity?
meaning continued in the Middle Ages.
The Opinion column offers
If its methodology comprises (say) dif
Authors such as Shakespeare, Milton,
ference and differential equations, bifur
mathematicians the opportunity to
Pope, Byron, and De la Mare had the
cations, Lyapunov exponents, fractals,
same understanding. A little-known ex
Fast Fourier Transforms, and wavelet
write about any issue of interest to
meant
confusion,
and
This would seem to be giving up
the characterisation by underlying order.
ception is Henry Adams's opinion in
analysis, it does not claim these as its ex clusive reserve. Demoting chaos theory
community. Disagreement and
1907 that chaos brings order. The author of the 1947 Marshall Plan again per
controversy are welcome. The views
ceived
and
theory would be possible, but could lead
and opinions expressed here, however,
present-day newspaper writers do like
to its disappearance as a separate entity.
the international mathematical
are exclusively those of the author, and neither the publisher nor the
chaos
with
foreboding,
wise. Contemporary dictionaries of ma jor European languages support them.
to a subset of another, better delineated
The demand to give it a separate identity is founded on claims for suc
In the mid-twentieth century, "chaos"
cesses in real-life situations: Ruelle
editor-in-chief endorses or accepts
and "chaos theory" began to be used
Takens phase space reconstruction,
responsibility for them. An Opinion
with technical meanings. By now these
mathematical modelling in control of
should be submitted to the editor-in chief, Chandler Davis.
words have appeared in more than
lasers and electronic circuits or car
7,000
diac arrhythmia. Unfortunately some
mathematical
and
scientific
books, dictionaries, and papers. Unfor
exaggerated claims, such as control of
tunately the technical meanings are
El Nifi.o climatic phenomena, under
themselves confused, and there is no
mine its credibility and provide sup
agreement on definitions. The contrast
port for those who regard chaos as
with literature and journalism, which
usurping territory of other theories.
give the word an unambiguous and
The justification of a clean-up effort
embar
is not based only on professional pride,
rassing: mathematics and science, in
on the desire to be at least as succinct
straightforward
meaning,
is
place of their vaunted rigour and pre
and definite as our literary and journal
cision, display ambiguities and even
istic counterparts. It is based on self-in
contradictions. The situation is chaotic.
terest. Techniques of chaos theory are
Confusion over the several technical
allegedly about to be applied effectively
meanings is so rife that some authors
in financial control. We had better re
are on record as abandoning attempts to
solve the present chaotic situation be
define "chaos." One relies on personal
fore we lose out on pecuniary benefits.
know-how to recognise its presence. Other authors depend upon their read ers' knowing the meaning already. Such definitions as have been pub lished have little in common. Is chaos a mathematical phenomenon or condi tion? or is it a range of phenomena? Is
Let me advance this phrasing as workable:
Chaos, the science of non-linear deterministic processes that appear stochastic. It insists on the scientific; it is inclu
chaos something occurring in space
sive enough to expand as the subject
and
it
develops; it may resolve the chaotic
bounded or unbounded? Non-linearity,
chaos dilemma sufficiently to protect
time,
or
irregularity,
just
in
space?
intermittency,
Is
unpredic
tability, transience, changeability, and
the image of mathematics as a rigor ous discipline.
complexity are all invoked in the tech nical literature as attributes of chaos.
1 2, Tulip Tree Avenue
In addition there is the feature of or
Kenilworth
der underlying the appearance of ran
Warwickshire CV8 2BU
domness-a
United Kingdom
usage
which
conflicts
© 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 3, 2000
5
MARCELO VIANA
What's New on Lorenz Strange Attractors?
esides its philosophical implications on the ideas of determinism and (un)predictability ofphenomena in Nature, E. Lorenz's famous article ''Deterministic nonperiodic flow" [ 1 7], published nearly four decades ago in the Journal of Atmospheric Sciences,
raised a number of mathematical questions that are among the
leitmotifs for the extraordinary development the field of DynamicalSystems has been going through.This paper is about those and related questions, and some remarkable recent answers.The first part is a general overview, mostly in chronological order.The four remaining sections con tain more detailed expositions of some key topics. Modeling the weather
Lorenz, a meteorologist at MIT, was interested in the foun dations of long-range weather forecasting. With the advent of computers, it had become popular to try to predict the weather by numerical analysis of equations governing the atmosphere's evolution. The results were, nevertheless, rather poor.A statistical approach looked promising, but Lorenz was convinced that statistical methods in use at the time, especially prediction by linear regression, were es sentially flawed because the evolution equations are very far from being linear.
