The Chaos Avant-Garde: Memoirs of the Early Days of Chaos Theory

THE CHAOS AYBNT-GAROE MEMORIES OF THE E ~ 0 ~~ OF~L~ H5~~ THE D SOR^

WORLD SCIENTIFIC SERIES ON NONLI~EARSCIENCE Editor: Leon 0.Chua University of California, Berkeley Series A. Volume 23:

MONOGRAPHS AND TREATISES

Nonlinear Dynamics in Particle Accelerators R. Diliio & R. Alves-Pires Volume 24: From Chaos to Order G. Chen & X. Dong Volume 25: Chaotic Dynamics in Hamiltonian Systems H. D a ~ k o w ~ ~ Volume 26: Visions of Nonlinear Science in the 21st Century J. L. Huertas, W.-K. Chen & R. N. Madan Volume 27: The Thermornechanics of Nonlinear Irreversible BehaviorsAn Introduction G. A. Maugin Volume 28: Applied Nonlinear Dynamics & Chaos of Mechanical Systems with Discontinuities Edited by M. Wercigroch & B. de Kraker Volume 29: Nonlinear & Parametric Phenomena* V. Damgov Volume 30: Quasi-ConservativeSystems: Cycles, Resonances and Chaos A. D. Morozov Volume 31: CNN: A Paradigm for Complexity L. 0, Chua Volume 32: From Order to Chaos II L. P. Kadanoff Volume 33: Lectures in Synergetics V. 1. Sugakov Volume 34: Introduction to Nonlinear Dynamics* I. Kocarev& M. P. Kennedy Volume 35: Introduction to Control of Oscillations and Chaos A. L. Fradkov & A. Yu. Pogromsky Volume 36: Chaotic Mechanics in Systems with Impacts & Friction B. Blazejczyk-Okolewska,K. Czolczynski, T. Kapitaniak & J. Wojewoda Volume 37: Invariant Sets for Windows - Resonance Structures, Attractors, Fractals and Patterns A. D. Morozov, T. N. Dragunov, S. A. Boykova & 0. V. Malysheva Volume 38: Nonlinear Noninteger Order Circuits & Systems -An Introduction P. Arena, R, Caponetto, L. Fortuna & D. Port0

*Forthcoming

--

-

Series Editor: Leon 0. Chua

MEMORIES OF THE EARLY DAYS OF ~ H ~ THE O SOR^ Editors

Ralph Abraham University of California, Santa Cruz

Y ~ s h i s ~ kUeda e Kyoto Unjversi~

World Scientific Singapore 'New Jersey. London *HongKong

Published by

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THE CHAOS AVANT-GARDE Memories of the Early Days of Chaos Theory

Copyright 02000 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereof; may not be reproduced in anyform or by any means, electronic or mechanical. including photocopying, recording or any information storage and retrieval system now known or to be invented, without wrigen p e ~ ~ s sfrom ~ o nthe Publisher.

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ISBN 981-02-4404-5

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In 1998 we decided to put together this volume of memoirs of chaos pioneers, before they disappear. After some thought, we decided to invite only the avant-garde, the people who had struggled with chaos concepts before the acceptance of the new paradigm. We limited the time span of this period to 1959-1974. The terminus was the attachment of the word chaos to dynamical systems theory, early in 1974, by Li and Yorke, which coincided with the beginning of Rdssler's writings on the significance of the word chaos in this context. We wrote to these individuals in the Spring of 1999, almost all responded with a memoir or two, which are published here, many for the first time. One exception is Ed Lorenz, who did not respond to our letter. We have included a short piece by him, taken from his own book, The Essence of Chaos. Another is David Ruelle, who replied, "I now leave it to other people to think and write about the early days of chaos theory." Finally, Robert May declined, due to pressures on his time. The order of the pieces in this collection is our rough approximation to the actual sequence of events. Various permutations might have been chosen equally. We apologize to the many pioneers not represented here. We hope that this volume is just the first of a sequence. Finally, we are grateful to Professors Akamatsu and Inagaki, former students of Professor Ueda, for suggesting this project. Ralph Abraham Yoshisuke Ueda Santa Cruz, March 6,2000

V

Contents Preface ...................................................................................................... 1. On How I Got Started in Dynamical Systems 1959-1962 Steve Smale

v

...............1

2. Finding a Horseshoe on the Beaches of Rio ..................................... Steve Smale

7

3. Strange Attractors and the Origin of Chaos .................................... Yoshisuke Ueda

23

4. My Encounter with Chaos .............................................................. Yoshisuke Ueda

57

5 . Reflections on the Origin of the Broken-Egg Chaotic Attractor ....65

Yoshisuke Ueda

.......................................

81

7. The Butterfly Effect ........................................................................

91

6. The Chaos Revolution: A Personal View Ralph Abraham Edward Lorenz

8. I. Gumowski and a Toulouse Research Group in the

“Prehistoric” Times of Chaotic Dynamics ...................................... Christian Mira

9. The Turbulence Paper of D. Ruelle and F. Takens Floris Takens

vii

.......................

95 199

viii

Contents

10. Exploring Chaos on an Interval .................................................... I: !I Li and James A. Yorke

201

11. Chaos, Hyperchaos and the Double-Perspective .......................... Otto E. Riissler

209

.................................................................................................

221

Sources

1. On How I Got Started in Dynamical Systems 1959-1962* Steve Smale City University of Hong Kong

1. Let me first give a little mathematical background. This is conveniently divided into two parts. The first is the theory of ordinary differential equations having a finite number of periodic solutions; and the second has to y solutions, or, roughly speaking, with do with the case of i n ~ n i t e ~many "homoclinic behavior." For an o r d i n a ~differential equation in the plane (or a second-order equation in one variable), generally there are a finite number of periodic solutions. The PoincarC-Bendixson theory yields substantial information. If a solution is bounded and has no equilibria in its limit set, its asymptotic behavior is periodic. The Van der Pol equation without forcing is an outstanding example which has played a large role historically in the qualitative theory of ordinary differential equations. There was some systemjzation of this theory by Andronov and Pontryagin in the 1930s when these scientists introduced the notion of structural stability. There is a school now in the Soviet Union, the "Gorki school" which reflects this tradition. Andronov's wife, Andronova Leontevich, was a member of this school, and I met her in Kiev in 1961 (Andronov had died earlier). A fine book ~transl~ted into Engijsh~by Andronov, Chaiken, and Witt gives an account of this mathematics. The work of Andronov-Pontryagin was picked up by Lefschetz after World War 11. Lefschetz's book and influence helped establish the study of structural stability in America. This essay is partly based on a talk given at a Berkeley seminar circa 1976. 1

2

Chaos Avant-Garde:Memories ofthe Early Days ofChaos Theory

The second part of the background mathematics has to do with the phenomenon of an ~nfinitenumber of periodic solutions (persistent under perturbation) or the closely related notion of homoclinic solutions. Again, Poincard wrote on these problems; Birkhoff found a deeper connection between the two concepts, hom~ciinicso~utionsand an i n ~ n i t en u ~ b e of r periodic points for transformations of the plane. From a different direction, ~ a ~ w r i g and h t Littlewood, in their extensive studies of the Van der Pol equation with forcing term, came across the same phenomena. Then Levinson simp1ifed some of this work. 2. Next, I would like to give a little personal back~round.I ~ n ~ s h emy d thesis in topology with Raoul Bott in 1956 at the University of Michigan; and that summer I attended my first mathemat~csconference, in Mexico City, with my wife, Clara. It was an international conference in topology and my introduction to the international mathematics community. There I met Rene Thom and two graduate students from the Uniyersity of Chicago, Moe Hirsch and EIon Lima. Thom was to visit the University of Chicago that fall, and I was start~ngmy first teaching job at Chicago then {not in the mathematics depa~ment,but in the College). In the fall I became good friends with all three. li attended Thorn's lectures on transversality theory and was happy that Thorn and Wirsch became interested in my work on ~mmers~on theory. My main interests were in topoiogy throughout these years, but already my thesis contained a section on ordinary differential equations, Also at Chicago, I used an ordinary differential equation argument to find the homotopy structure of the space of diffeormorphism~of the 2-sphere. Having both Dick Palais and Shlomo Sternberg at Chicago was very helpfu~ in getting some understanding of dynamical systems.

3, It was around 1958 that I first met ~ a u r i e i oPeixoto. We were introduced by Lima who was finishing his Ph.D. at that time with Ed Spanier. Through Lefschetz, Peixoto had become interested in structural stability and he showed me his own results on structural stability on the disk D2(in a paper which was to appear in the Annals of Mathematics, 1959). 1 was i m ~ e d ~ a t e ienthusiastic, y not only about what he was doing, but with the possibility that, using my topology background, I could extend his work to n dimensions. I was extremely naive about ordinary differential equations at that tirne and was also extremely presumptuous. Peixoto told me that he had

On How f Got Started in ~ ~ m iSystems c ~ l

3

met Pontryagin, who said that he didn't believe in structural stability in dimensions greater than two, but that only increased the challenge. In fact, I did make one contribution at that time. Peixoto had used the condition on D'of Andronov and P o n t ~ a g i nthat Ifno solution joins saddles." This was a necessary condition for structural stability. Having learned about transversality from Thom, I suggested the generalization for higher dimensions: the stable and unstable manifolds of the equilibria (and now also of the nontrivial periodic solutions) intersect transversally. In fact, this was a useful condition, and I wrote a paper on Morse inequalities for a class of dynamical systems incorporating it. However, my overenthusiasm led me to suggest in the paper that these systems were almost all (an open dense set) of ordinary differentia1 equations! If I had been at all familiar with the literature (Poincard, Birkhoff, Ca~wright-Littlewood),I would have seen how crazy this idea was. On the other hand, these systems, though sharply limited, would find a place in the literature, and were christened Morse-Smale dynamical systems by Thom.This work gave me entry into the mathematical world of ordinary differential equations. In this way, I met Lefschetz and gave a lecture at a conference on this subject in Mexico City in the summer of 1959. We had moved from Chicago in the summer of 1958 to the Institute for Advanced Study in Princeton; Peixoto and Lima invited me to Rio to finish the second year of my NSF postdoctoral fellowship. Thus, Clara and I and our two kids, Nat and Laura, left Princeton in December, 1959, for Rio. 4. With kids aged 112 and 2-112, and most of our luggage consisting of

diapers, Clara and I stopped to see the Panamanian jungles (from a taxi). I had heard about and always wanted to see the famous railway from Quito in the high Andes to the jungle port of Guayaquil in Ecuador. So, around Christmas of 1959, the four of us were on that train. Then we spent a few days in Lima, Peru, during which we were all quite sick. We finally took the plane to Rio, I still remember well arriving at night and going out several times trying to get milk for the crying kids, always returning with cream or yogurt, etc. We learned later that milk was sold only in the morning in Rio. But with the help of the Limas, the Peixotos, and the Nachbins, life got straightened out for us. In fact, it happened that just before we arrived the leader of an abortive coup, an air force officer, had escaped to Argentjna. We got his luxurious Copacabana apartment and maids as well.

4

Chaos Avant-Garde: Memories offhe Early Days of Chaos Theory

Shortly after arriving in Rio my paper on dynamical systems appeared, and Levinson wrote me that one couldn't expect my systems to occur so generally. His own paper (which, in turn, had been inspired by work of Cartwright and Littlewood) already contained a counter-example. There were an infinite number of periodic solutions and they could not be perturbed away. Still partly with disbelief, I spent a lot of time studying his paper, eventually becoming convinced. In fact this led to my second result in dynamical systems, the horseshoe, which was an abstract geometrization of what Levinson and Cartwright-Littlewood had found more analytically before. Moreover, the horseshoe could be analysed completely qualitatively and shown to be structurally stable. For the record, the picture I abstracted from Levinson looked like this:

When I spoke on the subject that summer (1960) at Berkeley, Lee Neuwirth said: "Why don't you make it look like this?"

On How I Got Started in Dynamical Systems

5

I said "fine" and called it the horseshoe. I still considered myself mainly a topologist, and when considering some questions of gradient dynamical systems, I could see possibilities in topology. This developed into the "higher-dimensional PoincarC conjecture" and was the genesis of my being quoted later as saying I did my best known work on the beaches of Rio. In fact, I often spent the mornings on those beaches with a pad of paper and a pen. Sometimes Elon Lima was with me. In June I flew to Bonn and Zurich to speak of my results in topology. This turned out to be a rather traumatic trip, but that is another story. I had accepted a job at Berkeley (at about the same time as Chern, Hirsch, and Spanier, all from Chicago) and arrived there from Rio in July, 1960. Except for a few lectures on "the horseshoe," I was preoccuped the next year with topology. But, in the summer of 1961, I announced to my friends that I had become so enthusiastic about dynamical systems that I was giving up topology, The explicit reason I gave was that no problem in topology was as important and exciting as the topological conjugacy problem for diffeomorphisms, already on the 2-sphere. This conjugacy problem represented the essence of dynamical systems, I felt. 5. During this year 1 had an irresistible offer from Columbia University, so

we sold a house we had just bought and moved to New York in the summer of 1961. But before taking up teaching duties at Columbia, I spoke at a conference on ordinary differential equations in Colorado Springs and then in September, 1961, went to the Soviet Union. At a meeting on nonlinear oscillations in Kiev, I gave a lecture on the horseshoe example, "the first structurally stable dynamical system with an infinite number of periodic solutions." I had a distinguished translator, the topologist, Postnikov, whom I had just met in Moscow. Postnikov agreed to come to Kiev and translate my talk in return for my playing go with him. He said he was the only go player in the Soviet Union. My roommate in Kiev was Larry Markus. 1 met and saw much of Anosov in Kiev. Anosov had followed the Gorki school, but he was based in Moscow. After Kiev I went back to Moscow where Anosov introduced me to Arnold, Novikov, and Sinai. I must say I was extraordinarily impressed to meet such a powerful group of four young mathematicians. In the following years, I often said there was nothing like that in the West.

6

Chaos Avant-Garde: Memories of the Early Days of Chaos Theory

1 gave some lectures at the Steklov Institute and made some conjectures on the structural stability of certain toral diffeomorphisms and geodesic flows of negative curvature. After 1 had worked out the horseshoe, Thom brought to my attention the toral diffeomorphisms as an example with an infinite number of periodic points which couldn't be perturbed away. Then 1 had examined the stable manifold structure of these dynamical systems.

6. After teaching dynamical systems the fall semester at Columbia, I was off again, with Clara and the kids, this time to visit AndrC Haefliger in Lausanne for the spring quarter. Besides lecturing in Lausanne, 1 gave lectures at the College de France, Urbino, Copenhagen, and, finally, Stockholm, all on dynamical systems and emphasizing the global stable manifolds, which I found more and more to lie close to the heart of the subject. In Stockholm, at the International Congress, 1 saw Sinai again and he told me that Anosov had proved all the conjectures I had made the preceding year in the Soviet Union. In Lausanne I had begun to start thinking about the calculus of variations and infinite-dimensional manifolds, and this preoccupation took me away from dynamical systems for the next three years.

2.

Finding a Horseshoe on the Beaches of Rio Steve Smale City U n j v e r s o~f~Hong Kong

This is an expanded version of a paper to appear in the proceedings of the ~ n t e r n ~ t Congress ~ ~ n u ~ of Science and Technology-45 years ofthe ~ a ~ i ~ n u l Research Council of Brazil.

What is Chaos? A ma~ematiciandiscussing chaos is featured in the movie Jurassic Park. James Gleick's book Chaos remains on the best seller list for many months. The characters of the celebrated Broadway play Arcadia of Tom Stoppard discourse on the meaning of chaos. What is chaos? Chaos is a new science which establishes the omnipresence of unpredictability as a f u n d ~ e n t afeature l of common experience, A belief in determinism, that the present state of the world determines the future precisely, dominated scientific thinking for two centuries. This credo was based on certain laws of physics, Newton's equations of motion, which describe the trajectories in time of states of nature. These equations have the mathematical property that the initial condition determines the solution for all time. Thus lies the mathematical and physical foundation for deterministic philosophy. One manifestation of determinism was the rejection of free will and hence even of human responsibility. At the beginning of this century, with the advent of quantum mechanics and the revelations of the German scientists, Heisenberg, Planck, and Schrbdinger, the great delusion of determinism was exposed. At least on the level of electrons, protons, and atoms it was discovered that uncertaint~ 7

8

Chaos Avant-Garde: Memories of the Early Days of Chaos Theory

prevailed. The equations of motion of quantum mechanics produce solutions which are probabilities evolving in time. In spite of quantum mechanics, Newton's equations govern the motion of a pendulum, the behavior of the solar system, the evolution of the weather and many situations of everyday life. Therefore the quantum revolution left intact many deterministic dogmas. For example, well after the Second World War, scientists held the belief that long range weather prediction would be successful when computer resources grew large enough. In the 1970s the scientific community recognized another revolution, called the theory of chaos, which deals a death blow to the Newtonian picture of determinism. As a consequence, the world now knows that one must deal with unpredictability in understanding common experience. The coin-flipping syndrome is pervasive. "Sensitive dependence on initial conditions'' has become a catchword of modern science. Chaos contributes much more than extending the domain of indeterminacy, just as quantum mechanics did so more than half a century earlier. The deeper understanding of dynamics underlying the theory of chaos has shed light on every branch of science. Its accomplishments range from an analysis of electrocardiograms to aiding the construction of computational devices. Chaos developed not with the discovery of new physical laws, but by a deeper analysis of the equations underlying Newtonian physics. Chaos is a scientific revolution based on mathematics. Deduction rather than induction is the methodology. Chaos takes the equations of Newton, analyses them with mathematics, and uses that analysis to establish the widespread unpredictability in the phenomena described by those equations. Via mathematics, one establishes the failure of Newtonian determinism by using Newton's own equations!