To test his ideas, he decided to compare different meth ods applied to a simplified non-linear model for the weather.The size of the model (number and complexity of the equations) was a critical issue because of the limited computing power available in those days. 1After experi menting with several examples, Lorenz learned from B.Saltzmann of recent work of his [35] concerning ther mal fluid convection, itself a crucial element of the weather.A slight simplification
:i; = - ax + ay iJ = rx - y - xz z = xy - bz
(]" =
10
r = 28 b = 8/3
(1)
of a system of equations studied bySaltzmann proved to be an ideal test model. I'll outline later howSaltzmann ar rived at these equations, and why he picked these particu lar values of the parameters O", r, b. The key episode is recalled by Lorenz in [18].At some
1 Lorenz's computer, a Royal McBee LGP·30, had 1 6Kb internal memory and could do 60 multiplications per second. Numerical integration of a system of a dozen differential equations required about a second per integration step.
6
THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK
.
· .
difference in the final state. 2A similar point was stressed early in the twentieth century by H. Poincare [30], includ ing the very setting of weather prediction:
Why have meteorologists such difficulty in predicti'YI;_{J the weather with any certainty? Why is it that showers and even storms seem to come by chance, so that many people think it is quite natural to pray for them, though they would consider it ridiculous to ask for an eclipse by prayer? [ . . . ] a tenth of a degree more or less at any given point, and the cyclone wiU burst here and not there, and extend its ravages over districts that it would other wise have spared. If they had been aware of this tenth of a degree, they could have known it beforehand, but the observations were neither sufficiently comprehensive nor sufficiently precise, and that is the reason why it aU seems due to the intervention of chance.
Figure 1 . Lorenz strange attractor.
stage during a computation he decided to take a closer look at a particular solution. For this, he restarted the integra tion using some intermediate value printed out by the com puter as a new initial condition. To his surprise, the new calculation diverged gradually from the first one, to yield totally different results in about four "weather days"! Lorenz even considered the possibility of hardware fail ure before he understood what was going on. To speed things up, he had instructed the computer program to print only three decimal digits, although the computations were carried out to six digits. So the new initial condition en tered into the program didn't quite match the value gener ated in the first integration. The small initial difference was augmented at each integration step, causing the two solu tions to look completely different after a while. This phe nomenon, first discovered in a somewhat more compli cated system of equations, was reproduced in (1). The consequences were far-reaching: assuming the weather does behave like these models, then long-range weather prediction is impossible: the unavoidable errors in determining the present state are amplified as time goes by, rendering the values obtained by numerical integration meaningless within a fairly short period of time. Sensitivity and unpredictability
This observation was certainly not new. Almost a century before, J. C. Maxwell [22], one of the founders of the ki netic theory of gases, had warned that the basic postulate of Determinism-the same causes always yield the same effects-should not be confused with a presumption that similar causes yield similar effects, indeed there are cases in Physics where small initial variations may lead to a big
On the other hand, gas environments and, particularly, the Earth's atmosphere, are very complicated systems, in volving various types of interactions between a huge num ber of particles. Somehow, it is not surprising that their evolution should be hard to predict. What was most strik ing about Lorenz's observations was the very simplicity of equations (1), combined with their arising in a natural way from a specific phenomenon like convection. That the so lutions of such a simple set of equations, originating from a concrete important problem, could be sensitive with re spect to the initial conditions, strongly suggested that sen
sitivity is the rule in Nature rather than a particularfea tu1·e of complicated systems. Strange attractors
A few years later, in another audacious paper [33], D. Ruelle and F. Takens were questioning the mathematical inter pretation of turbulent fluid motion. It had been suggested by L. Landau and E. Lifshitz [16], and by E. Hopf before them, that turbulence corresponds to quasi-periodic mo tions inside tori with very large dimension (large number of incommensurable frequencies) contained in the phase space. However, Ruelle and Takens showed that such quasi-periodic tori are rare (non-generic) in energy dissi pative systems, like viscous liquids. Instead, they main tained, turbulence should be interpreted as the presence of some strange attractor. An attractor is a bounded region in phase-space, in variant under time evolution, such that the forward tra jectories of most (positive probability), or even all, nearby points converge to it. Ruelle and Takens did not really try to define what makes an attractor strange. Eventually, the notion came to mean that trajectories converging to the
attractor are sensitive with respect to the initial condi tions. Lorenz's system (1) provides a striking example of a strange attractor, and several others have been found in
2Even earlier, the same idea appeared in E. A Poe's The mystery of Marie Roget, in the context of crime investigations ... So much for priorities on this matter.