Taxpayers Money In 1960 in Rio de Janeiro I was receiving support from the National Science Foundation (NSF) of the United States as a postdoctoral fellow, while doing research in an area of mathematics which was to become the theory of chaos. Subsequently, questions were raised about my having used U.S. taxpayers' money for this research done on the beaches of Rio. In fact none other than

Finding a Horseshoe on the Beaches of Rio

9

President Johnson's science adviser, Donald Hornig, wrote on this issue in 1968 in the widely circulated magazine Science: This blythe spirit leads mathematicians to seriously propose that the common man who pays the taxes ought to feel that mathematical creation should be supported with public funds on the beaches of Rio . . . What happened during the passage of time from the work on the beaches to this national condemnation? This was the era of the turbulent '60s in Berkeley where I was a professor; my students were arrested, tear gas frequently filled the campus air, dynamics conferences opened under curfew; Theodore Kaczynski, the suspected Unabomber, was a colleague of mine in the math department. The Vietnam War was escalated by President Johnson in 1965, and I was moved to establish with Jerry Rubin a confrontational anti-war force. Our organization, the Vietnam Day Committee (VDC), with its teach-ins, its troop train demonstrations and big marches, put me onto the front pages of the newspapers. These events led to a subpoena by the House Unamerican Activities Committee (HUAC), which was issued while I was enroute to Moscow to receive the Fields Medal (the main prize in mathematics) in 1966. The subsequent press conference I held in Moscow attacking U. S. policies in the Vietnam War (as well as Russian intervention in Hungary) created a long lasting furore in Washington, DC. Now we are going to see what actually happened in that spring of 1960 on those beaches of Rio de Janeiro.

Flying Down to Rio Topology is the part of mathematics which is sometimes nicknamed "rubber sheet geometry" since a topologist is allowed to bend rigid objects. In the 1950s there was an explosion of ideas in this subject. Topology, with its developments dominating all of mathematics, caught the imagination of many young research students such as myself. I finished a Ph.D. thesis in that domain at the University of Michigan in Ann Arbor in 1956. During that summer I, with my wife, Clara, attended in Mexico City, a topology conference reflecting this great movement in mathematics with the world

10

Chaos Avant-Garde: Memories of the Early Days of Chaos Theory

wide notables in topology present and giving lectures. There I met a Brazilian graduate student, Elon Lima, writing a thesis in topology at the University of Chicago, and as I was about to take up the position of instructor at that university, I became good friends with Elon. A couple of years later, Elon introduced me to Mauricio Peixoto, a young visiting professor from Brazil. Mauricio was from Rio, although he came from a northern state of Brazil where his father had been governor. A good humoured pleasant fellow, Mauricio, in spite of his occasional bursts of excitement was conservative in his manner and in his politics. As was typical for the rare mathematician working in Brazil at that time, he was employed as teacher in an engineering college. Mauricio also helped found a new institute of mathematics (IMPA) and his aspirations brought him to America to pursue research in 1957. Subsequently he was to become the President of Brazilian Academy of Sciences. Mauricio was working in the subject of differential equations or dynamics and showed me some beautiful results. Before long I myself had proved some theorems in dynamics. In the summer of 1958, Clara and I with our newly born son Nat, moved to the Institute for Advanced Study (IAS) in Princeton, New Jersey. This was the locale made famous as Einstein's workplace in America, with Robert Oppenheimer as its director. I was supposed to spend two years there with an NSF postdoctoral fellowship. However, due to our common mathematical interests, Mauricio and Elon invited me to finish the second year in Rio de Janeiro. So Clara and I and our children, Nat and newly arrived Laura, left Princeton in December, 1959, to fly down to Rio. The children were so young that most of our luggage consisted of diapers, but nevertheless we were able to realize an old ambition of seeing Latin America. After visiting the Panamanian jungle, the four of us left Quito, Ecuador, Christmas of 1959, on the famous Andean railroad down into the port of Guayaquil. Soon we were flying into Rio de Janeiro, recovering from sicknesses we had acquired in Lima. I still remember vividly, arriving at night, going out several times trying to get milk for our crying children, and returning with a substitute as cream or yogurt. We later learned that, in Rio, milk was sold only in the morning, on the street. At that time Brazil was truly part of the "third world." However our friends soon helped us settle down into Brazilian life. We arrived in Brazil just after a coup had been attempted by an air force colonel. He fled the country to take refuge in Argentina, and we were able to rent,

Finding a Horseshoe on the Beaches ofRio

11

from his wife, his luxurious 1 1-room apartment in the district of Rio called Leme. The U.S. dollar went a long way in those days, and we even were able to hire the colonel's two maids, all with our fellowship funds. Sitting in our upper story garden veranda we could look across to the hill of the favela (called Babylonia) where Black Orpheus was filmed. In the hot humid evenings preceeding Carnaval, we would watch hundreds of the favela dwellers descend to samba in the streets. Sometimes I would join their wild dancing which paraded for many miles. In the front of our apartment, the opposite direction from the hill, lay the famous beach of Copacabana. I would spend my mornings on that wide, beautiful, sandy beach, swimming and body surfing. Also I took a pen and paper and would work on mathematics.

Mathematics on the Beach Very quickly after our arrival in Rio, I found myself working on mathematical research. My host institution, Instituto da Matematica, Pura e Aplicada (IMPA), funded by the Brazilian government, provided a pleasant office and working environment. Just two years earlier IMPA had established itself in its own quarters, a small colonial building in the old section of Rio called Botafogo. There were no undergraduates and only a handful of graduate mathematics students. There were also a very few research mathematicians, notably Peixoto, Lima, and an analyst named Leopoldo Nachbin. Also there was a good math library. But no one could have guessed that in less than three decades IMPA would become a world center of dynamical systems housed in a palatial building as well as a focus for all of Brazilian science. In a typical afternoon I would take a bus to IMPA and soon be discussing topology with Elon, dynamics with Mauricio or be browsing in the library. Mathematics research typically doesn't require much, the most important ingredients being a pad of paper and a ballpoint pen. In addition, some kind of library resources, and colleagues to query are helpful. I was satisfied. Especially enjoyable were the times spent on the beach. My work was mostly scribbling down ideas and trying to see how arguments could be put together. Also I would sketch crude diagrams of geometric objects flowing through space, and try to link the pictures with formal deductions. Deeply

12

Chaos Avant-Garde: Memories oJthe Early Days of Chaos Theory

involved in this kind of thinking and writing on a pad of paper, the distractions of the beach didn't bother me. Moreover, one could take time off from the research to swim. The surf was an exiting challenge and even sometimes quite frightening. One time when Lima visited my ''beach office," we entered the surf and were both caught in a current which took us out to sea. While Elon felt his life fading, bathers shouted the advice to swim parallel to the shore to a spot where we were able to return. [It was 34 years later just before Carnaval, that once again those same beaches almost did me in. This time a special wave bounced me so hard on the sand it injured my wrist, tore my shoulder tendon, and then that same big wave carried me out to sea. I was lucky to get back using my good arm.]

Letter from America At that time, as a topologist, I prided myself on a paper that I had just published in dynamics. I was delighted with a conjecture in that paper which had as a consequence (in modern terminology) "chaos doesn't exist"! This euphoria was soon shattered by a letter I received from an M.I.T. mathematician named Norman Levinson. He had coauthored the main graduate text in ordinary differential equations and was a scientist to be taken seriously. Levinson wrote me of an earlier result of his which effectively contained a counterexample to my conjecture. His paper in turn was a clarification of extensive work of the pair of British mathematicians Mary Cartwright and J. L. Littlewood done during World War 11. Cartwright and Littlewood had been analyzing some equations that arose in doing war-related studies involving radio waves. They had found unexpected and unusual behavior of solutions of these equations. In fact Cartwright and Littlewood had proved mathematically that signs of chaos could exist, even in equations that arose naturally in engineering. But the world wasn't ready to listen, and even today their important contributions to chaos theory are not well-known. I never met Littlewood, but in the mid-sixties, Dame Mary Cartwright who was head of a women's college (Girton) at Cambridge invited me to high table. I worked day and night to try to resolve the challenge to my beliefs that the letter posed. It was necessary to translate Levinson's analytic arguments

Finding a Horseshoe on the Beaches of Rio

13

into my own geometric way of thinking. At least in my own case, understanding mathematics doesn't come from reading or even listening. It comes from rethinking what I see or hear. I must redo the mathematics in the context of my particular background. And that background consists of many threads, some strong, some weak, some algebraic, some visual. My background is stronger in geometric analysis, but following a sequence of formulae gives me trouble. I tend to be slower than most mathematians to understand an argument. The mathematical literature is useful in that it provides clues, and one can often use these clues to put together a cogent picture. When I have reorganized the mathematics in my own terms, then I feel an understanding, not before. Through these kinds of thought processes, I eventually convinced myself that indeed Levinson was correct, and that my conjecture was wrong. Chaos was already implicit in the analyses of Cartwright and Littlewood. The paradox was resolved; I had guessed wrongly. But while learning that, I discovered the horseshoe!

The Horseshoe The horseshoe is a natural consequence of a geometrical way of looking at the equations of Cartwright-Littlewood and Levinson. It helps understand the mechanism of chaos, and explain the widespread unpredictability in dynamics. Chaos is a characteristic of dynamics, and dynamics is the time evolution of a set of states of nature. Thus one can think of dynamics as solutions of the equations of motion as given for example by Newton. While time is usually considered as a continuous entity we will suppose that time is measured in discrete units as seconds or minutes. A state of nature will be idealized as a point in the two dimensional plane. We will start by describing a non-chaotic linear example. The idea is to take a square, Figure 1, and to study what happens to a point on this square in one unit of time, under a transformation to be described. The vertical dimension is now shrunk uniformly towards the center of the square and the horizontal is expanded uniformly at the same time. Using dots to outline the domain obtained by this process superimposed over the original square yields Figure 2.

Chaos Avant-Garde: Memories of fhe Early Days of Chaos Theory

14

A

B

C

D Figure 1

B

A

D

C

Figure 2

Here superscripts are used to denote the motion so that the corner A moves to A*. We have also shaded in the set of points which don't move out of the square in this process.

15

Finding a Horseshoe on the Beaches of Rio

The second of our three stages in understanding is the perturbed linear example. Now the square is moved into a bent version of the elongated rectangle of Figure 2. Thus Figure 3 describes the motion of our square obtained by a small modification of Figure 2.

B

A

A',

.............................. :. ......... .................

C*

C

D

Figure 3

The horseshoe is the fully nonlinear version of what happens to points on the square, by an extension of the process expressed in Figures 2 and 3. This is the situation when motion is expressed by a qualitative departure from the linear model. See Figure 4. The horseshoe is the domain surrounded by the dotted line. Instead of a state of nature evolving according to a mathematical formula, the evolution is given geometrically. The full advantage of the geometrical point of view is beginning to appear. The more traditional way of dealing with dynamics was with the use of mathematical, e.g. algebraic, expressions. But a description given by formulae would be cumbersome. It would unlikely lead to insights or to a perceptive analysis, since that form of a description wouldn't communicate as efficiently the information in the figure. My background as a topologist, trained to bend objects as squares helped to make it possible to see the horseshoe.

Chaos Avant-Garde: Memories of the Early Days of Chaos Theory

16

B

A

D

C

Figure 4

The dynamics of the horseshoe is described by moving a point in the square to a point in the horseshoe according to Figure 4. Thus the corner marked A moves to the point marked A* in one unit of time. The motion of a general point x in the square is a sequence of points xo XI, x2 , . . Here xo = x is the present state, XI is that state a unit of time later, x2 that state two units of time later, etc. Now imagine our visual field to be just the square itself. When a point is moved out of the square we will discard that motion. Figure 5 shades in the points which don't leave the square in one unit of time. We will call a motion which never leaves the square a visual motion. Our results in the next section concern visual motions. In summary, a fully nonlinear motion finds its realization in the horseshoe. In the next section we will see the consequences to chaos that this picture carries.

17

Finding a Horseshoe on the Beaches of Rio

B

A

D

C

Figure 5

The Horseshoe and Chaos: Coin Flipping The laws of chance, with good reason, have traditionally been expressed in terms of flipping a coin. Guessing whether heads or tails is the outcome of a coin toss is the paradigm of pure chance. On the other hand it is a deterministic process that governs the whole motion of the coin, and hence the result, heads or tails, depends only on very subtle factors of the initiation of the toss. This is the concept called "sensitive dependence on initial conditions." A coin-flipping experiment is a sequence of coin tosses each of which has as outcome either heads (H) or tails (T). Thus it can be represented in the form HTTHHTTTTH. . . A general coin flipping experiment is thus a sequence s 0 s I s 2 . . . where each of so, s 1, s 2 . . , is either H or T. Here is the result of the horseshoe analysis that I found on that Copacabana beach. Consider all the motions of the horseshoe construction which stay in the square, i.e., don't drift out of our field of vision. These motions correspond precisely to the set of all coin flipping experiments! This

.'

' To give a complete picture in this section, one needs to reverse time and consider sequences of heads and tails which go back in time as well.

18

Chaos Avant-Garde: Memories ofthe Early Days of Chaos Theory

discovery demonstrates the occurence of unpredictability in fully nonlinear motion and gives a mechanism of how determinism produces uncertainty. The demonstration is based on the following construction. To each visual motion there is an associated coin flipping experiment. If xo X I x2 . . . is a visual motion of the horseshoe dynamics, at time i = 0, 1, 2, 3, . . . associate H or heads if x , lies in the top half of the square and T or tails if it lies in the bottom half. Moreover, and this is the crux of the matter, every possible sequence of coin flips is represented by a horseshoe motion. Therefore the dynamics is as unpredictable as coin-flipping. In the natural one-one correspondence xo XI x2.

. . +so

SI s2.

.. ,

xo xIx2 . . . is a motion lying in the square and so sIs2 . . . is a sequence of H's and T's. On the left is a deterministically generated motion and on the right a coin-flipping experiment.

The analysis shows that the above association is a complete natural identity between the set of motions lying in the square and the set of all sequences of heads and tails.* We have seen how deterministic fully nonlinear motion, the horseshoe, can be represented as unpredictable coin-tossing experiments. This is chaos.

The Hidden Origins of Chaos As chaos is a mathematically based revolution, it is not surprising to see that a mathematician first saw evidence of chaos in dynamics. Henri PoincarC was (with David Hilbert) one of the two foremost mathematicians in the world active at the end of the last century. Poincart was an originator of topology who had written an article claiming that a manifold with the same algebraic characteristics as the n-dimensional sphere was actually the n-dimensional sphere. When he found a mistake in his proof, restricting himself now to 3 dimensions, he formulated the assertion as a problem, now called PoincarC's Conjecture. This problem is the biggest problem in topology and even one of the three or four great unsolved Mathematicians say that we have an isomorphism preserving the dynamics, or conjugacy, between the horseshoe motions lying in the square and the coin-flipping experiments.

Finding a Horseshoe on the Beaches ofRio

19

problems in mathematics today. What concerns us here however is this scientist's contribution to the theory of chaos. PoincarC made extensive studies in celestial mechanics, that is to say, the motions of the planets. At that time it was a celebrated problem to prove the solvability of those underlying equations, and in fact PoincarC at one time thought that he had proved it. Shortly thereafter however he became traumatized by a discovery which not only showed him wrong but showed the impossibility of ever solving the equations for even three bodies. This discovery was a motion he christened "homoclinic point." A homoclinic point is a motion tending to an equilibrium as time increases and also to that same equilibrium as time recedes into the past. See Figure 6. Here p is an equilibrium and h marks the homoclinic point. The arrows represent the direction of time.