VOLUME 22, NUMBER 3, 2000
7
various models for experimental phenomena as well as in theoretical studies. However, not many examples were available at the time [33] was written.Still unaware of the work of Lorenz, which came to the attention of the math ematical community only slowly,R uelle andTakens could only mentionS.Smale's hyperbolic solenoids [38], which, although very important from a conceptual point of view, had no direct physical motivation. Hyperbolic systems
Throughout the sixties,Smale was much interested in the concept of structurally stable dynamical system, intro duced byA.Andronov and L. Pontryagin in [2].The reader should be warned that the word "stability" is used in DynamicalSystems in two very different senses.One refers to trajectories of a system: a trajectory is stable (or at tracting) if nearby ones get closer and closer to it as time increases.Another applies to systems as a whole: it means that the global dynamical behavior is not much affected if the laws of evolution are slightly modified.3 Structural sta bility belongs to the second kind: basically, a system is structurally stable if small modifications of it leave the whole orbit structure unchanged, up to a continuous global change of coordinates. In an insightful attempt to identify what the known sta ble systems had in common,Smale introduced the geo metric notion of hyperbolic dynamical system. I will give a precise definition of hyperbolicity later; for now let me just refer to Figure 2, which describes its basic flavor: ex istence, at relevant points x in phase-space, of a pair of sub-manifolds that intersect transversely along the trajec tory of x, such that points in one of them (the horizontal "plane") are forward asymptotic to x, whereas points in the other submanifold (the vertical "plane") are backward asymptotic to x. Most remarkably, hyperbolicity proved to be the crucial ingredient for stability: the hyperbolic sys tems are, essentially, the structurally stable ones. More over, a beautiful and rather complete theory of these sys tems was developed in the sixties and the seventies: hyperbolic systems and their attractors are nowadays well understood, both from the geometric and the ergodic point of view.The reader may find precise statements and ref erences to a number of authors, e.g., in the books [6, 28, 36]. Yet, not every system can be approximated by a hy perbolic one. . . The flow described by equations (1) is not hyperbolic, nor structurally stable, so it doesn't fit into this theory.On the other hand, its dynamical behavior seems very robust. For instance: Figure 1, which represents a solution of (1) integrated over a long period of time,4 would have looked pretty much the same if one had taken slightly different values for the parameters u, r, b. How can this be, if these systems are unstable (and sensitive with respect to initial data!)?
Figure 2. Hyperbolicity near a regular trajectory.
It was probably a fortunate thing thatSmale and his stu dents and colleagues did not know about this phenomenon at the time they were laying the foundations of the theory of hyperbolic systems: it might have convinced them that they were off in a wrong direction. In fact, a satisfactory theory of robust strange attractors of flows would come to existence only recently, building on several important ad vances obtained in the meantime. But I'm getting ahead of myself! Lorenz-like flows
For now, let us go back to the mid-seventies, when the work of Lorenz was finally becoming widely known to dy namicists, and, in fact, attracting a lot of attention.So much so, that by the end of that decade C.Sparrow could write a whole book [39] about the dynamics of equations (1) over different parameter ranges. Understanding and proving the observations of Lorenz in a rigorous fashion turned out to be no easy task, though. A very fruitful approach was undertaken, independently, by V. Afraimovich, V. Bykov, L.Shil'nikov [1], and by J. Guckenheimer andR . Williams [8, 44]. Based on the be havior observed in (1), they exhibited a list of geometric properties such that any flow satisfying these properties
must contain a strange attractor, with orbits converging to it being sensitive with respect to initial conditions. Most important for the general theory, they proved that such flows do exist in any manifold with dimension 3. These examples came to be known as geometric Lorenz
models. The strange attractor has a complicated geometric struc ture like the "butterfly" in Figure 1.5Sensitivity corre sponds to the fact that trajectories starting at two nearby states typically end up going around different "wings" of the butterfly.There are orbits inside the strange attractor that are dense in it.This means that the attractor is dy namically indecomposable (or transitive): it can not be split into smaller pieces closed and invariant under the flow.Another very important feature of these models is that the attractor contains an equilibrium point.
3For instance, by altering the values of parameters appearing in the evolution equations. 4An initial stretch of the solution is discarded, so that the part that is plotted is already close to the strange attractor. 5The figure was produced by numerical integration of the original equations (1).
8
THE MATHEMATICAL INTELLIGENCER
Now, one might expect that small modifications of the flow could cause such a complicated behavior to collapse. For instance, the attractor might break down into pieces displaying various kinds of behavior, or the different types of trajectories (regular ones and equilibria) might be set apart, if one changes the system only slightly. Surprisingly, this is not so: any flow close enough to one of these also has an attractor containing an equilibrium point and ex hibiting all the properties I described, including sensitivity and dynamical indecomposability. A theory of robust strange attractors
Vz
VI
�""""'"-
�
L
I
!.
IP(VJ) J
-
-
P(\-S)
r Figure 3. Suspended horseshoe.
As 111
explain later, the presence of equilibria accumulated by regular orbits of the flow implies that these systems can not be hyperbolic. On the other hand, they can not be dis regarded as a pathology because, as we have just seen, this kind of behavior is robust. Indeed, these and other situa tions, often motivated by problems in the Natural Sciences, emphasized the need to enlarge the scope of hyperbolicity into a global theory of Dynamical Systems. Profiting from the success in the study of specific classes of systems like the geometric Lorenz flows or the Henon maps (M. Benedicks and L. Carleson [3], after pioneer work of M. Jakobson [14]), as well as from fundamental advances like the theory of bifurcations and Pesin's non-uniform hy perbolicity [29], a new point of view has been emerging on how such a global theory could be developed. In this direc tion, a comprehensive program was proposed a few years ago by J. Palis, built on the following core corijecture: every
smooth dynamical system (diffeomorphism or flow) on a compact manifold can be approximated by another that has only finitely many attractors, either periodic or strange. I refer the reader to [27] for a detailed exposition. In the context of flows, decisive progress has been ob tained recently by C. Morales, M. J. Pacifico, and E. Pujals [25, 26], whose results provide a unified framework for ro bust strange attractors in dimension 3. While robust at tractors without equilibria must be hyperbolic [11], they prove that a ny robust attractor that contains some equi
librium poi nt is Lorenz-like: it shares aU the fundamen tal properties of the geometric Lorenz models. A key in gredient is a weaker form of hyperbolicity, that Morales, Pacifico, and Pujals call singular hyperbolicity. They prove that any robust attractor containing an equilibrium point is singular hyperbolic [26]. 6 This is a crucial step lead ing to a rather complete geometric and ergodic theory, ap plying to arbitrary robust attractors of 3-dimensional flows. More on this will come later. Back to the original equations
While they were catalysing such fundamental develop ments in Dynamical Systems, equations (1) themselves kept resisting all attempts at proving that they do exhibit a sensitive attractor.
On the one hand, no mathematical tools could be de vised to solve such a global problem for specific equations like (1). For instance, M. Rychlik [34] and C. Robinson [32] considered systems exhibiting certain special configura tions (codimension-2 bifurcations) and, using perturbation arguments, proved that nearby flows have strange attrac tors like the geometric Lorenz models. This enabled them to exhibit the first explicit examples (explicit equations) of systems with strange attractors of Lorenz type: those special configurations occur in some families of polyno mial vector fields, with degree three, for appropriate choices of the parameters. However, it has not been pos sible to find parameter values u, b, and r, for which (1) sat isfy the assumptions of their theorems. Another approach was through rigorous numerical cal culations. Here a major difficulty arises from the presence of the equilibrium: solutions slow down as they pass near it, which means that a large number of integration steps are required, resulting in an increased accumulation of in tegration errors. This could be avoided in [9, 10, 23, 24], where all the relevant solutions remain far from the equi librium point (error control remains delicate, neverthe less). In these works, the authors gave computer-assisted proofs that the Lorenz equations have rich dynamical be havior, for certain parameters. More precisely, they used numerical integration with rigorous bounds on the inte gration errors, to identify regions V = V1 U V2 inside some cross-section of the flow, such that the image of V under the first-return map consists of two pieces that cross V as in Figure 3. By a classical result of Smale, see [38], this con figuration ("suspended horseshoe") implies that there is an infinite set of periodic trajectories. Still, the original question remained: do equations (1) re ally have a strange attractor? These equations have no par ticular mathematical relevance, nor do the parameter val ues (10, 28, 8/3): their great significance is to have pointed out the possibility of a new and surprising kind of dynam ical behavior that we now know to occur in many situa tions. Nevertheless, many of us felt that answering this question, for parameters near the original ones, was a great challenge and a matter of honor for mathematicians. 7
6See [4, 7] for related results about discrete-time systems, in any dimension. These and other important recent developments are surveyed in my article [43]. 7See Smale's list of outstanding open problems for the next century in vol. 20, no. 2 of The Mathematical lntelligencer, and his contribution to the book Mathematics: Frontiers and Perspectives, tube published under the aegis of the International Mathematical Union as part of the celebrations of the World Mathematical Year 2000.