P

Figure 6

This definition sounds harmless enough but carries amazing consequences. PoincarC wrote concerning his di~covery:~

' my own translation from the French

20

Chaos Avant-Garde: Memories ofthe Early Days of Chaos Theory

One will be struck by the complexity of this figure which I won't even try to draw. Nothing can more clearly give an idea of the complexity of the three body problem and in general of all the problems of dynamics . . . In addition to showing the impossibility of solving the equations of planetary motion the homoclinic point has turned out to be the trademark of chaos; it is found in essentially every chaotic dynamical system. It was in the first half of this century that American mathematics came into its own, and traditions s t e m ~ ~ nfrom g Poincark in topology and dynamics were central in this development. G. D. Birkhoff was the most well known American mathematician before World War 11. He came from Michigan and did his graduate work at the University of Chicago, before settling down at Harvard. Birkhoff was heavily influenced by Poincar6's work in dynamjcs, and he developed these ideas and especially the properties of homoclinic points in his papers in the '20s and '30s. Unfortunately, the scientific community soon lost track of the important ideas surrounding the homoclinic points of PoincarC. In the conferences in differential equations and dynamics that I attended in the late ' ~ O S there , was no awareness of this work. Even Levinson never showed in his book, papers, or correspondence with me that he was aware of homoclinic points. It is astounding how important scientific ideas can get lost, even when they are exposed by leading mathematicians of the preceeding decades. I learned about homoclinic points and PoincarC's work from browsing in Birkhoffs collected works which I found in IMPA's library. It was because of the recently discovered horseshoe that the homoclinic landscape was to sink into my consciousness. In fact there was an important relation between horseshoes and homoclinic points. If a dynamical system possesses a homoclinic point then I proved that it also contains a horseshoe. This can be seen in Figure 7. Thus the coin-flipping syndrome underlies the homoclinic phenomenon, and helps to comprehend it.

21

Finding a Horseshoe on the Beaches of Rio

c \

D

Figure 7

The Third Force I was lucky to find myself in Rio at the confluence of three different historical traditions in dynamics. These three cultures, while dealing with the same subject, were isolated from each other, and this isolation obstructed their development. We have already discussed two of these forces, Cartwright-Littlewood-Levinsonand Poincare-Birkhoff. The third had its roots in Russia with the school of differential equations of A. Andronov in Gorki in the 1930s. Andronov had died before the first time I went to the Soviet Union, but in Kiev, in 1961, I did meet his wife, Andronova Leontovitch, who was still working in Gorki in differential equations. In 1937, Andronov teamed up with a Soviet mathematician L. Pontryagin. Pontryagin had been blinded at the age of 14, yet went on to become a pioneering topologist. The pair described a geometric perspective of differential equations they called "rough," subsequently called structural

22

Chaos Avant-Garde: Memories ofthe Early Days of Chaos Theory

stability. Chaos, in contrast to the two previously mentioned traditions, was absent in this development because of the restricted class of dynamics. Fifteen years later the great American topologist Solomon Lefschetz became enthusiastic about Andronov-Pontryagin's work. Lefschetz had also suffered an accident, that of losing his arms, before turning to mathematics, and this perhaps generated some kind of bond between him and the blind Pontryagin. They first met at a topology conference in Moscow in 1938, and again after the war. It was through Lefschetz's influence, in particular, via an article of his student, De Baggis, that Mauricio Peixoto in Brazil learned of structural stability. Peixoto came to Princeton to work with Lefschetz in 1957 and this is the route which led to our meeting each other through Elon. After this meeting, I studied Lefschetz' book on a geometric approach to differential equations and eventually came to know Lefschetz in Princeton. Via Pontryagin and Lefschetz there was the specter of topology in the concept of structural stability of ordinary differetial equations. I believe that was why I listened to Mauricio.

Good Luck Sometimes a horseshoe is considered an omen of good luck. The horseshoe I found on the beach of Rio certainly seemed to have such a property. In that spring of 1960 I was primarily a topologist, mainly motivated by the problems of that subject, and most of all driven by the great unsolved problem posed by PoincarC. Since I had started doing research in mathematics, I had produced false proofs of the 3-dimensional PoincarC Conjecture, returning again and again to that problem. Now on those beaches, within two months of finding the horseshoe, I found to my amazement an idea which seemed to succeed provided I returned to PoincarC's original assertion and then restricted the dimension to 5 or more. In fact the idea produced not only a solution of PoincarC's Conjecture in dimensions greater than 4, but it gave rise to a large number of other nice results in topology. It was for this work that I received the Fields Medal in 1966. Thus . . the mathematics created on the beaches of Rio . . (Hornig) was the horseshoe and the higher-dimensional Poincark's conjecture. 'I.

.I'

3.

Strange Attractors and the Origin of Chaos* Yoshisuke Ueda Kyoto University

Prologue I am greatly honored to have been given this wonderful theme, "Strange Attractors and the Origin of Chaos" for my presentation today. First I would like to take this opportunity to offer a special thanks to each one of the people who planned and made this symposium possible. At present, people say that the data I was collecting with my analog computer on the 27th of November, 1961, is the oldest example of chaos discovered in a second-order non-autonomous periodic system. Around the same time, it was Lorenz who made the discovery of chaos in a third-order autonomous system. At that time I was simply frustrated with this seemingly mysterious phenomenon which I accidentally came upon during my experiments. For my part, it was nothing as glorious as an act of discovery-all I did for a long period of time was to keep on pursuing my stubborn desire to understand this unsettling phenomenon. "What are the possible steady states of a nonlinear system?"-this has always been my question. And my paradigm has always been the phenomena themselves, not papers with their abstractions, but something we can actually observe or quantify.

* This article was first presented at the international symposium entitled "The Impact of Chaos on Science and Society," where I was invited as a guest speaker. The symposium was organized by the United Nations University and the University of Tokyo, and held in Tokyo between 15-17 April, 1991.

23

24

Chaos Avant-Garde: Memories of the Early Days of Chaos Theory

In this report, I would like to reflect upon the circumstances of my research and the general conditions of Japanese science around 1978 before the study of chaos began. As I prepared this talk, I kept asking myself what propelled me to pursue my question so relentlessly, but I must confess that I don't know the answer. I have not, in my wildest dreams, imagined that I would be given an opportunity to speak on this very subject. It was so unexpected, my mind was reduced to total chaos! As an academic term, I do not find the word "chaos" very appropriate. But what shall we call it then? My proposal has been "randomly transitional phenomena"; I will explain this below. The characteristics of chaos in a physical system can be summarized as follows: Random phenomena that occur in deterministic systems. Random phenomena whose short-term behavior is predictable. Random phenomena whose long-term behavior is unpredictable. Although the phenomena are irregular and unpredictable, chaos does have a definite structure. The original meaning of chaos, I feel, is a "total disorder and ultimate unpredictability." But as scientific terminology, the word "chaos" seems to overemphasize the unpredictability alone. This symposium provides an opportunity to clear this misunderstanding and to inform the non-specialist of the correct meaning of the word. Even so I have to use the word "chaos1' here, instead of ''randomly transitional phenomena." It is a concise expression which has already filtered into people's minds, and therefore I have decided it is rather pointless to resist it.

The Oldest Chaos in a Non-Autonomous SystemA Shattered Egg The 27th of November, 1961 became a memorable day for me, although I did not have any joyous realization that something wonderful had happened, or any vivid memory of the events of that particular day. As I remember, I had just finished writing the narrative to accompany the data I was going to publish at the Special Committee on Nonlinear Theory of the Institute of

25

Strange Attractors and the Origin of Chaos

Electrical Communication Engineers, to be held on December 16th, and was carrying out some analog computer experiments with the help of Susumu Hiraoka, who was two years my junior, in order to test the applicability of the approximate computation I quoted in my paper. Had I not had the date on the printout of that old analog computer, which was destined for a wastebasket, I would never have been able to recall the date (Fig. 1).

Fig. 1 Output of an analog simulation of the equation

ii - ~ ( 1 p2)zi+ u3 = ~~~~~t

(1)

with p = 0.2, y = 8and B = 0.35 obtained on 27 November 1961 is shown. A continuous orbit is drawn lightly on the Uzi plane and points in the Poincard section at phase zero are given by heavy dots; five dots near the top are fixed points for a sequence ofvalues at v = 1.01, 1.012, 1.014, 1.016

26

Chaos Avant-Garde: Memories of the Early Days of Chaos Theory

and 1.018, the remaining points are on the chaotic attractor at v = 1.02. (Courtesy of Dr. H. B. Stewart, Brookhaven National Laboratory)

At that time, 1 was a third-year graduate student at Kyoto University, working on the phenomenon of frequency entrainment under the guidance of Professor Chihiro Hayashi. When a circuit (oscillator) which would, if left alone, keep on generating an electrical (self-sustained) oscillation with a certain frequency and amplitude, is driven externally with signals whose frequency is different from that of the self-oscillator, its self-oscillating frequency is drawn to and synchronized with that of the driving frequency. This phenomenon is called frequency entrainment. There are exceptions, of course-depending on the value of the driving frequency and amplitude, entrainment sometimes does not occur. Instead, an aperiodic "beat oscillation" with drifting frequencies would appear. I was receiving direct guidance from Professor Hiroshi Shibayama of Osaka Technological University, who was visiting our laboratory as a guest scholar several times a week. Warm and gentle, he let me do whatever I wanted to do, while introducing me to the basics of the research. Even now I look up to him as my senior friend and receive all kinds of advice. The main purpose of my computer experiment was to simulate the nonautonomous nonlinear differential equation describing frequency entrainment, and to examine the range of the frequency and amplitude of the driving signals which cause synchronization, as well as the amplitude and the phase of its oscillation. Allow me to go into a little technical detail here. The approximate computation was done by rewriting the non-autonomous equation into an autonomous equation using the averaging method (approximation enters the process at this stage, with the result that chaos is suppressed). The aim was to approximate the steady state of the original system with an equilibrium point or limit cycle of the autonomous equation. In this method, the stable equilibrium point corresponds to synchronized frequency entrainment, and a stable limit cycle corresponds to asynchronous drift conditions. Actually there are two kinds of asynchronous oscillations-quasi-periodic oscillation (represented by a limit cycle in the averaged equation) and chaotic oscillation (represented by strange or chaotic attractors): but the common sense of the day failed to recognize the chaotic oscillation. In those days only the equilibrium point and limit cycle were known to exist as steady states of a (second order) autonomous system, so it was understandable for everyone to have possessed the preconceived notion

Strange Attractors and the Origin of Chaos

21

that asynchronous condition meant quasi-periodicity. On that day, the 27th of November, when I changed the parameter (frequency of the driving input), and the condition shifted from frequency entrainment to asynchronization, the oscillation phenomena portrayed by my analog computer was chaotic indeed. It was nothing like the smooth oval closed curves in Fig. 1, but was more like a broken egg with jagged edges. My first concern was that my analog computer had gone bad. But I soon realized that that was not the case. It did not take long for me to recognize the mystery of it all-the fact that during asynchronous phase, the shattered egg appeared more frequently than the smooth closed curves, and that the order of the dots which drew the shattered egg was totally irregular and seemingly inexplicable. As I watched my professor preparing the report without a mention of this shattered egg phenomenon, but rather replacing it with the smooth closed curves of the quasi-periodic oscillation, I was quite impressed by his technique of report writing. But at the same time, I realized that one needs to be very careful in reading reports of this sort 113. I am getting a little sidetracked here, but the analog computer had been developed and created as his research project by Minoru Abe who was three years my senior. My deep respect goes to him who so laboriously and meticulously handbuilt this practical computer with vacuum tubes, Figure 2 is an example of the waveform data made by the computer. It reminds me of the long hours patiently sitting in front of the analog computer, and of my wonder at its accuracy-a testament to its maker's unique skill. To obtain a sheet of data as shown in Fig. 2, it took the computer about 60 to 100 minutes. Most of them have been discarded, but we had accumulated at least 1000 sheets of data during my five years in graduate school. I would like to mention that I had not contributed very much in creating this computer beyond helping him build several operational amplifiers and learning to repair the chopper amplifier and the recorder. As I look back, I feel that after those long exhausting vigils in front of the analog computer, staring at its output, chaos had become a totally natural, everyday phenomenon in my mind. People call chaos a new phenomenon, but it has always been around. There's nothing new about it-only people did not notice it.

28

Chaos Avant-Carde: Memories of the Early Days of Chaos Theory

Fig. 2 Waveforms of an analog simulation of Eq. (1) with varying v are shown for B = 0.3. A pulse on the waveform which is generated by an auxiliary circuit independent of the main analog computation circuit for Eq. (1) shows a time mark at every period of the external periodic forcing; when this pulse sequence forms a straight line (after transient), we can infer that periodic motion appears with the system being entrained by the external periodic signal. These data progress from entrained state to asynchronized state and vice versa exhibiting a narrow hysteresis zone. (Courtesy of Dr. H. B. Stewart, Brookhaven National Laboratory)

29

Strange Attractors and the Origin of Chaos

2-

1-

I

O-

.>

-I

-

-2

-

-3

unuble fhud point I

-2

0

-1

I

I

1

2

-V

0:tlUmmic

:1/3-lhnmniC

Fig. 3 Domains of attraction for harmonic and 1/3-harmonicresponses. Points 1 and 2 show fixed points representing non-resonant and resonant harmonic oscillation, respectively, and point 3 is a saddle on the basin boundary. Points 4 , 5 and 6 are completely stable 3-periodic points representing 1/3-harmonicoscillation. This attractor-basin phase-portrait is obtained by exhaustive checking of initial points on the VV plane by executing analog computer experiments for the Duffing equation

ii+kzi+u3 =Bcost

(2)

with k = 0.1, B = 0.15. (Reproduced with the courtesy of McGraw-Hill Book Company and Nippon Printing and Publishing Company [2,3, 171)

30

Chaos Avant-Garde: Memories of the Early Days of Chaos Theory

When 1 think of those long hours in front of the analog computer, I also think of my classmate, Toshiaki Murakami. He had completed his master's degree and came to work in April of 1961. During our graduate school days, we used to compete for the use of the analog computer. He and his instructor, Yoshikazu Nishikawa, four years ahead of us, used to keep long vigils, too, painstakingly examining point by point the domain of the initial conditions of the Duffing equation. Their main results appear in Fig. 10.6 in [2], and Fig. 4 of [3] in the bibliography, reproduced here as Fig. 3. In Figure 3 of [3], reproduced below as Fig. 4, we can clearly see the curve (alphabranch) which swirls down from the saddle point 3 on the basin boundary to point 1. At the time, the pertinence of the figure was controversial among the specialists. Dr. Hayashi himself voiced his doubts and did not include it in his book published by McGraw-Hill in 1964. I will discuss this point later in "From the Harmonic Balance Method to the Mapping Method."

Fig. 4 The loci of some image points on the invariant curves of the directly unstable fixed point 3, for the same equation and parameter values as Fig. 3. (Reproduced with the courtesy of Nippon Printing and Publishing Company [3, 171)

Strange Attractors and the Origin of Chaos

31

In the research data [l] published in December of 1961, the saddle-mode point corresponding to the boundary of the frequency entrainment domain was plotted on the limit cycle, but a careful examination proved that it was actually not on the limit cycle. This error was corrected immediately after the publication. The fact that these data survived was by itself a miracle. It symbolizes my packrat tendency-a typical trait for those who grew up during the war. Naturally I cannot verify the dates for those which were not dated at the time, but the following account can be corroborated from other data such as those of the nonlinear research group, to within a few months. I will touch upon the original data at the time of the experiments later in "The Original Data that were Preserved." Thus ended 1961. In 1962, the nonlinear oscillation group of Hayashi students was studying frequency entrainment phenomenon of the selfoscillatory system with driving periodic signals (forced self-oscillatory system) and the oscillatory phenomenon of a system which holds a steady equilibrium if no external periodic signals are added (forced oscillatory system). The former research centered around Prof. Shibayama, focusing mainly on the analyses of Van der Pol equations with added forced terms, or of a mixture of Van der Pol and Duffing equations, while the latter, the study on the Duffing equation, centered around Prof. Nishikawa.

Encounter with the Japanese Attractor One day in the fall or winter of 1962, Prof. Chihiro Hayashi was waiting for me in the laboratory room. At that time he was working on the manuscript for his forthcoming book [2] which was eventually published in 1964 by McGraw-Hill. The laboratory room was probably more convenient for his work: he left his office vacant and occupied the laboratory room. "I want you to do this computation in a hurry," he told me. It was to solve third order simultaneous equations with four unknowns and draw the amplitude characteristic curve of the periodic solution of the Duffing equation, taking into consideration the components up to the third harmonic component. Let me explain the circumstances of Prof. Hayashi's request. When he sent chapter 6 of his book to McGraw-Hill, it was returned with an objection from a reviewer. The contents of chapter 6 were exactly the same as chapter 3 of his earlier book (1953) [4]. The reviewer, as I recall, was Prof. Higgins

32

Chaos Avant-Garde: Memories of the Early Days of Chaos Theory

of the University of Wisconsin. In the book, Prof. Hayashi showed the amplitude characteristic curve for the periodic solution represented by cosine components alone, of the Duffing equation with zero as a damping term, (Fig. 5), but not the curve with non-zero damping coefficient. There was a reason for this. When the damping coefficient is not zero, sine components become necessary in the periodic solution, in addition to the cosine components, thus doubling our workload. It was particularly difficult on the hand-cranked calculator. I was somehow trusted by Prof. Hayashi as someone having fast and accurate computation and careful drawing skills. So on that occasion, I felt quite proud of myself and accepted the challenge, knowing it was a difficult job. Even if I did not want to do it, I would not have been able to refuse him anyway. I was a little amused to find this most exalted professor high up in the clouds a bit flustered, especially with the objection written in English no less!