VOLUME 22, NUMBER 3, 2000
9
The Lorenz attractor exists!
Remarkably, a positive solution was announced about a year ago, by W. Tucker, then a graduate student at the University of Uppsala, Sweden, working under L. Carleson's supervi sion. Theorem 1 (Tucker [41, 42]) For the classical parame
ters, the Lorenz equations (1) support a robust strange attractor. Tucker's approach is a combination of two main ingre dients. On the one hand, he uses rigorous numerics to find a cross-section I and a region N in I such that orbits start ing in N always return to it in the future. After choosing reasonable candidates, Tucker covers N with small rec tangles, as in Figure 4, and estimates the forward trajec tories of these rectangles numerically, until they return to I. His computer program also provides rigorous bounds for the integration errors, good enough so that he can safely conclude that all of these rectangles return inside N. This proves that the equations do have some sort of attractor. A similar strategy is used to prove that the attractor is sin gular hyperbolic in the sense of [26]. The other main ingredient, normal form theory, comes in to avoid the accumulation of integration errors when trajectories are close to the equilibrium sitting at the origin. Tucker finds coordinate systems near the equi librium such that the expression of the vector field in these coordinates is approximately linear. Thus, solu tions of the linear flow (which are easily written down in analytical form) can be taken as approximations of the true trajectories, with efficient error estimates. Accord ingly, Tucker instructs the computer program to switch the integration strategy when solutions hit some small neighborhood of the equilibrium: instead of step-by-step integration, it uses approximation by the linear flow to estimate the point where the solution will exit that neigh borhood. Verifying that a computer-assisted proof is correct in volves both checking the algorithms for logical coherence, and making sure that the computer is indeed doing what it is supposed to do. The second aspect is, of course, less familiar to most mathematicians than the first. In fact, as
Figure 4. Covering a possibly invariant region with small rectangles.
10
THE MATHEMATICAL INTELLIGENCER
computer-assisted proofs are a rather new tool, there hasn't been much time to establish verification standards for computer programs. A basic procedure is to have the codes recompiled and rerun on different machine archi tectures. Preferably, beforehand the algorithm should be reprogrammed by different people. As far as I know, such a detailed independent verification of Tucker's computer programs has not yet been carried out. The first version did contain a couple of "bugs," which Tucker has fixed in the meantime [42], and which I will mention briefly near the end. He has also made the text of his thesis and the computer codes, as well as the initial data used by his pro grams, available on his web page [41]. An outline of Tucker's arguments is given in the last sec tion of this work. Right now, let us go back to where it all started, for a closer look. From Thermal Convection to the Equations of Lorenz
Most of the motion in the Earth's atmosphere takes the form of convection, caused by warming of the planet by the Sun: heat absorbed by the surface of the Earth is trans mitted to the lower layers of the atmosphere; warmer air being lighter, it rises, leaving room for downward currents of cold air. A mathematical model for thermal convection was pro posed early in the twentieth century by the British physi cist R. J. Stutt, better known as Lord Rayleigh. This model [31] describes thermal convection inside a fluid layer con tained between two infinite horizontal plates that are kept at constant temperatures Ttop and Tbot · It is assumed that the bottom plate is hotter than the top one, in other words, Tbot > Ttop · If the temperature difference 11T = Tbot - Ttop is small, there is no fluid motion: heat is transmitted up wards by conduction only. In this case, the fluid tempera ture Tsteady varies linearly with the vertical coordinate YJ. As 11T increases, this steady-state solution eventually be comes unstable, and the system evolves into convective motion. Convection cells are formed, where hot fluid is cooled down as it rises, and then comes down to get heated again. See Figure 5. B. Saltzmann [35] analyzed a simplified version of Rayleigh's model. Firstly, he assumed that the system is in variant under translations along some favored direction, like the direction of convection rolls in Figure 5, so that the corresponding dimension in space may be disregarded.
Figure 5. Convective motion.
This brings the problem down to two spatial dimensions;
metical integration showed that, for convenient choices of
the evolution equations reduce to
the parameters, all but three of the dependent variables are
a( 'I',V2'1')
aV2'1' at
-- =
a(g, TJ) a('¥,8)
a8 at where
g
TJ
and
(2)
D.T a'l' 2 -+KV8 H ag
( 3)
+
are the spatial coordinates,
'I'
and
8
t
is time, and
are interpreted as fol
lows: •
although the phase-space has dimension
tor" contained in a three-dimensional subset of the phase space. On the other hand, these three special non-transient modes seemed to have a rather complicated (nonperiodic) evolution with time. Lorenz took the system of equations obtained in this manner, by truncating the original infmite system right
'l'(g, TJ, t)
is a stream function: the motion takes place
along the level curves of
( •
other words,
seven, many solutions seem to converge to some "attr3;c
-
a(g,TJ)
the dependent variables
transient: they go to zero as time increases to infmity. In
a8 +vV4'I'+g aag
8(g, TJ, t) = T(g, TJ, t)
'I',
a'lt a'¥
-
aTJ' ag -
from the start to only these three variables. This corre sponded to looking for solutions of
with velocity field
).
'l'(g, TJ, t)
Tsteady(g, TJ, t) is
the temperature
ecg, TJ, t)
departure from the steady-state solution mentioned above. The other letters represent physical parameters: height of the fluid layer,
g
H
is the
is the constant of gravity,
a
is
the coefficient of thermal expansion, v is the viscosity, and
K
is the thermal conductivity. If the Rayleigh
However, as observed by Rayleigh, as
Ra
where
A0, Eo, C0,
where X0 and
are created, of the form
t) = Yo cos
( ;; g) (; TJ) ( ;; g) (; TJ}
( 4)
sin
(5)
constants. They describe the motion
5, the parameter a
being related to the eccentricity of the cylinders. They are
stationary solutions,
as time
on the right hand side of
( 4)
t
does not appear explicitly
and
g, a,
v,
K,
and
D.T.
a and H.
The parameters
cr,
r, bin (1)
v
b
cr =
K
=
a
To obtain
and
(1)
D o depending
are given by
4
. 1 +a2
Some simple facts about equations ( 1) are easy to check
sin
in cylindrical convection cells in Figure
Z = C oZ,
one also rescales time, by another constant on
t) = Xo sin
Yo are
one obtains three
are constants depending only on
the physical quantities H,
Ra r =Rc
TJ,
(2), ( 3),
Y = B oY
X = A oX
crosses a threshold
8 (g,
sin
rescaling:
is small, the system remains in the steady-state equilibrium
TJ,
sin
X(t), Y(t), Z(t) which are equivalent to equations ( 1). Actually, X, Y, Z are not exactly the same as x, y, z in (1), but the
VK
'l'(g,
sin
two sets of variables are related to each other, simply, by
number
(2) -( 3)
sin
ordinary differential equations in the coefficients
D.T
new solutions of
X(t)
Inserting these expressions in
R a = g a/{J 'I' = 0, 8 = 0.
of the form
( ;; g) (; TJ) (7: g) (; TJ) +Z(t) (� TJ) ·
=
Y(t) cos
=
(2), ( 3),
(5).
For instance, of
(0,0,0) is an equilibrium point,
for every value
r. This equilibrium is stable (attracting) when r < 1, cor
responding to the stability of the steady-state solution. As
r crosses the value 1, the origin becomes unstable,
and two
new stable equilibrium points P 1 and P 2 arise. They corre spond to the stationary solutions given by one further increases
r,
( 4)
and
(5). If
these two solutions become un
stable. This suggests that, for large Rayleigh number, the convection motion described by
( 4) and (5), is replaced by
some different form of dynamics. Let me also comment on the particular choice of para
Nonperiodic behavior
Aiming to understand what happens when
D.T is further in
creased, Saltzmann looked for more general solutions space-periodic in both dimensions. For this, he expanded
'I'
and
8
as formal Fourier series in the variables
g and TJ,
with time-dependent coefficients. Inserting this formal ex pansion into
(2), ( 3),
(1). Saltzmann took a = 1/ v2, which is the a for which R c is smallest. This gives b = 8/3. The Prandtl number cr = 10 is typical of liquids (cr = 4. 8 for water); for air cr 1 . Finally, the relative Rayleigh num ber r = 2 8 is just slightly larger than the transition value r = 2 4. 7368 at which the two equilibria P 1, P 2 become meter values in
one fmds an infmite system of ordi
value of
=
unstable.
nary differential equations, with the Fourier coefficients as unknowns. Saltzmann truncated the system, keeping only
Geometric Lorenz Models
a finite number of these equations.