B-

Fig. 5 Amplitude characteristicof the periodic solution

v = y, cost + y3cos3t

(3)

of the Eq. (2) for k = 0, obtained by using harmonic balance method. (Reproduced with the courtesy of Nippon Printing and Publishing company [4])

33

Strange Attractors and the Origin of Chaos

As it was impossible to solve the third order simultaneous equations with four unknowns by sheer brute force, I eliminated two variables from the four expressions to lead to high order simultaneous equations with two unknowns, gave the amplitude of the periodic solution ahead of time, with a bit of a "manipulation," returned to the computation of the amplitude of the external force, and finally, by hand calculations, solved approximately fifty cases of third order equations with the Newton-Raphson method, and drew the curves shown in Fig. 6 (Fig. 6.1 in the reference [2]). I fondly remember this crash project completed within a few weeks. The damping coefficient was set at 0.2. 4

k-0.2

/--

5

LO

15

20

5

Fig. 6 Amplitude characteristicof the periodic solution

v = x, sint + y , cost + x3 sin 3t + y3 cos31 with r,' = x: + y:, r: = x: + y:, of the Eq.(2) for k = 0.2. (Reproduced with the courtesy of McGraw-Hill Book Company [2]) The amplitude characteristic curves drawn with only two frequency components contained in the periodic solution, are nothing but an approximation. The standard procedure, therefore, was to verify the results with the analog computer. I struggled again for several days in a row, in front of the computer. When the amplitude of the external force increased,

34

Chaos Avant-Gar& ~ e ~ oof$the~Early e ~Days of Chuos Theory

the high frequency components in the periodic solution also increased, accelerating the response time, thus making it extremely difficult for the servo multiplier-that represented the nonlinear term-to follow up. Cons~quently,one had to extend the computer's time scale so as to slow down the response time. I repeated the experiment with one cycle of the external force 2n corresponding to 3 1.4...seconds. During the process, I encountered enough chaotic oscillations (the source of the Japanese attractor) to make me sick. But Prof. Hayashi told me, "Oh, it's probably taking time to settle down to the subharmonic oscillations. Even in an actual series resonance circuit, such a transient state lingers for a long time." When I look back, though, I seemed to have sensed at that time that chaos was not a phenomenon unique to forced self-oscillatory systems in which quasiperiodic oscillation appeared. Only a fragmental record of the original data of my work with the analog computer and manual calculation remains now. I may have been worried at that time that, had Prof. Hayashi seen this data, he would have told me to repeat the analog computer experiments until the transient state settled to a more acceptable result. Sensing that no matter how long I continued the simulation, I would never be able to come up with the data he wanted, short of making up some false data, I must have suggested the larger damping coefficient, 0.4, and done away with the problematical data, or burnt them in 1978 when we moved. These were the circumstances of obtaining the amplitude characteristic curve for my doctoral dissertation (Figs. 7 and 8). By the summer of 1963, the amplitude characteristic curves for the unsymmetrical periodic solution had already been obtained as well, but since they did not appear in the ~ c ~ r a w - H ibook, l l the manuscript editing must have been completed by that time. The Figures 7 and 8 were included in reference [ 5 ] , but the description of chaos does not appear anywhere, Based on this data, Prof. Hayashi wrote his paper on higher-harmonic oscillation, which he reported during the International Congress at Marseilles [ 6 ] . I finished my doctorate in the spring of 1964, or more accurately, completed the units requirement, left school, and was hired as a research assistant in the Dept. of Electrical Engineering. During that year, Prof. Hayashi's book, Nonlinear Oscillations in Physical Systems, came out [2]. Students are often critical of their teachers. I was thinking "the book may be nothing but a databook of harmonic balance methods." Now the table is turned. As I write this, I chide myself who has not yet published a book.

Strange Attractors and the Origin af Chaos 4

35

-

k 0.4

8-

Fig, 7 Amplitude characteristic of the periodic solution

v = x1sin t + yl cost + x:, sin 3t + y3cos3t

+

+

with ti2 = X: y ; , .," = x: y:, of the Eq. (2) for k = 0.4. (Reproduced with the courtesy of Nippon Printing and Publishing Company (6, 171)

I

Fig. 8 Amplitude ch~a~teristic of the periodic solution 2) = z

+ x1sin t + yl cost + x2 sin 2t + y2 cos 2t

with q 2 = X 12 + y ~ , ~ = x ~ + y ~ , ~ f t h e E q . ( 2 ) f o r k = O . 4 . (Reproduced with the courtesy ofNippon Printing and Publishing Company [6, 171)

36

Chaos ~vani-Garde:~ e m o r ~oef sthe Early Days of Chaos Theory

The Hayashi Laboratory at the Time of the "McGraw-Hill Book" The above was the condition of the Hayashi Laboratory up until the publication of Prof. Hayashi's "McGraw-Hill Book." The laboratory was overflowing with chaotic data produced by the analog simulations, and yet they were overlooked as a quasi-periodic oscillation or transient state. I would like to touch upon the goings-on in the laboratory around that time. Prof. Shibayama, having finished his dissertation, had stopped coming around. Prof. Nishikawa had left nonlinear oscillations and changed his field to control engineering, a few years earlier. Prof. Abe was focusing his attention mainly on the application of analog techniques to control systems, and was concerned with nonlinear oscillations only indirectly. This made me the senior researcher on nonlinear oscillation in the laboratory. Even after the publication of his book, Prof. Hayashi used to ask me to do all kinds of work for him. He was especially strict with me, always demanding "Don't give this work to anyone, be sure to do it yourselfl" Even if he did not say that, there were several other senior researchers in the laboratory who used most of the students' help, leaving no one to help me. Let me mention a group that started from then on. "From then on" because it continues even now. It includes Nobuo Sannomiya, three years my junior, and Masami Kuramitsu, four years my junior. Since 1963, we three took turns at holding seminars a couple of hours every week, reading Smirnov's A Course of Higher Mathematics (the original was in Russian). Starting in 1963, it took five years to finish studying the Japanese version from Vols. 5 through 12. It was truly helpful. Without the background thus developed by this seminar, I, who had been trained with the engineering school curriculum, would never have been able to understand papers such as Birkhoff's. Prof. Hayashi did not seem to like the idea of our round-robin seminar, and told us often to work on calculations if we had time to read books. But we kept ignoring his admonitions. He also had an extreme suspicion of and dislike for the digital computer KDC-I which finally became available around that time. But I used it to compare results such as the periodic solution of the Duffing equation with the analog data. The KDC-I was a machine built with transistors, and took about 60 seconds to integrate the Duffing equation along the time axis from t=O to 2n, using the Runge-Kutta-Gill method that set the size of integration step at 2~160.It wouldn't even make a toy today, but at that time I was deeply impressed by

37

Strange Attractors and the Origin of Chaos

the fact that such a calculation-impossible possible.

to do by hand-was

finally

From the Harmonic Balance Method to the Mapping Method Dr. Hayashi's "McGraw-Hill Book" was highly regarded, and he was invited to be a guest professor at Columbia University and the Massachusetts Institute of Technology from the fall of 1965 through the summer of 1966. I was his trusted student, and was even given the honor of handling his paychecks during his absence. As he was always present in the laboratory before and after his stay in the United States, I naturally cherished a year's freedom from his watchful eyes. That fall, during Prof. Hayashi's absence, Hiroshi Kawakami from Tokushima (five years my junior) was in the second year of his masters program, studying analog techniques and their applications to control systems. But he came to me for help in studying nonlinear oscillation which he apparently found interesting. It was impossible to change his research project in his second year of graduate school especially in such a small laboratory, but I drew him into the calculation and analog simulation of the amplitude characteristics of nonlinear oscillation with two saturable cores. Soon after, Prof. Hay ashi returned. By then Kawakami was already in his doctorate program and his determination to pursue nonlinear oscillation research was already established. He was trying to select a thesis for his research. In our laboratory, as I have already mentioned, mapping or the stroboscopic method had been used for analog computer experiments, and technical terms such as completely stable fixed point were familiar to us. The knowledge came mostly from W. S. Loud's Papers [7]. Prof. Hayashi was friendly with Prof. Loud, regularly corresponding with him. Around that time we got a copy of N. Levinson's paper [S], although I am not sure when and how it fell into our hands. Probably Kawakami, being a diligent student, copied it from somewhere. When I was studying the paper, I came upon the Figs. 2 and 3 on p. 735. The moment I understood the meaning of these figures, I thought "This is it!" (Fig. 9). It solved a long-standing mystery for me. Figure 4 was clearly an error, and the key to the correct answer was in these figures. I thought it through for several days, and figured it out also in numerical terms. This was my first encounter with the concept of heteroclinic point. Of

38

Chaos Avant-Garde: Memories of the Early Days of Chaos Theory

course I did not know the term then. After several days, I was confident. I drew a rough sketch and presented it to Prof. Hayashi (Fig. 10). He did not agree with me immediately, but eventually did, and we decided to investigate all the cases in which subharmonic oscillation of order 1/3 appeared. I recommended the subharmonic oscillation of order 7/3 which I had seen during my earlier analog simulation of higher harmonic oscillation. It used to appear where the external force was large and things got complicated, and I used to feel relieved. We immediately asked Kawakami to draw the invariant curves of the directly unstable fixed point in this case. At the same time we asked Prof. Abe to create an automatic mapping device. Figure 7(b) in the reference [9] shows our results. Figures 11, 12 and 13 were completed right after the abstracts for the 4th Conference on Nonlinear Oscillations held at Prague in 1967 had been mailed, and therefore became its appendix. Since these figures have never been published either in the proceedings of the Conference or in Kawakami's doctorate dissertation, except in the NLP research data [lo], I would like to include them here. Please refer to references [ 101 and [ l 11. The first results we obtained from Prof. Abe's automatic mapping device are shown in Fig. 6 in reference [12] (reproduced here as Fig. 14), in which the error of Fig. 4 has been corrected. The device itself is summarized in the appendix of reference [ 131. In modern terms, the automatic mapping device is a device which uses the analog computer to create PoincarC maps. In other words, the device plots on the recording paper a sampling of the analog signal at the same instant during each cycle of the external force. Although Kawakami and myself helped Prof. Abe build the trial device, it would have been extremely difficult without Prof. Abe's remarkable expertise. Thanks to the device, drawing invariant curves, or outstructures in more modern terms, became much easier. It goes without saying that as a result, the application of discrete dynamical systems theory to nonlinear oscillation was speeded up considerably. Through this experience I learned the importance of examining the original texts as well as making the tools and devices on our own, which are necessary in achieving one's research goals.

Strange Attractors and the Origin of Chaos

Fig. 9 Examples of maximum finite invariant domains. (a) 0:completely stable fixed point A, €3, C: completely stable 3-periodic points D, E, F: directly unstable 3-periodic points (b) 0, D, E, F: the same as in (a) A, B, C: inversely unstable 3-periodic points G , H, I, J, K, L: completely stable 6-periodic points (Reproduced with the courtesy of Annals of Mathematics [8])

39

40

Chaos Avant-Garde:Memories of the Early Days of Chaos Theory

Fig. 10 Corrected schematic diagram of the invariant curves corresponding to Fig. 4.

41

Strange Attractors and the Origin of Chaos

0

Fig. 1 1 Fixed points and invariant curves of the mapping T for equation x =Y,

j , = -ky - x 3 + Bcost

with k = 0.2 and B = 10.0. (Unpublished supplements to Ref. [12])

(7)

42

Chaos Avant-Garde: Memories of the Early Days of Chaos Theory

I

2.0

I

I

I

I

2.5

3.0

3.5

0

I-

Fig. 12 Fixed points and invariant curves of the mapping ? for Eq. (7) with k = 0.2 and B = 10.0. (Unpublished supplements to Ref. [ 121)

43

Strange Attractors and the Origin of Chaos

I

x

2.0

3.0

2.5

3.5

0

-X

Fig. 13 Domains of attraction of the completly stable fixed points of the mapping la for Eq. (7) with k = 0.2, B = 10.0. (Unpublished supplements to Ref. [ 121)

44

Chaos Avant-Garde: Memories of the Early Days of Chaos Theory

-x

Fig. 14 Computed fixed points and invariant curves corresponding to Figs. 3 , 4 and 10. Note: Numbering of the points is different, but the equation and the parameter values are the same. (Reproduced with the courtesy of Nippon Printing and Publishing Company [17])

In the spring of 1966, Norio Akamatsu (seven years my junior) came into the masters program. Another Tokushima native, he chose nonlinear oscillation as the subject of his research from the very start and came directly under my guidance. I asked him to begin with the application of the mapping method to the research in synchronization phenomenon, for which the harmonic balance method had been the standard method in the past. A part of his master's thesis is described in the latter half of reference [9]. These two students from Tokushima were both extremely sharp, and contributed greatly to nonlinear oscillation research. I owe them a great deal. I did not force them to do anything under Prof. Hayashi's authority, but instead built a cooperative atmosphere so as to let their unique abilities grow to the fullest. I pride myself for helping to build the golden era of the Chihiro Hayashi Laboratory during the latter half of the 1960s. In fact, I tried to point out to them the particulars of bibliographic research and interpretations, while gathering data with them, examining their parameter settings, and at the same time acted as their protecting wall, or more accurately, I completed or did over their results which did not meet Prof. Hayashi's approval. Akamatsu

Strange Attractors and the Origin of Chaos

45

was the last graduate student to complete his doctorate program in the Hayashi Laboratory. Kawakami left us in 1970, and Akamatsu left in 1971, leaving Masami Kuramitsu, Kenji Ohshima and myself, who were studying nonlinear oscillations in a postdoctorate capacity. Because of our research themes, we three maintained a parallel relationship. Kuramitsu was working on a nonlinear system with many degrees of freedom, while Ohshima was undertaking experimental research on actual electrical circuits.

The True Value of an Advisor: A Scion of Chaos I cannot forget to mention my mentor, Dr. Michiyoshi Kuwahara, to whom my work is deeply indebted, Dr. Kuwahara was my senior fellow in the Hayashi Laboratory, and was the very first graduate student who completed his program in our Lab. He understood my position well, and had paid me a visit once a month for the past thirty some years, always ready to give me good advice and suggestions. To this day, I still listen to his wisdom. He does not mince his words-a frankness I always like and admire. Sometimes his opinion struck close to home and was quite painful to my ear. But when I received a lashing from Prof. Hayashi, Dr. Kuwahara used to listen to me patiently and comfort me. I remember those occasions very vividly indeed. I received much valuable advice from Dr. Kuwahara. Among his innumerable suggestions, I am most thankful for his insistence concerning the art of paper-writing. "Write papers and send them to appropriate academic journals." he advised. "Oral reports (nonlinear research group data, etc.) do not help you. Send them off yourself, so that it will be credited to you by the referees." I did not receive this kind of advice from Prof. Hayashi. Most of the data in the nonlinear research group reports were published by Prof. Hayashi at the international meetings, so I had to be careful not to duplicate his material. Around the time when the university was in turmoil with student protest, I sent a paper of my own for the first time, to the Journal of the Institute of Electronics and Communication Engineers, which had referees. The democratic atmosphere that prevailed on campus because of the protest might have prompted me to take this action. The paper passed the review smoothly and was accepted. Figure 6 in the paper [14], demonstrates quasi-periodic oscillation and chaos in the Rayleigh-Duffing mixed type equation (Fig. 15). Though it hadn't taken a clear shape yet, the concept of chaos was already established in my mind by that time.

Chaos Avant-Garde: Memories ofthe Early Days ofchaos Theory

46

a5

t

-

O-

I 1

-1.0

-0.5

On

I

I

I

I

0

0.5

1.0

-X

(a)

Fig. 15 Point sequences representing beat oscillation for equation

i =y,

y = p(1- w 2 ) y- x 3 + Bcos vt

(8)

with p = 0.2, y = 4.0, showing the difference between almost periodic oscillation and chaotic oscillation. (a) B = 0.1, v = 1 . 1 : invariant closed curve representing almost periodic oscillation. (b) B = 0.3, v = 1 . 1 : chaotic attractor. (Reproduced with the courtesy of the Institute of Electronics and Communication Engineers of Japan [ 141)

Strange Attracforsand the Origin of Chaos

47

This fact can be supported by the following circumstance. A seminar called "Ordinary differential equation and nonlinear dynamics" was held at the Research Institute for Mathematical Sciences of the Kyoto University from the 17th through the 19th of December, 1970, under the leadership of the research director, Prof. Minoru Urabe. At the seminar, I volunteered to make an oral report entitled "A steady solution of the nonlinear ordinary differential equations of the second order." The record of the following comments I made at the end of my research report still remains. . . However, according to my observation of the phenomenon with the use of a computer, each of the minimal sets which make up the set of central points are all unstable, and the steady state seems to move randomly around the vicinity of the minimal sets, influenced by small fluctuations in the oscillatory system or external disturbances." [ 151 These minimal sets are of course the unstable periodic motions in the attractor; the above description led to my proposed name "randomly transitional phenomena." The reason 1 made the report in Prof. Urabe's seminar was because I hoped for the mathematicians to hear and possibly support my ultimate interpretation of the random oscillations. I was hoping especially because in those days Prof. Hayashi did not welcome my mention of "set of central points," or "minimal set," etc. during our seminars. Everytime my interpretation of the random oscillations was mentioned, he kept pressing me to examine the errors further. Despite my ardent hope, however, my gamble backfired that day. I cannot be sure of the date, since Prof. Urabels seminar, although small in scale, was held every year. After my report, at any rate, Prof. Urabe admonished me personally. "What you saw was simply the essence of quasiperiodic oscillations," he said. "You are too young to make conceptual observations." The existence of random oscillations (chaos) was so obvious in my mind, that the negative comment did not crush me. Even so, I was deeply disappointed that no one understood it no matter how hard I tried to explain. From then on, I became even more careful in my research efforts. I think it was 1971 when I sent off the second paper based on my December 1970 oral presentation at the Research Institute for Mathematical Sciences mentioned above. Although I have very little record of it on hand, the Electronics and Communication Society must have some data. The paper was rejected. My only collaborator in random oscillation research, Akamatsu, had gone back to Tokushima, and I was all alone. I spent nearly a year rewriting it and sent it again on Sept. 7, 1972. After going through the review process, it was finally accepted [16]. The summary of the discrete 'I.