Here is an outline of the main facts about the geometric
He tested several possibilities, but one particular case, involving seven equations, was especially interesting. Nu-
Lorenz models in
[ 1 , 8, 44],
that are also relevant for the
next two sections.
VOLUME 22, NUMBER 3, 2000
11
r
:::J
)P(I\f)
c: I
Figure 6. A geometric Lorenz flow.
Figure 8. The image of the first-return map P.
Equilibrium point
A first condition in the definition of the geometric models
is that the flow should have an equilibrium point 0. If X de notes the vector field associated to the flow, the derivative
DX(O) should have one positive eigenvalue A1 > 0 and two negative eigenvalues -A2 < -A3
1 such that dist(ftx), .fty))
:.::::
T
dist(x, y)
(7)
for any two points x, y located on the same side of the dis continuity point c. It is interesting to point out that Lorenz computed such a map numerically for (1) in [17). The cross-section he was considering (implicitly) is not quite the same as our I, so that he got a seemingly different picture, shown on the right-hand half of Figure 10. Nevertheless, the information provided by either of the two maps is equivalent. A sensitive attractor
Under these assumptions, [1] and [8, 44) prove that the flow exhibits a strange attractor A, that contains the equilibrium point 0. The attractor is the closure of the set of trajecto ries that intersect the cross-section I infinitely many times in the past (as well as in the future). Any forward trajec tory that cuts I accumulates in A, and these form a whole neighborhood of the attractor.
r
1'2
'}'1
P( 'YI) I
I
X
c
Figure 9. Definition of the interval map f.
f(x)
c
c Figure 10. Interval maps related to Lorenz flows.
Moreover, these trajectories are sensitive with respect to initial conditions. Indeed, suppose you are given two nearby points z and w, whose forward flow trajectories in tersect I. Typically, the intersection points will be in two different leaves Yz and Yw of the invariant foliation (corre sponding to nearby points x =I= y in /). The next time the two flow trajectories come back to I, the new intersection points will be in leaves y� and y;_,, corresponding to the points.ftx) and .fty) in I. Now, because of the expansive ness property (7), the distance dist( y�, y;_,) is larger than dist( Yz, Yw). Thus, the distance between the two flow tra jectories at successive intersections with I keeps increas ing; one can check that it eventually exceeds some uniform lower bound that does not depend on how close the initial points z and w are.9 Another important conclusion is that the attractor con tains dense orbits (dynamical indecomposability): there ex ists some z0 E A whose forward trajectory visits any neigh borhood of any point of A. (For this one requires the constant r in (7) to be larger than V2.) In particular, the trajectory of z0 accumulates on the equilibrium point 0. Lorenz models without invariant foliations
Numerical investigations of the Lorenz equations carried out by M. Henan andY. Pomeau [12, 13], showed that when the relative Rayleigh number increases beyond r 30 the image of the first-return map develops a "hooked" shape, described in Figure 11. This indicates that for such para meter values there is no longer an invariant foliation, as as sumed in the geometric Lorenz models. As a simple, easy to-iterate model of the behavior of the first-return map near the "bends," they introduced the family of maps of the plane =
(x, y) � (1 - a:i2
-
y,
bx),
(8)
that is now named after Henon. Based on their computations, and especially on Chapter 5 of Sparrow's book [39], S. Luzzatto and I proposed an ex tended geometric model for Lorenz equations, including the creation of the hooks. The main result [21, 20) is that
the attractor survives the destruction of the invariant fo liation, at the price of losing its robustness: after the hooks are formed, a strange attractor exists for a positive-
90f course, the growth must stop at some point: these distances cannot exceed the order of magnitude of the attractor's diameter.
VOLUME 22. NUMBER 3, 2000
13
r
A=
{x :
'Pt(x) E U for all t E IR}.
Secondly, given any vector field Y close to the original one X, the set
Ay : =
{x : 'f' Kx)
E U for all t E IR}
of 'P �-trajectories that never leave U is dynamically inde composable for the flow 'P � associated to Y. Here close ness means that the two vector fields X and Y, and their first derivatives, are uniformly close over M. Finally, A is a robust attractor if the neighborhood U can be chosen to be trapping (or forward invariant): 'Pt(U) c U for all t > 0. Hyperbolicity
Figure 1 1 . Hooked return maps in Lorenz equations.