48

Chaos Avant-Garde: Memories offhe Early Days ofchaos Theory

dynamical system theory I included in the appendix was by far the most difficult work. I will never forget how nervous I was, wondering whether or not I had a full and accurate grasp of the concept. The mathematicians who valued rigorous proofs were, in a way, my bane. They can set up any unrealistic assumptions in their heads and live in their world of abstractions, but we are living in the real world. While I wanted them to hear me out a little more sympathetically, I also idolized mathematics. In the appendix, I summarized the essence of the papers by Birkhoff, Nemytskii and Stepanov, etc., but I simply could not understand the transfinite ordinal of the second ordinal class. As I reread this paper [ 161 now, I am a bit embarrassed by my poor writing. And yet this was where I advocated the existence of chaotically transitional oscillation. I feel that this paper holds true even today, except for a seemingly erroneous description of structural stability, and for the fact that the numerical examples in Figs. 8 and 9 obtained from the analog computer data, turned out to be non-chaotic, according to some later digital simulations which showed periodic oscillations instead. (These periodic oscillations are exceptional, and at typical nearby parameter values, both digital and analog simulations show chaotic oscillations.) Or rather, I should say my qualitative understanding of the steady chaotic phenomenon has not changed or advanced since then. Akamatsu came from Tokushima to write a clean copy of this paper for me. I had intentionally sent off the paper during Prof. Hayashi's absence, so that I had a good excuse for not having it reviewed by him, and it probably lacks certain fine editing because of it. But there was no way I could show the paper to Prof. Hayashi, since I knew he would make drastic changes and cut out what seemed essential to me. I could not compromise, however. It would have been more troublesome to show it to him and then clash with him than not show it at all. I was truly desperate. Even so, I had to consider the protocol of our research, as well as Akamatsu's position since he had not submitted his dissertation yet. Prof. Hayashi's name had to be included as a co-author. Ignoring such protocol may have been a lot simpler, but I could not do that. I have published several papers since then, and I always try to be very careful in selecting data. I would like to continue this practice in the future as well. I also try not to forget that there are junior colleagues who are plagued with insecurity but are clinging to the hope of a better future. It is regrettable that there are some people who are not ashamed to use them to their own advantage. If you have a chance to read Prof. Hayashi's paper on the Duffing equation [17] written around this time, the credibility of my account should be clear to you. In

Strange Attractors and the Origin of Chaos

49

September of 1972, Prof. Hay ashi was attending the Sixth International Conference on Non~inearOscil~ationheld at P o z n h It sounds silly now, but I learned the hard way that changing the already established order in this world was truly a difficult task. Prof. Hayashi kindly disregarded this incident, but my paper [16] was not included in his Selected Papers on ~ ~ ~ l ~~n~ cei lal art i o published ns in 1975 [ 173. The circumstance was such that I could not personally translate or proofread the English version of the paper [161 published in Scripta.

The End of the Chihiro Hayashi Laboratory Prof. Hayashi retired in the spring of 1975. I was just an Associate Professor at that time and had no idea what had transpired. But the Hayashi Laboratory was dismantled several years later. I have never been clear on the reason for this. We were a group of lost souls without a leader, but I have fond memories of being able to do my work freely for the first time. In addition to the chaos research, I was studying nonlinear systems with time delay with Yoshiaki Inoue. I was also doing calculation of the power spectrum of chaos with the collaboration of the Institute of Plasma Physics, Nagoya University. After that I was invited into the laboratory of Prof. Chikasa Uenosono of Kyoto University where I have been allowed to stay, in the Engineering Dept., to this day. This period was also a chaotic time in my own life. In the spring of 1978, at the time of my move to the Uenosono Laboratory, my paper 1181 was published in the Journal of the Institute of Electrical Engineers. This was the paper in which the strange attractor (Fig. 16)-the one for which Prof. David Ruelle coined the name "Japanese attractor"-was reported [19,20]. What was new in the paper was the addition of the power spectrum of random oscillation and related properties to my earlier paper [16], but considering the five years between the two papers, I don't find much progress in it. As I was planning to present this paper to mark the end of my nonlinear oscillation research, I took special care in writing the whole paper. It was around the time that I first heard the term "chaos." But I had never imagined that the enormous amount of the data on the Duffing equation system, etc. which I had accumulated, actually represented chaos itself, and that they would draw such wide attention later on.

50

Chaos Avant-Garde: Memories of the Early Days of Chaos Theory

Fig. 16 Japanese attractor.

The Original Data that were Preserved My original data concerning the Japanese attractor, including the response curves of the shattered egg mentioned earlier, calculated by hand, and the output diagram of the analog computer, are preserved at present in the Brookhaven National Laboratory. In order to explain how they got there, I have to start with my meeting with Dr. Hugh Bruce Stewart. It was June 1986 when I met Dr. Stewart for the first time. He had come all the way to Henniker, New Hampshire to see me, and attend SIAM's Conference on Qualitative Methods for the Analysis of Nonlinear Dynamics held under the leadership of F. M. A. Salam and M. L. Levi. For that entire week at Henniker, we had long talks. His interest was not limited to

Strange Attractors and the Origin of Chaos

*

0 I.r

technical discussions. He was also interested in the origin of my chaos research. So I began with the shattered egg and reminisced about how it came about. On the last day of our meeting, he handed me a hand-written note which said, "I have gotten approval to acquire and preserve all of your chaos research material. Would you give it in its entirety to the Brookhaven National Laboratory? We will take the full responsibility for preserving it properly." He eventually typed up this letter and sent it to me in Japan. To me, the material had only some sentimental value, but it would have been worthless in Japan. Thinking that I would probably throw it away when leaving the University, I selected some relevant material and sent a boxful to Brookhaven. That gave me a good excuse to put in order my original data that had filled my office all these years. Except for a few things, the whole mess got cleared away, making much needed room in my office. In April of 1988, Bruce came to Japan. While we were talking, he popped the question of when had I seen the Japanese attractor? I began telling him about the circumstances I had mentioned above, but he did not seem to be convinced. His usual warmth and friendliness gradually turned into a skeptical demeanor. I felt like I was being put on the spot. The day was April llth, 1988. I fished out some files that I had not sent to BNL, and showed them to him. He studied the rumpled report papers thoroughly and checked my scribbled numbers with hi5 calculator. Finally he seemed to be convinced, and returned to his usual self. I thought this was a good opportunity to clean up some more of the mess, and tossed the files into the wastebasket. But Mr. Bruce said, "Here, give them to me!" I wasn't particularly proud of my rumpled messy files, but agreed. He immediately attached a memo and sent them to Prof. Ralph Abraham at the University of California Santa Cruz. I have seen Bruce many times since then, but haven't seen that stern face ever again. He was probably thinking, "no matter what Yoshi says, I have to examine the actual evidence myself before deciding whether or not to believe it"-this must have been his credo, and I deeply respect that. Many of my results have been cited in his books and papers, but from what I heard, he decided to select them only after checking them carefully until he was absolutely convinced [21]. I believe that every scientist should practice such rigor.

' (Lv

51

52

Chaos Avant-Garde:Memories of fhe Early Days of Chaos Theory

Epilogue What I have been telling you is nothing but my subjective account, Nevertheless, I ~ntentionallymentioned real names, as I wanted this memoir to be accountable and traceable by the data behind it. Regrettably, Profs. Chihiro Hayashi and Minoru Urabe are no longer with us and won't be able to dispute my account. In order to keep the context intact, however, I could not avoid telling this obviously one-sided story. As for Prof. Chihiro Hayashi, for sixteen long years, from my graduate student days to the day of his retirement, I received his guidance. He was the last of the true breed of Meiji Era imperial university professors in the Electrical Engineering Dept. of the School of Engineering, Kyoto University. I tried to tone down the descript~onof his pe~onality,but as you can see in this account, he had a very strong personal it^. He was the emperor of his laboratory, and yet outwardly he was a miId-manne~dgentleman. I believed at that time that his was the most feudalistic of any laboratory in the world, and the wall of his authority was impenetrable during his reign, and I still believe it. But because of that, we were not swept away by w ~ r ~ d ~ y concerns and could concentrate on our research, being faithful to our own ideas. For this I am truly thankful. It is obvious that my research was not possible without Prof. Hayashi's presence. The rigor, will, and courage needed for one's own research-I trust that these were Prof, Hayashi's legacy. However, I was hounded by his obsession for cleanliness in diagrams. Later I received ~uidancefrom Prof. Chikasa Uenoson~.Under his guidance, I learned the rigor and intensity of the world of technology, At the same time I had an opportunity to reaffirm my belief in the importance of experiments as well as accurate perception of phenomena. Prof. Uenosono also taught me the rules and order of the human world and how to handle them. These things have greatly enriched my life. Furthermore, I would not have been sitting in my present position, had it not been for Prof. Uenosano. Needless to say, I am deeply indebted to these two professors. However, the greatest influence came from Prof. Michiyoshi Kuwahara, to whom I am most grateful. As I have focused this account on my chaos research alone, you may have had the impression that I reached the idea of randomly transitional phenomena directly without going astray. in reality, however, it was a long, meandering and groping process, as you can see if you draw the time line of my career.

53

Strange Attractors and the Origin of Chaos

What I have been working on during the period I described can be called research in nonlinear oscillation, or from a larger point of view, in basic electrical engineering or applied mathematics. I have a feeling that the people who are involved in these fields are more or less criticized continually by both mathematicians and engineers just the way I have been. The criticisms can be summarized as follows: "Has it been proven?" "Is it useful in some way?" These are reasonable questions indeed, and I have always been at a loss for an answer. I would like to take this opportunity to bring forth a counter-argument. To mathematicians, I would like to ask "How sound is your proposition?" and to engineers I would like to recommend a reading of Bob Johnstone's article entitled "Research and Innovation: No chaos in the classroom" in the Far Eastern Economic Review [22]. In the article he quotes Tien-Yien Li's observation (Li of the Li-Yorke's theorem), "If the applicability of chaos becomes apparent, the Japanese will show a fierce interest in it." Whatever the case may be, I believe it is most important, especially for Japanese researchers, to make an unbiased evaluation of one another's positions based on accurate communication. In this sense, the role of this symposium is truly momentous.

References [I] C. Hayashi, H. Shibayama and Y. Ueda, Quasi-periodic oscillations in selfoscillatory systems with external force (in Japanese), IECE Technical Report, Nonlinear Theory (Dec. 16,1961). [2] C. Hayashi, Nonlinear Oscillations in Physical Systems. McGraw-Hill, New York, 1964; Reissue, Princeton Univ. Press, Princeton, NJ, 1984. [3] C. Hayashi and Y. Nishikawa, Initial conditions leading to different types of periodic solutions for Duffing's equation, Proc. Int. Symp. Nonlinear Oscillations, Kiev, 2 (1 963), 377-393. [4] C. Hayashi, Forced Oscillations in Nonlinear Systems. Nippon Printing and Publishing Co., Osaka, Japan (1953). [5] C. Hayashi, Y. Nishikawa and Y. Ueda, Higher-harmonic oscillations in nonlinear circuits (in Japanese), IECE Technical Report, Nonlinear Theory (Nov. 29,1963). [a] C. Hayashi, Higher harmonic oscillations in nonlinear forced systems, Colloq. Znt. CNRS, Marseilles, 148 (1965), 267-285. [7] K.W. Blair and W.S.Loud, Periodic solutions of x" cx' g ( x ) = ,5j(t) under variation of certain parameters, J. SOC.Ind Appl. Math. 8, (1 960), 74-10 1.

+

+

54

Chaos Avant-Garde: Memories of the Early Days ofchaos Theory

[8] N. Levinson, Transformation theory of nonlinear differential equations of the second order, Ann. Math. 45 (l944), 723-737. [9] C. Hayashi, Y. Ueda and H. Kawakami, Transformation theory as applied to the solutions of non-linear differential equations of the second order, Int. J. NonLinear Mech. 4 (1969), 235-255. [lo] C . Hayashi, Y. Ueda, H. Kawakami and S. Shirai, Analysis of Duffing's equation by using mapping procedure (2) (in Japanese), IECE Technical Report, NLP67-13 (Dec. 8 , 1967). [ I I] T. Endo and T. Saito, Chaos in electrical and electronic circuits and systems, Trans. IEICE 73-E (1990), 763-771. [12] C. Hayashi, Y. Ueda and H. Kawakami, Solution of Duffing's equation using mapping concepts, Proc. Fourth Int. ConJ on Nonlinear Oscillations, Prague, (1968), pp. 25-40. [13] C . Hayashi and Y. Ueda, Behavior of solutions for certain types of nonlinear differential equations of the second order, Proc. Sixth Int. ConJ on Non-linear Oscillations, Poznkn, 14 (1 973), 34 1-35 1. [14] C. Hayashi, Y. Ueda, N. Akamatsu and H. Itakura, On the behavior of selfoscillatory systems with external force (in Japanese), Trans. IECE Japan 53-A, (1970), 150-158. [15] C. Hayashi, Y. Ueda and N. Akamatsu, On steady-state solutions of a nonlinear differential equation of the second order (in Japanese), Research Report RIMS, Kyoto Universi& No. 113 (1971), pp. 1-27. [16] Y. Ueda, N. Akamatsu and C. Hayashi, Computer simulation of nonlinear differential equations and non-periodic oscillations (in Japanese), Trans. ZECE Japan 5 6 4 (1 973), 2 18-225. [17] C. Hayashi, Selected Papers on Nonlinear Oscillations. Nippon Printing and Publishing Co., Osaka, Japan, 1975. [ 181 Y. Ueda, Random phenomena resulting from nonlinearity-in the system described by Duffing's equation (in Japanese), Trans. IEE Japan 98-A (1978), 167-173. [19] D. Ruelle, Les attracteurs Btranges, La Recherche 11, (1980), 132-144. [20] D. Ruelle, Strange Attractors, The Mathematical Intelligencer 2, (1980), 126137. [21] J. M. T. Thompson and H. B. Stewart, Nonlinear Dynamics and Chaos. John Wiley and Sons, Chichester, 1986. [22] B. Johnstone, No chaos in the classroom, Far Eastern Economic Review (22 June 1989), 55. [23] Y. Ueda, Steady motions exhibited by Duffing's equation: A picture book of regular and chaotic motions, New Approaches to Nonlinear Problems in Dynamics (edited by P. J. Holmes). SIAM, Philadelphia, 1980, pp. 3 11-322. [24] J. Gleick, CHAOS: Making a New Science. Viking Penguin Inc., New York, 1987.

Strange Attractors and the Origin of Chaos

55

Acknowledgments This article originally written in Japanese was translated by Mrs. Masako Ohnuki. The sequence of the events is as follows. The book CHAOS: Making a New Science by James Gleick was published in 1987 by Viking Penguin Inc. [24]. This book was very popular and stayed on the New York Times best-seller list for more than half a year, and was widely admired for its skillful explanations. The present author received a letter from Mr. James Gleick when this book was to be translated into Japanese by his nominated translator, Mrs. Masako Ohnuki. He requested me to supervise her translation, When the author read her translation, he was completely struck with admiration. The translation is accurate and there is no sense of incompatibility peculiar to a translation. The author lacks the skill to write on delicate matters in English, and he asked her to translate the article, and she kindly agreed. He also requested Dr. Hugh Bruce Stewart to review her translation and to check the contents of this article for historical accuracy. The author would like to express his sincere thanks to Mrs. Masako Ohnuki and Dr. Hugh Bruce Stewart. He also expresses his thanks to the following colleagues: Chihiro Hayashi Minoru Urabe Chikasa Uenosono Hiroshi Shibayama Michiyoshi Kuwahara Yoshikazu Nishikawa Minoru Abe Nobuo Sannomiya Masami Kuramitsu Hiroshi Kawakami Norio Akamatsu

deceased deceased Professor Emeritus, Kyoto University Professor Emeritus, Osaka Institute of Technology Professor Emeritus, Kyoto University Kyoto University Kyoto University Kyoto Institute of Technology Kyoto University Tokushima University Tokushima University

4,

My E ~ ~ o with u ~Chaos* ~ ~ r Yoshisuke Ueda Kyoto U ~ i v e r s ~ ~

By the Time It's Popular, It's Too Late It has been several years since many people began to give me credit for discovering Chaos, claiming that the output data I got from my old analog computer on Nov. 27, 1961 was in fact one of the earliest examples of chaos, and that the theory of the randomly transitional phenomenon I had proposed was in fact Chaos theory. But while I was toiling alone in my laboratory, I was never trying to pursue such a grandiose dream as making a revolutionary, new discovery, nor did I ever anticipate writing a memoir about it. I was simply trying to find an answer to a persistent question, faithfully trying to follow the lead of my own perception of a problem. Was it really such a great feat? I still am not sure. In any case, when I asked myself this question in 1979, everything about it was already a thing of the past. It is true that some friends and acquaintances had warned me earlier: "You're still messing around with van der Pol and Duffing? It's about time you quit!" For myself, however, persistence seemed to have paid off. My advice, then, to the young readers of this memoir is to faithfully pursue your own questions, and build solid work day by day, without giving in to peer pressure. Many scientists all over the world flock to popular research topics

' This Memoir was translated by Mrs. Masako Ohnuki and Dr. H. Bruce Stewart from the article in Japanese published in the Transactions of the Institute of Electronics, Information and Communication Engineers of Japan, Vol. 77, May 1994, with kind permission of the Institute. 57

58

Chaos Avant-Garde: Memories ofthe Early Days ofChaos Theory

of the day, but one should realize that by the time they become popular, it may already be too late for newcomers to begin working on them.

The Oldest Attractor was a Result of Perseverance, Not of Wisdom When I entered graduate school, Professor Hiroshi Shibayama (presently an Emeritus Professor at Osaka Institute of Technology) encouraged me to work on s ~ n c hr o n i z a t ~o ~henomena. ~ S~nchronizat~on occurs when periodic (external) signals with a frequency different from that of self-oscillation are injected into an oscillatory circuit. The self-oscillatory frequency can be entrained by the external frequency, and becomes synchronized with it. If synchronization is achieved, periodic oscillation appears; if it fails, a beat is sustained. ~sci~lation Prof. Shibayama was a research fellow at the time who came to our laboratory several times a week and helped me with my project. He was a very kind man who worked meticulously, although he did not force others to follow suit. I tried to follow his example and work as carefully as possible. I am grateful, as I realize now what he had given me was of iasting value. He was working on synchronization based on Van der Pol's equation, to which a term for a forced sinusoidal (external) frequency was added:

Deriving an averaged equation that approximates the equation (I), its point of equilibrium and limit cycle were obtained numerically using a Tiger Calculator (the commercial name for the most popular mechanical desk-top hand calculator of the day). In this approximation, the point of equilibrium represents the periodic oscillation, while the limit cycle represents the best oscillation. Our main job was to compare the approximate results with those of the original equation obtained through an analog computer. In the analog computer experiments, the periodic oscillation shifts to a beat oscillation when the amplitude B and frequency v are moved outside of the boundary of frequency entrainment. The boundary of frequency entrainment was drawn on the (B, v) plane for each of the numerous (a, v)

My Encounter with Chaos

59

values in this manner. Today it is well known that the limit of entrainment is represented by a saddle-node bifurcation (fold bifurcation) if the external amplitude B is small, and by a Hopf bifurcation if it is large. In those days, however, these phenomena had to be carefully tracked down with Tiger Calculators and low-speed analog computers. The analog computer experiments were extremely cumbersome and took an infernally long time-almost impossible to imagine today when computer technology is so highly advanced. Our work in those days proceeded as follows: first, you would switch on the analog computer and wait for 20 to 30 minutes until the vacuum tubes heated up and a stationary state was achieved. Next, you would carefully perform a drift adjustment on each of the operational amplifiers. Since the multiplier was operated by a servo mechanism, there would be a time delay if you quickened the response speed. Consequently, the period could not be any shorter than 6.28 seconds even in the case of nearly pure-sinusoidal waves with very little higher harmonic components. During the calculations, the chopper amplifier occasionally balked and had to be coaxed back to show normal wave patterns by cleaning the contact points. Then, at the crucial moment, the thread holding the recorder pen would break! If you sent it out for repair, it would take a few days, so you had to fix it on the spot yourself. If you were unlucky, it took half a day just to repair it. If you were using a recorder with a fountain pen, it was absolutely necessary to wash the tip of the pen after each experiment. And in order to prevent others from using your clean pen, you needed to hide it. In those days the University used to cut off direct current at 8 p.m. sharp. Since a part of our analog computer was powered by DC, we had to rush to the substa~ionaround 750 to plead for a time extension. The old man at the substation got to know us rather well, and did not seem to mind our frequent requests ..... So this was the way we worked. By the time I entered the Ph.D. program, I was working on a forced selfoscillatory system containing a nonlinear restoring term-in other words, a Van der Pol/Duf~ngmixed-type equation:

d' x --

dt2

dx p(1- p2)- + x3 = Bcosvt dt

I used this equation because when one chose an intermediate value of external amplitude B, that was neither too large nor too small, both the periodic and beat oscillation would coexist, thus making the boundary of

60

Chaos Avant-Garde: Memories of the Early Days of Chaos Theory

entrainment extremely complex, causing hysteresis and jump phenomena. The parameter range within which such phenomena could occur was too narrow in the case of equation (1) for analog computer experiments. But with equation (2), analog computer experiments were possible because the response curve had a greater incline, thus widening the parameter range. Thus it was through these analog computer experiments, based on equation (2), when I was studying the hysteresis between periodic and beat oscillations, that 1 happened to come across, what is reputedly the earliest observation of chaos (in a non-autonomous system, to be exact). The beat oscillation in equation (1) is called "almost periodic oscillation." Stroboscopic sampling (observation of a representative point once during each cycle of the external signals) filled out a smooth closed curve very similar to the approximating limit cycle. In contrast, when the beat oscillation based on equation (2) was viewed by stroboscopic sampling, a jagged shape resembling a broken egg shell appeared. The first time I saw it, I attributed it to malfunction of my analog computer. But within several days, I had no doubt that the computer was normal and that whenever the external parameters B and v were outside of the boundary of entrainment, this annoying and perplexing shape would appear. After spending several weeks staring day and night at the output data from my analog computer, I felt I could hear the phenomenon itself (or at least the fact revealed through these computer experiments) insisting that "the result can't be any other way but this!" In this way came the answer to my long-held question. "What are the steady states of a nonlinear system?" It just came to me-I did not plan it. I just got the answer by staring at the data for a long time. After the hysteresis between the periodic and beat oscillation based on equation (2) was largely understood, I became concerned about further bifurcations that occur only in the system based on equation (2), but not on (1). (These bifurcation occur if the angular frequency v of the external force is smaller than the self-oscillatory angular frequency.) In this case, once again, the parameter range based on equation (2) would become too narrow for analog computer simulation. The equation I found (or rather, created) was a RayleighIDuffing mixed-type equation:

My Encounter with Chaos

61

Using this equation, I repeated the averaging and the analog computer experiments. It proved to be an extremely cumbersome problem. The phenomena were too complex to produce a satisfactory approximation using a single frequency component. Moreover, it was difficult to distinguish between transient and steady states because the effect of damping was almost negligible. This made it hard to draw the limit cycle of the averaged equation using the isocline method. And the results of analog computer experiments were not reliable. What more or less saved us was the new digital computer KDC-1, which had just come out. When the external frequency was greater than that of the self-oscillatory frequency, a phenomenon similar to that of equation (2) was observed in the system based on equation (3). The illustration in the Transactions of the Institute of Electronics and Communication Engineers of Japan (53-A, p.155, Fig.6) describes the two types of beat oscillation that occur in a system based on equation (3): the almost periodic oscillation, and the strange oscillation (chaos). Although this is out of sequence, I must mention the following Duffing equation, which was thrust upon us during our work on equations (2) and (3):

d2x dx -+k-+x dt2 dt

3

= Bcost

(4)

Equation (4)was the trademark of Prof. Chihiro Hayashi's laboratory. I cannot pinpoint the exact date of my encounter with chaos within this system: it occurred sometime during the latter semester of my fourth year in graduate school (between the fall of 1962 and the spring of 1963). Prof. Hayashi was at the time writing a book to be published by McGraw-Hill. Rather than work in his office, he was sitting in the laboratory, frequently calling on his staff and students for assistance. Each chapter was sent to the publisher as it was completed. Chapter 6, entitled "Higher Harmonic Oscillation," was returned to him with a reviewer's criticism. In order to counter this, it was necessary to draw the amplitude characteristic curve using the harmonic balance method, and then confirm the result with the analog computer without delay. This urgent task fell on my lap. It was a tough job, especially under pressure of time, but somehow I completed it for the deadline. During the course of this work, I again encountered a whole slew of chaos. These studies were the origin of the strange attractor which

62

Chaos ~ v a n ~ - G u r~~eem : o ~ ioef sthe EurfyDays of Chaos Theory

Prof. D. Ruelle of the Institut des Hautes Etudes Scientifiques later called "The Japanese Attractor." The experience with this Duffing equation made me confident that chaos was not just a freak ~ h e n o m e ~ ounn ~ ~ to u ea forced self-oscillato~system (a system that sustains itself oscillation after the periodic external force has been removed), but one that also occurs in forced oscillatory systems-in other words, rather than being a variation of an almost periodic oscillation, it was a phenomenon in itself.

Professor's Principles and ~ u t u aTrust l In writing this memoir, I cannot but touch on Prof. Hayashi's work habits, and the standards he maintained in his work. He was thorough, meticulous and strict. I had a deep respect for his care and perfectionism in experiment. He also had an extraordina~passion for the aesthetics of the diagrams in his papers. It was I who took the direct onslaught of this obsession. First I resisted it, but after a long exposure to this habit, it has become my second nature. Early on, partly being proud of Prof. Hayashi's trust in my work, I even took on drawing diagrams for other people's papers and books totally unrelated to my own work. As I got older, however, I realized it was the most inefficient use of my precious time. Even so, I did not protest and kept drawing diagrams for Prof. Hayashi albeit begrudgingly until his retirement. I did lose a lot of my time for his principles, but what I gained from him amply compensates it. I absorbed Prof. Hayashi's principles by osmosis so to speak-so much so that I believed him without questioning when he told me about his experimental observation of a sustained transient condition where subharmonic oscillation with higher harmonic components appears, as the voltage is increased. He was telling me, in essence, "chaos was observed in an actual circuit," although he himself never accepted the concept of chaos.

Publication of My First "Chaos" Paper The fact that work long pursued by a single researcher has gained recognition only after twenty-some years might seem to imply a lack of awareness or interest by other scientists. During my long pursuit of the

My Encounter with Chaos

63

subject, I never hid my data from others or kept them secret. On the contrary, I kept reporting my results and observations in papers as well as at scientific meetings. In discussing the subject with others, however, I often found myself retreating after failing to convince my powerful opponents. I was even subjected to reprimands. Eventually I resigned myself to believing that meetings weren't important as long as I understood what I believed to be true, and that no one would lend an ear to a powerless research fellow's opinions until the right time arrived, This was my feeling in 1972 when I submitted my second "chaos" paper, proposing my understanding of randomly transitional phenomena (chaos) to the Transactions of the IECE whose motto was to accept papers both worthy and seemingly unworthy (the Trans. of IECE, 56-A, pp.218-225, 1973). My only ally in this chaos research was Norio Akamatsu (presently a Professor of Engineering, School of Engineering, Tokushima University) who was a graduate student at the time. The first person who gave recognition to my chaos research was Hiromu Momota (associate professor, Kyoto University, School of Engineering at the time: presently a professor at the Nuclear Fusion Science Laboratory). It was in the early morning hours (3 a.m.) of March 3, 1974, the first day of entrance examination in the midst of the student unrest. Ten or so of us faculty were captured by the students during the meeting of associate professors and lecturers, and were forced to spend the night locked up in a classroom. Since we had nothing to do, we decided to talk about the work each of us was doing at the time. Impressed by my work on chaos, Momota encouraged me. I was delighted. He later introduced me to the Plasma Laboratory at Nagoya University. My liaison with the Laboratory eventually opened the way to our cooperative work in computer use for experiments. Although Kyoto University also had computers, we could not use too much computer time there, since our funding was scarce. At Nagoya, I could use the computers to calculate the power spectra of chaos. This work was published in the Transactions of the Electrical Engineers of Japan (98-A, pp. 167-173, 1978).

64

Chaos Avant-Garde: Memories of the Early Days of Chaos Theory

Since Chaos Became Fashionable So far I have written of the dark hours before chaos was universally recognized. After that, however, I have had many happy experiences. Thanks to the popularity of chaos, I can now write a memoir like this. When I was asked to recount my experience, I wasn't sure if I had anything left to say, since I had poured everything into my invited speech at the 1991 International Symposium, "The Impact of Chaos on Science and Society," organized by the United Nations University together with the Tokyo University; the Proceedings of the Symposium will be published soon. I decided to write this piece, however, at the urging of the former chairman of the Non-Linear Problem Study Group of this organization, Prof. Masami Kuramitsu, who reminded me that it was the duty of us old timers to show the reality of scientific research to young scholars who are eager to do good work, but who are unsure how to proceed. I have tried not to duplicate what has already been published. The reference material for this memoir is included in my book entitled The Road to Chaos (published in 1992 by Aerial Press, P.O. Box 1360, Santa Cruz, CA 9506 1). In closing, I would like to express my deep gratitude to the editorial members of this journal for their effort in planning and publishing this memoir.

5.

Reflections on the Origin of the Broken-Egg Chaotic Attractor Yoshisuke Ueda Kyoto University

Abstract We return to the context of nonlinear oscillation theory in which we began, in 1961, our experience with chaotic behavior. This context involves the systems of forced self-oscillators arising in electronic circuits. In this paper we combine earlier examples into a single mixed system, transform into a form suitable for perturbation analysis, and obtain information on the bifurcation diagram of this system. In particular, the parameter regime of the broken-egg chaotic attractor, which we discovered in analog simulation in November 1961, is mapped.

Introduction We consider the classical context of nonlinear oscillation theory: a periodic signal is applied to a self-oscillator. The behavior of this forced system depends on the amplitude of the forcing signal, and its frequency relative to the frequency of the unforced self-oscillator. In case the forcing frequency is sufficiently close to the natural frequency of the unforced self-oscillator, and with moderate amplitude, we observe synchronous behavior: the forced system oscillates with the forcing frequency. With greater difference between the forcing and the free frequencies, we may find harmonic 65

Chaos Avant-Garde: Memories of the Early Days of Chaos Theory

66

oscillations (subharmonics, superharmonics, or m/n harmonics), almostperiodic motions, or even chaotic behavior. Beginning in 1959, we were studying particular systems of this type arising in electronic circuits with triode vacuum tubes. Generalizations or combinations of the Rayleigh, Duffing, and Van der Pol systems were extensively studied, using early analog computers made in our laboratory. And in 1961, in this context, we discovered a chaotic attractor, the brokenegg attractor, which has since been found to be characteristic of many systems [ 1,2]. In this paper we revisit this context, with the power of modern digital computers, and some new analytical ideas, to obtain further insight into the bifurcation diagram, in which parameter regimes of the m/n harmonics and broken-egg chaotic attractor may be clearly visualized. In particular, we use the methods of perturbation theory as an analytical aid to the extensive simulations and their interpretation. The equations reduce, when damping is reduced to zero, to a particular conservative system we call the generating system.

The broken-egg chaotic attractors We begin now with the specification of two different examples of the type of systems described above, in both of which, broken-egg attractors are observed.

DufJingIVan der Pol mixed type equation Consider the equation

d2v 2 - p(1- y”)’--&

dV

f

v 3 = Bcosw

This is a combination of Duffing forced pendulum and Van der Pol selfoscillator systems [1,3]. Extensive simulations of this mixed system, led to the discovery of the broken-egg attractor shown in Fig. 1 on November 27, 1961, with parameter values p = 0.2, y = 8, B = 0.35 and v = 1.02.

Reji'ections on the Origin ofthe Broken-EggChaotic Attractor

67

Fig. 1. Output of an analog simulation of Equation (1) with IJ. = 0.2, and B = 0.35, obtained on 27 November 1961. A continuous orbit is drawn lightly on the vv plane and points in the Poincare section at phase zero are indicated by heavy dots. Five dots near the top are fixed pointsforasequenceofvaluesat V = 1.01, 1.012, 1.014, 1.016, and 1.018. The remaining points are on the chaotic attractor with V = 1.02.

y

= 8,

RayleigWDuBng mixed type equation Consider next the equation

This is a mixture of the Rayleigh and Duffing systems. Simulations in 1963 led to the Figs. 2, 3, and 4, published in 1970, with parameter values p = 0.2, y = 4, B = 0.3 and v = 1 . I [I]. By hindsig~t,we recognize a chaotic attractor of the broken-egg type in the ring domain, shown hatched in Fig. 4.It is the closure of the -branch of the directly and inversely unstable 2-periodic points marked ' D2j , iZj2 ( i , j = 1,2) in Fig, 3.