Lebesgue-measure set of parameter values. This set is nowhere dense, which is related to the lack of robustness: near the parameter values for which the strange attractor exists there are others corresponding only to attracting pe riodic orbits. Moreover, the conclusions of [2 1], which treats a version of the problem for interval maps, have been further extended by Luzzatto and Tucker in [19].
Now I define when an invariant set A is hyperbolic. At each point x of A it should be possible to decompose the tan gent space into three complementary directions (sub spaces)
depending continuously on x, such that •
•
Robust Strange AHractors
In order to state the results of Morales, Pacifico, Pujals, mentioned before, I need to introduce the precise defini tion of a robust attractor. It makes sense also for discrete time systems (diffeomorphisms, or just smooth transfor mations), but here I restrict myself to flows.
1. qP(x) = x for every x E M, and 2. 'fit qf(x) = 'Pt+s(x) for all x E M and o
s, t E IR.
Denote by X the associated vector field on M, which is de fined by
D'Pt E/1 = E�'Cx)
is dense in A. This property is often called transitivity in the specialized literature. An invariant set A is called robust if it admits a neigh borhood U such that the following two conditions are sat isfied. Firstly, A consists of the points whose trajectories under 'fit never leave U:
14
THE MATHEMATICAL INTELUGENCER
and
D'Pt E!f = Eif'Cx)
and there are constants C > 0 and A > 0 such that
IID'Pt
I E/1 11 ::s ce - At
and
IICD'Pt
I EJ;')- 111 ::s ce - At
for all t > 0 and all x E A. I already mentioned the following important conse quence, described in Figure 2. If an invariant set A is hy perbolic, in the sense of the previous paragraph, then every point x in it is contained in a pair of local sub-manifolds of M, the stable manifold WS(x) and the unstable manifold WU(x) such that •
We say that a subset A of M is invariant if trajectories starting in A remain there for all times: if x E A then 'Pt(x) E A for all t E IR. In what follows I always consider compact invariant sets. An invariant set A is dynamically indecomposable if there exists some point x E A whose forward orbit
tion of the vector field X at x; the linearized flow ll'f't preserves the subbundles E8 and Eu (this is clear for Ffl); moreover, it contracts E8 and expands Eu exponentially fast.
This last condition means that
Robust invariant sets
Let q/ : M � M, t E IR, be a flow on a manifold M, that is, a one-parameter group of diffeomorphisms satisfying
EJ is the one-dimensional subspace given by the direc
•
if y is in W8(x), then d('f!t(x), 'Pt(y)) :::; ce - At, for every t > 0; if z is in wu(x), then d('f!-t(x), 'P-t(z)) :::; ce- At, for every t > 0.
The existence of these manifolds also determines the local behavior of solutions close to the one passing through x. So far, I have implicitly assumed that x is a regular point of the flow, that is, X(x) is not zero. When, on the contrary, x is an equilibrium point, then the definition has to be re formulated: in this case, the directions Ef:; and E!f; alone must span the tangent space:
Figure 7 describes stable and unstable manifolds through an equilibrium point.
Observe that if x is an equilibrium then dim EJ
+ dim EJ:: =
dim M,
whereas in the regular case dim Ei: + dim EJ:: = dim M
-
1.
This has a simple, yet important, consequence: an invariant indecomposable set containing equilibria is never hyperbolic (except in the trivial case when it consists of a unique point). This is because the dimension of either Ei: or Ei'would have to have a jump at the equilibrium, contradicting the require ment of continuity. In particular, the geometric Lorenz at tractors that I described above can not be hyperbolic. Singular hyperbolicity
Morales, Pacifico, and Ptijals propose a notion of singular hyperbolic set, which plays a central role in their results. Let A be an invariant set for a flow q/. We say that A is singular hyperbolic if at every point x E A there exists a decomposition TxM Ex EB Fx of the tangent space into two subpsaces Ex and Fx such that the linear flow con tracts Ex exponentially, and is exponentially volume ex panding restricted to Fx: =
-
det (D
0. The decomposition must depend continuously on the point x. Note that the linear flow is allowed to ei ther expand or contract Fx, or both. However, we also re quire that whatever contraction there is in this direction, it should be dominated by the one in the Ex direction:
IID'Pt(x)e[[ IID'Pt(x)f[[
-