A

3

89

Rejections on the Origin of the Broken-Egg Chaotic Attractor

69

Fig. 4. Ring domain within which the chaotic broken-egg attractor for Equation (2) is confined. The parameters are p = 0.2, y = 4, B = 0.3, and V = 1.1,

Parameter regimes of the broken-egg attractors In both cases above, the broken-egg attractor is found in the regime of parameter plane of frequency and amplitude belonging to the area of the doubly periodic, or 1/2 harmonic, motion.

Modified RayleighlDuffingNan der Pol equations Next we combine the two systems discussed above into a single system. The first step is a modification of the damping term of the Van der Pol equation, introducing a fourth degree term. Then this is combined with the RayleigWDuffing system, resulting in these equations.

" I

dt

70

Chaos Avant-Carde: Memories of the Early Days of Chaos Theory

Note that this is no longer a second order equation, due to the new term in the first equation. The parameter p is considered a small parameter, or pe~urbationparameter, as we shall see below. Without the forcing term, F = 0, the limit cycle representing selfoscillation is expressed by u4 3- 2v2 = 1

(4)

Its period is given by

This artificial system (3) is now the target of our analysis. We shall study the system by digital s~mulationto discover its qualjtative phenomena.

Regimes of the various harmon~co~ci~~ations We are now ready to present the results of the simulations of the modified system, and all of these are shown in a single map of the (v,F)parameter plane with p fixed at 0. I, in Fig. 5. In this map of the (v, F ) parameter plane, the effects of hysteresis are suppressed. This has been done by marking only points at which a bifurcation occurs, that is an harmonic disappears, as the parameter point moves from inside to outside of a regime. Thus, what is indicated for each m1n harmonic may be regarded as the outer envelope of its regime. Note that the regimes all have a roughly similar shape, which we call a question-mark regime. These are extensions of the tongue-shaped regimes well known in related contexts for smali values of amplitude. Note also that each question-mark regime contains a shaded horned crescent-shaped regime. For example, the shaded horned crescent within the 112 harmonic regime is a 2/4 harmonic regime, and a parameter change from the 1/2 regime into the shaded horned crescent causes a period-doubling bifurcation. Indeed, continuing a parameter motion in this direction leads, by a perioddoubling cascade, to a broken-egg chaotic attractor.

d

vl

Reflections on the Origin of the Broken-Egg Chaotic Attractor

L

II

0 Y

9 I

0

71

Chaos Avant-Garde: Memories of the Ear& Days of Chaos Theory

12

Preliminary transformation of the equation into perturbation form Having in mind the procedures of perturbation theory, we now wish to transform our modified equation by a transformation of variables, as follows.

z = vt, x = U I V , y = V I V2

(6)

Substituting into equation (3) we obtain

I

-=pvdx

c

7 - x -2y

x+y 2,

F -= -x3 + pv-?(71 x 4 - 2 y 2 ) y + - c V0 s3. r

(7)

Proposal of a periodically forced two-dimensional system Inspired by the new form of our system (7), we now pose a similar (but not exactly equivalent) system,

:I

-dx= € ( a

4

- x 4 -2y2)x+y

3= -x3 + € ( a

4

- x4

-2y2)y+ajFcost

(8)

in which the new parameter a is analogous to 1 I v in (7). Note that, roughly speaking, the parameter a determines the average amplitude of the resulting oscillation. The new parameter E is analogous to pv3 of (7). However, we now regard E as a small parameter, to be reduced gradually to zero. And in case E = 0, and replacing u3F with B ,and we obtain the system, =Y

(9) - = -x3 3- Bcost

dt

Reflections on the Origin of the Broken-Egg Chaotic Attractor

73

which we call the generating system. We proposed this new system in order to simplify the analysis of equation (3). Note that equation (3) is also in perturbation form. But as the small parameter p goes to zero, we obtain a generating system with two parameters, F and V . But now we have a generating system with only one parameter, B. With all this preparation, we may now present the results of our simulations of the new system (8). These are shown in Fig. 6, in which we show only three parameter regimes of the mln harmonics, with mln = 1/1, 112, and 1/3, in the parameter plane of ( u , F ) ,with E = 0.1. Within the 1/2 harmonic regime we also show, shaded, one crescent of the 214 harmonic. Finally, we show as dotted curves, the loci of constant values of the new parameter, B(= u'F) . On each curve, we have the same generating system.

Numerical experiments based on perturbation theory In this section we are going to explore the behavior of the system in the perturbation form, as given in (8) above. Note in Fig. 6 that one of the contour curves of the parameter B , namely that for B = 0.2, is shown as a slightly heavy broken curve. We are going to explore our system along the part of this contour where the broken-egg attractor is found.

The chaotic sea within the phase portrait of the generating system Recall that as the perturbation parameter E decreases, we imagine the behavior of the system tending to that of the generating system, see equations (9), which is a conservative, second-order system for each value of B . But now the value of B is fixed at 0.2, and this specifies a single case of the generating system. We now consider the behavior of this particular conservative system. Figure 7 shows a "chaotic sea'' in the phase portrait of this system, corresponding to a fixed phase of the forcing oscillation, as found by simulation. That is, a computed (discrete, or stroboscopic) orbit densely fills this region. In this figure, the structure of the chaotic sea is shown as follows. First of all, the points N , 0, R are fixed points:

" 0-4

0.6

0.8

1.0

a Fig. 6. Regimes of the 1 4 , 112, and 1!3 harmonic oscillations observed in the system governed by Equation (8) with E = 0.1.

1.2

75

Reflections on the Origin of the Broken-Egg Chaotic Attractor

-21

I

-2

-1

1

I

0

1

X

Fig. 7. Chaotic sea within the phase portrait of the generating system (9) with B = 0.2.

N is a stable fixed point, corresponding to a Nonresonant fundamental harmonic oscillation, D is a Directly unstable fixed point of saddle type, and R is a stable fixed point, corresponding to a Resonant fundamental harmonic oscillation. Next, we see in the figure that the chaotic sea is roughly a ring domain. That is, it is topologically an anchor ring, bounded by two cycles, from which some islands have been deleted. The outer boundary, cardioid-shaped, is a nongeneric homoclinic curve. That is, the inset ( W-branch) and outset (a-branch) of the saddle point, D, coincide. The inner boundary is a necklace generated two 19-periodic orbits. One is stable center, the other is of saddle type with nongeneric homoclinic limit sets. The necklace, that is the inner boundary of the chaotic sea, is this homoclinic set.

Chaos Avant-Garde: Memories ofthe Eady Days ofchaos Theory

76

Finally, we note the appearance of many deleted islands, in which are found stable periodic orbits. These islands are in fact organized in finite sets, which are permuted periodically by the discrete dynamics of the generating system. We have observed, in particular, some of the larger islands, having periods 1, 3, 5 , 9, and so on. These aspects of the generating system are important to our study, as will appear later.

B = 0.2

0 -

1

1.o

0.5

a Fig. 8. Bifurcation diagram in the (a,E ) parameter plane for the Equation (8) with

B(= a’F) = 0.2.Here

E plays role of perturbation parameter.

Bifurcations of the broken-egg attractor We now consider the perturbation system as the perturbations parameter, E , tends to zero. We will see that its portrait does not tend to that of the

Reflections

on the Origin of the Broken-Egg Chaotic Attractor

77

generating system, but it does comes into an interesting relationship with it. We will organize our many simulations into a bifurcation diagram, showing the bifurcations of the portrait or the perturbation system, (8), as & decreases from 0.125 to 0, for various values of a in the range 0.5 to 1.0. This bifurcation diagram is shown in Fig. 8, which we now explain. First of all, the (a,&) parameter regime in which the broken-egg attractor occurs is shown shaded in this figure. This is divided into two parts by a curve. In the upper part, there is a single attractor, the broken egg. And in the lower part, there coexist two attractors, the broken egg and a resonant fixed point, R, which tends to the point R described above for the generating system. Next, consider a sequence of four points in the parameter plane of Fig. 8. These are all on the line defined by Q = 0.75, and have E values of E , ( = 0.075), &, ,,,,e r ( = 0.0579), &*(= 0.050), and &,,,(= 0.0404). The portraits of the attractor of the perturbation system corresponding to these parameter values are given in a tableau in Fig. 9. Note that at Eupper, which is on the upper bifurcation curve, the second attractor, R, appears. And at which is on the lower bifurcation curve, the broken egg vanishes in a blue sky catastrophe, and only the attractor R persists. Note that the two bifurcation curves meet at a codimension-two bifurcation point in the upper right of this diagram. Finally, consider a sequence of points in the diagram of Fig. 8 of constant value E = 0.1, but decreasing values of a,a3(=0.647), aN(= 0.544). As a decreases, the broken egg vanishes by implosion to a 3periodic point at u3.And decreasing below aN,there appears a nonresonant fixed point, N , corresponding to the point N discussed above for the generating system. However, when we increase a above a3 with the same value of E , we observe at aR(=0.977) the broken egg disappears, and only the attractor R appears. Note. The regime of existence of the 3-periodic point attractor is shown in Fig. 6 with the label 1/3. The regimes of N and R belong to the area marked 1/1. Note that aN, a3, and aR marked on this figure correspond exactly to the points labelled thus on Fig. 8. The locations of the points aN and u3 are nearly independent of the parameter &, but aR descends with decreasing &. Thus the left-hand boundary of the 1/1 regime in Fig. 6 moves further left with decreasing E .

78

Chaos Avant-Garde: Memories of the Early Days of Chaos Theory

broken egg

broken egg

broken egg

Fig. 9. Bifurcation of the broken-egg attractor and the resonant fixed point R, the parameters being a = 0.75 and B(= a 3 F )= 0.2,

79

Reflections on the Origin of the Broken-Egg Chaotic Attractor

21

1 '

Y

0 .

-1

-2 1

I

-2

-1

1

0

1

2

X

Fig. 10. The broken-egg attractor with a = 0.75, a'F(= B ) = 0.2, and & = 0.041 for Equation (8) is shown inside the chaotic sea of the generating system (9) with B = 0.2.

Discussion We may summarize our results in experimental perturbation theory as follows. The chaotic sea in the limit, or generating system, is the mother of the broken-egg attractor in the perturbation system, with small dissipation, & . The discontinuous limit behavior is dissected in the bifurcation diagram, Fig. 8. A further detail regarding the approach of the baby broken egg to the mother chaotic sea is shown in Fig.10. This shows the broken egg superimposed over the chaotic sea, just before the disappearance, with a = 0.75, and E = 0.041,

80

Chaos Avant-Garde:Memories of the Early Days of Chaos Theory

Conclusion We now step back and regard what we have done. We began with a mixed system evolved from our earliest work in nonlinear dynamics. But we ended up with an exercise in experimental perturbation theory. We have exploring here an example in a classical theme, the near-Hamiltonian system. Experimental results such as these might well lead the way to further theoretical developments. One further direction, for example, might be a qualitative theory of perturbation for a nonlinear dynamical system. Another direction might be the inverse problem, to construct a dynamical system in analytical form corresponding to a given chaotic attractor, perhaps determined from data.

Acknowledgments We are very grateful to Dr. Hirofumi Ohta for assistance with computer programming, Associate Professor Takashi Hikihara for various advices and Professor Ralph Abraham for discussions.

References [ I ] Y. Ueda, The Road to Chaos (Aerial Press, Santa Cruz, CA 95061, 1992). [2] Y. Ueda, Strange Attractors and the Origin of Chaos, in J. A. Yorke and C. Grebogi, Eds. The Impact of Chaos on Science and Society (United Nations University Press, 1997), pp. 324-354, Proc. Int. Symp. on The Impact of Chaos on Science and Society, Tokyo, Apr. 15-17, 1991. Also published in Nonlinear Science Todqv, Vol. 2 , No. 2, pp. 1-16, Springer-Verlag, 1992. [3] C. Hayashi, Forced Oscillations in Nonlinear Systems (Nippon Printing and Publishing Co., Osaka, Japan, 1953).

6.

The Chaos Revolution: A Personal View* Ralph Abraham University of Calfornia at Santa Cruz

Introduction To me, The Chaos Revolution was one of the most important social transformations of all time (Abraham, 1994). To others, it was a passing fad. Whatever the judgment on this issue, it has been a big factor in my life. And due perhaps to the occurrence of my name in an extraordinarily popular book by a journalist, I have been consulted by many scientists involved in the paradigm shift from order to chaos when it first appeared in their own fields (Gleick, 1987). These first-hand experiences are the grist of this chaos story.

The Discovery of the Homoclinic Tangle by Poincarb, 1889 This story has been told only recently (Barrow-Green, 1997; Peterson, 1993). Dedekind (a professor at Berlin) claimed to have proved the stability of the solar system, then he died. Weierstrass (also professor at Berlin) tried to provide the proof, without success. His students Mittag-Leffler and Kovalevskaya (professors at Stockholm) persuaded King Oscar I1 of Sweden and Norway to offer a prize for the solution of this unsolved problem, the winner to be announced on his 60th birthday, January 12, 1889. Weierstrass, Mittag-Leffler, and Hermite (professor at Paris) were the judges. Poincard (a Based on a lecture at Kyoto University, March 20, 1998 81

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student of Hermite) was judged the winner. But before the prize paper had been published, questions raised by PhragmCn (professor at Stockholm) who was editing the prize paper for publication in July of 1889 led to the discovery by PoincarC of a mistake in his proof. Due to professional jealousies, ethics, and royal interest, there was a lot of pressure on PoincarC to repair the error. He succeeded, and the paper he originally submitted for the prize was hidden away, and replaced by a new one, in which the discovery of the homoclinic tangle first appeared. The discovery must have been around December 1, 1889. The replacement prize paper was submitted in January of 1890, and published later that year. See Barrow-Green (1997) for support of these dates, and Abraham (1992; Part HI) for an extensive visual introduction to homoclinic tangles.

From Paris to Mexico City: The First 70 Years of Chaos Theory Chaos theory has been known variously as dynamical systems theory, the theory of nonlinear oscillations, the qualitative theory of systems of ordinary differential equations, and the mathematical theories of chaotic attractors and their bifurcations. Its peregrinations from the big bang of PoincarC in 1889 to me in 1960 involves the mathematical histories of France, Russia, the United States, and Mexico. The main sequence of events may be pieced together from historical essays by Okan Gurel (Gurel, 1979; Introduction), Christian Mira (Abraham, Gardini, and Mira, 1997, Appendices 5, 6; Mira, 1997), and others (Hirsch, Marsden, and Shub, 1993, Chs. 1 - 10, 17). The short version of the story, as I know it, is this. The new ideas of PoincarC, following his death in 1912, were continued by the young American mathematician, George David Birkhoff (professor at Harvard). This American line of heritage, however, soon died out. Meanwhile, perhaps inspired by Liapunov, the Russian contemporary of PoincarC and independent pioneer of the qualitative theory, the Moscow-Gorki school, beginning with Mandelstham around 1925 and his outstanding student Andronov, followed parallel lines, which continue to the present day. Concurrently, a European tradition evolved, including the engineers Duffing in Berlin and Van der Pol in Holland. The revival of dynamical systems theory in the Americas was due to an intentional intervention by Solomon Lefshetz, the great pioneer of algebraic

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topology. Born in Russia and trained in electrical engineering, he switched to mathematics after an accident which claimed both of his forearms. Following his original work in algebraic topology during the 1930s and 40s, he returned to applied topics. Through his familiarity with the Russian language, he became aware of the work of the Moscow-Gorki school. In order to stimulate work in dynamics in the United States and Europe, he translated one of the main works of that school into English (Andronov and Chaikin, 1949). Another factor might have been the publication of a text in English on the Russian developments (Minorsky, 1947). At the same time, in an altruistic effort to promote mathematics in Latin America, Lefshetz began spending half of each year at the National Autonomous University of Mexico, in Mexico City. It was there that I first met him, in the summer of 1959.

Among his students were several excellent topologists, including Mauricio Peixoto. The 1958 work of Peixoto on structurally stable systems in the plane, published in 1959, triggered the revival of the PoincarC tradition in the United States. Stephen Smale met Peixoto in 1958, and learned of this work. (Hirsch, Marsden, and Shub, 1993; Ch. 2) A meeting in Mexico City in the summer of 1956 brought together Lefshetz, Smale, and others, who would figure prominently in chaos theory.

The Golden Years of Global Analysis: Berkeley, 1960-1968 With my thesis finished in early 1960, I looked for a job. My thesis advisor, Nathaniel Coburn, was very helpful, but by the late Spring, I had only one offer. I prepared to move to Milwaukee. But at the last minute, I received another offer, for a special position with reduced teaching, from the University of California at Berkeley. I accepted at once and moved to Berkeley, not knowing yet that a substantial number of leading mathematicians were moving there at the same time. When I arrived, I began to meet these people-Chern, Spanier, Hochschild, Smale, Hirsch-without fully realizing the level on which they were working. In October, 1960, I met Smale in the daily math tea, and innocently asked him what he was doing. This began a working partnership and friendship which spanned several years. Our subject was then called global analysis, and included dynamical systems theory, the calculus of variations, manifolds of mappings, nonlinear functional analysis, partial differential equations, and so on. We would speak

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regularly-over a span of eight years, moving separately around Berkeley, Columbia, and Princeton-on all areas of global analysis, and especially about homoclinic tangles. While I obtained results on manifolds of mappings and transversality theory, Smale obtained a number of results on dynamical systems theory which dominated the advance of the subject at the time, and were summarized in his fantastic survey paper of 1967. Among these results my favorite was his spectacular work on the horseshoe map, in which the homoclinic tangle of PoincarC was untangled for once and for all. Here are some highlights of the eight golden years. 1960-61. In Berkeley, Smale and I reread Birkhoff's collected works, particularly a paper written jointly with Paul Smith in 1928, and had many discussions on homoclinic tangles of diffeomorphisms of the plane. We were also very interested in Hamiltonian dynamical systems. Rend Thom was in Berkeley at this time, and I attended his course on singularity theory. At the end of this year we had a Summer conference on dynamics in Berkeley. Shlomo Sternberg and Sol Schwartzman were among the active participants. Then we went to Urbino, where Smale presented his work on global stable manifolds, and we went on to Bonn, where I presented my version of transversality theory for the first time. 1961-62. Smale transferred to Columbia University, where ironically Paul Smith was the chair, while I stayed on for a second year in Berkeley. During this time I frequently spent the lunch hour with Moe Hirsch and Ed Spanier at the swimming pool on the Berkeley campus, and we enjoyed long and fruitful discussions on global analysis, transversality theory, and the like. Among other things, these swims resulted in some of the main ideas presented in my 196'7 book, Transversal Mappings and Flows, and Morris Hirsch's 1976 book, Differential Topology. These happy times are still vivid in my memory. 1962-63. I followed Smale to Columbia, where we resumed working together. He was interested in quantum mechanics and variational calculus. I gave lectures on transversality theory in his course, especially during his absence occasioned by the Bay of Pigs crisis, as recounted in his book, The Dynamics of Time. My notes for these lectures were published as a pamphlet in 1963, entitled Lectures of Smale on Differential Topologv, even though they covered primarily my own lectures. One goal of my lectures was to firmly establish Birkhoffs signature of a tangle. During this year we met Bob Williams, Mike Shub, and Charlie Pugh, who eventually joined our

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group. I returned to Berkeley for the summer to find Smale and Hirsch very occupied with an antiwar movement, the Vietnam Day Committee. 1963-64. My second year at Columbia, Smale had gone back to Berkeley. I continued to work on features of homoclinic tangles, such as the signature of Birkhoff. (This work was not published until 1985.) In the Spring, I accepted an offer from Princeton. 1964-65. My first year at Princeton, I was assigned a course called Honors Calculus. The 15 students were extremely good, most of them followed my lectures for four years, then went on to graduate school and became professors. Harold Abelson, Michael Buchner, Len Fellman, Carl Morgenstern, and Lee Rudolph were in this group. 1965-66. My second year at Princeton. I was asked by Arthur Wightman of the physics department to offer a course in mechanics, including the new results of Kolmogorov, Arnold, and Moser. The course began in February, and attracted an excellent audience of visitors and graduate students, including Jerrold Marsden. By the end of July, Marsden's lecture notes had become the manuscript of a book, Foundations ofMechanics. At the same time, Joel Robbin's notes of my lectures for another graduate course became a book, Transversal Mappings and Flows. In addition, Thom was sending me draft chapters for his book on structural stability and morphogenesis. Through Wightman, I arranged to have Thom's book published by Bill Benjamin. 1966-67. Most of this year was spent on sabbatical in Paris, where I renewed my friendship with Thom, and met David Ruelle and Harold Levine. Smale's very influential article on his program for dynamics appeared in the Bulletin of the American Mathematical Society, in which he introduced basic sets and strange attractors. 1967-68. My last year in Princeton. After a party with some of my great undergraduate students, I was passing outside my ground floor office on the way home when I heard the phone ringing. I ran in to find Steve Smale calling from Berkeley, with a hypothesis on generic properties of dynamical systems. A counterexample came immediately to mind, which we presented in a joint paper in the climactic event of the golden years, the Summer Institute on Global Analysis of the American Mathematical Society, held in Berkeley in 1968 (Chern and Smale, 1970). Following this event, I moved to the University of California at Santa Cruz.

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1971. A conference in Brazil brought many of these people together during the twilight of our golden age. At this time I noted the beginning of a backlash movement against Thom and catastrophe theory (Peixoto, 1973). 1990. The cast of characters met in Berkeley for a reunion in honor of Smale's 60th birthday (Hirsch, 1993). 1998. Another reunion, this time i n Cincinnati, in honor of Bob Williams' 70th birthday.

The Chaos Revolution: 1968-1998 An amazing aspect of our work during the early 1960s was our total ignorance of the experimental work on chaotic attractors of Yoshisuke Ueda in Kyoto, from 1961, Edward Lorenz in Cambridge, from 1963, and Christian Mira in Toulouse, from 1965. Ueda, in particular, had painstakingly drawn the homoclinic tangle which formed the skeleton of his chaotic attractor. I do not remember how or when this exciting news penetrated our circle. For myself, I believe that during my Princeton course on mechanics in the Spring of 1966, I began to hear about chaos in the solar system, and later, about the Lorenz attractor. But certainly by the 1968 Berkeley Summer Conference of the American Mathematical Society, we all knew that our particular line of work had come to a halt, for our favorite hypotheses were not satisfied by the chaotic attractors recently discovered by the experimentalists. At this time, many of us turned to applications, in an effort to reground and to reorient our work. Personally, I was excited by the work of RenC Thom on catastrophe theory. In this context, Thom had introduced the basic concepts of a radically new direction in applied mathematics, with his ideas of attractors and basins, and catastrophic bifurcations. I went to Paris (that is, to IHES, the Institut des Hautes Etudes Scientifiques in Buressur-Yvette) to study with him in 1967, and again in 1972. It was during these years, after I had moved from Princeton to Santa Cruz in 1968, that the chaos revolution began in earnest. The crux, I believe, was the work of Takens and Ruelle on turbulence. Fluid dynamical turbulence, before chaos theory, was an embarrassment to theoretical physics, so it was downloaded to the engineers. Floris Takens had finished his thesis on dynamical systems theory (Berkeley style) in Holland in 1969, and went to IHES for a year. There he met the resident theoretical physicist,

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David Ruelle, a uniquely capable mathematician. Together they made a new model for the onset of turbulence in fluids through a sequence of bifurcations. Their paper was rejected by various journals as heretical, but finally, in 1971, it appeared in a journal of which Ruelle himself was an editor. And that, I believe, was the turn~ngpoint: chaos theory triggered the chaos revolution, a sequence of paradigm shifts in the various branches of science. First physics, then astronomy, chemistry, biology, medical physiology, and then economics, and the social sciences. In 1972, Thom had introduced me to the work of Hans Jenny of Basel, showing fluid dynamical forms created by vibrations in transparent liquids and powders. I hastened to Basel to meet Jenny, who showed me films and photos of his results. In 1974 I returned to UCSC from Amsterdam, India, and the casinos of Nevada, to initiate a program of research on vibrations, chaos, and spatial bifurcat~onsin transparent fluids. In the student machine shop, I built a device, modeled on that of Jenny, the Jenny Macroscope. Along with ideas of Thom and Zeeman, it contributed greatly to my marriage of ideas from chaos theory and from Hindu cosmology. A number of students worked with me on this project, Paul Kramerson in particular. At this time, through Terence McKenna, I met Erich Jantsch, who encouraged me to record my ideas in articles for his books. In 1975, just as Li and Yorke were bringing the word chaos into the picture, I turned completely towards experimental and applied chaos theory. The trigger was the arrival on our campus of the newly developed computer graphic device, the Tektronix 4006 "green screen." This new direction got a boost from Richard Palais, who wrote a BASIC program for dynamical systems research, ORBIT, during his brief stay in Santa Cruz. Our program, the Visual Math Project, was supported by state and federal funds until 1982. It was aimed primarily at suppo~ingthe lower division math courses with interactive computer graphics, for which we created extensive software in C. By 1983, the Visual Math Project had morphed into a thriving graduate program in applied and computational mathematics (read chaos theory) at UCSC. This program came to an end with my early retirement from UCSC in 1994, During the same period, the 1970s and '~OS,an autonomous group of graduate students in the physics depa~ment,later known as the Santa Cruz Chaos Cabal, began publishing significant experimental results. A meeting at the New York Academy of Sciences in 1979 brought many of the chaos pioneers together for the first time. An historic summer school on chaos

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theory in Les Houches, in 198 I , brought chaos theory to the attention of the international physics community. And during these years, I created, with Chris Shaw, the lengthy picture books of dynamical systems theory, Dynamics: The Geometry of Behavior. As my usual publisher, AddisonWesley, refused to publish these books in four colors as I wanted, Aerial Press sprang into existence to self-publish them. Time proved AddisonWesley correct, as Aerial income was rather less than out-go. It was in medical physiology, from 1978 to 1986, that I made my first serious efforts to make a difference with chaos theory. Walter Freeman (neurophysiology) and Gene Yates (endocrinology) were among my early collaborators. But it was the enormous popularity of Gleick's book, Chaos, from 1987 to about 1990, which brought me into contact with scientists from many fields, most notably: Richard Goodwin and his group in dynamical economics, Ervin Laszlo and his General Evolution Research Group (GERG), and William Irwin Thompson and his Lindisfarne Association. At a conference on Goodwin-style economics in 1991, I met Christian Mira and Laura Gardini, and heard for the first time of the method of critical curves, which became the subject of our joint book in 1997. I might end here with a list of some of the paradigm shifters whose stories I knew first hand, as a result of this brief wave of popularity, or otherwise. I am not sure of all the dates. Mauricio Peixoto, structural stability theory, 1960 Yoshisuke Ueda, electrical engineering, the first sighting, 1961 Christian Mira, control theory, 1965 Rene Thom, math, catastrophe theory, 1966 - David Ruelle and Floris Takens, physics, turbulence, 1969 * Otto R(issler, chemistry, taffy pulling, 1974 * Erich Jantsch, systems theory, 1975 * Christopher Zeeman, math, catastrophe theory, 1977 * Santa Cruz Chaos Cabal, physics, attractor reconstruction, 1978 - Eugene Yates, endocrinology, 1979 Walter Freeman, neurophysiology, olfactory bulb, 1979 * Alan Garfinkel, medical physiology, heart muscle, 1979 - Hermann Haken, physics, synergetics 1980 . Benoit Mandelbrot, math, fractal geometry, 1985 - Michael Mackey and Leon Glass, medical physiology, 1985 - Heinz-Otto Peitgen, math, fractal geometry, 1985 - Ervin Laszlo, social dynamics, 1985 *

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Fred Abraham, Psychology, 1988 Kate Hayles, literature, 1990 * William Irwin Thompson, history, 1990 * Herbert Shaw, Earth Sciences, 1991 . Richard Goodwin, economics, 1991 Robert Langs, Psychiatry, 1993

-

Conclusion Triggered by a mathematical discovery, the Chaos Revolution is a bifurcation event in the history of the sciences, comprised of sequential paradigm shifts in the various sciences. Perhaps it is also a major transformation in world cultural history: time will tell. Meanwhile, we are struck with the personal observation of the similarity in the sociological and psychological experiences of the various pioneers who have suffered from the novelty or their ideas, and the bravery of their convictions. We are deeply in their debt.

Bibliography Abraham, Ralph, Lectures of Smale on Direrential Topology. New York: Columbia University, 1963. Abraham, Ralph, Foundations of Mechanics: A Mathematical Exposition of Classical Mechanics with an Introduction to the Qualitative Theory of Dynamical Systems and Applications to the Three-body Problem. New York: W . A. Benjamin, 1967. Abraham, Ralph, and Joel Robbin, Transversal Mappings and Flows. New York: W . A. Benjamin, 1967. Abraham, Ralph, Chaos, Gaia, Eros. San Francisco, CA: Harper-Collins, 1994. Abraham, Ralph, Laura Gardini, and Christian Mira, Chaos in Discrete Dynamical Systems: A Visual lntroduction in 2 Dimensions. New York, NY: Springer-

Verlag, 1997. Abraham, Ralph H., and Christopher D. Shaw, Dynamics: The Geometry of Behavior, 2nd ed. Reading, MA: Addison-Wesley, 1992. Andronov, Aleksandr Aleksandrovich, and C. E. Chaikin, Theory of Oscillations. English language ed. edited under the direction of Solomon Lefschetz. Princeton, NJ: Princeton University Press, 1949.

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Barrow-Green, June, Poincare' and the Three Body Problem. Providence, RI: American Mathematical Society, 1997. Chern, Shiing-Shen, and Stephen Smale, eds., Symposium in Pure Mathematics. University of California at Berkeley, 1968; Providence: American Mathematical Society, 1970. Gleick, James, Chaos: Makinga New Science. New York, NY: Viking, 1987. Gurel, Okan, and Otto E. Rossler, eds., Bifurcation Theory and Applications in Scientific Disciplines. New York, NY: New York Academy of Sciences, 1979. Hirsch, Morris W., Di$erential Topology.New York: Springer-Verlag, 1976. Hirsch, M. W., J. E. Marsden, and M. Shub, eds., From Topology to Computation: Proceedings of the Smalefest. New York, NY: Springer-Verlag, 1993. Iooss, G., et al., Chaotic Behavior of Deterministic Systems, Les Houches, 1981. Amsterdam: North Holland, 1983. Minorsky, Nicolai, Introduction to Non-linear Mechanics: Topological Methods, Analytical Methods, Non-linear Resonance, Re/axation Oscillations. Ann Arbor: J.W. Edwards, 1947. Mira, Christian, Some historical aspects of nonlinear dynamics: possible trends for the future, Int. J. Bifurcation and Chaos, 7(9) (1997) 2 145-2 173. Peixoto, M. M., Proceedings of a Symposium Held at the Vniversiry of Bahia, Salvador, Brazil, July 26 - August 14, 1971. New York: Academic Press, 1973. Peterson, Ivars, Newton's Clock: Chaos in the Solar System. New York, NY: W. H. Freeman, 1993. Ruelle, D., and F. Takens, On the nature of turbulence, Commun. Math. Phys., 20, 167-192; 23,343-344, 1971. Smale, Stephen, Diffeomorphisms with many periodic points, in: Diflerential and Combinatorial Topology. Princeton, N J : Princeton University Press, 1964, 63-80. Smale, Stephen, The Mathematics of Time: Essays on Dynamical Systems, Economic Processes, and Related Topics. New York: Springer-Verlag, 1980. Thom, Rene, Stabilite Structurelle et Morphogenese: Essai d'une theorie generale des modeles. Reading, Mass.: W. A. Benjamin, 1973. Ueda, Yoshisuke, The Roadto Chaos, Santa Cruz, CA: Aerial Press, 1992.

7.

The Butterfly Effect* Edward Lorenz

Massachusetts Institute of Technology

Predictability: Does the Flap of a Butterfly's Wings in Brazil Set off a Tornado in Texas? Lest I appear frivolous in even posing the title question, let alone suggesting that it might have an affirmative answer, let me try to place it in proper perspective by offering two propositions. 1. If a single flap of a butterfly's wings can be instrumental in generating a tornado, so also can all the previous and subsequent flaps of its wings, as can the flaps of the wings of millions of other butterflies, not to mention the activities of innumerable more powerful creatures, including our own species. 2. If the flap of a butterfly's wings can be instrumental in generating a tornado, it can equally well be instrumental in preventing a tornado. More generally, I am proposing that over the years minuscule disturbances neither increase nor decrease the frequency of occurrence of various weather events such as tornadoes; the most that they may do is to modify the sequence in which these events occur. The question which really interests us is whether they can do even this-whether, for example, two particular weather * This is the text of a talk that was presented in a session devoted to the Global Atmospheric Research Program, at the 139th meeting of the American Association for the Advancement of Science, in Washington, D.C.,on December 29, 1972, as prepared for press release. It was never published, and it is presented here in its original form. [Le., in (Lorenz, 1993).] 91

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situations differing by as little as the immediate influence of a single butterfly will generally after sufficient time evolve into two situations differing by as much as the presence of a tornado. In more technical language, is the behavior of the atmosphere unstable with respect to perturbations of small amplitude? The connection between this question and our ability to predict the weather is evident. Since we do not know exactly how many butterflies there are, nor where they are all located, let alone which ones are flapping their wings at any instant, we cannot, if the answer to our question is affirmative, accurately predict the occurrence of tornadoes at a sufficiently distant future time. More significantly, our general failure to detect systems even as large as thunderstorms when they slip between weather stations may impair our ability to predict the general weather pattern even in the near future. How can we determine whether the atmosphere is unstable? The atmosphere is not a controlled laboratory experiment; if we disturb it and then observe what happens, we shall never know what would have happened if we had not disturbed it. Any claim that we can learn what would have happened by referring to the weather forecast would imply that the question whose answer we seek has already been answered in the negative. The bulk of our conclusions are based upon computer simulation of the atmosphere. The equations to be solved represent our best attempts to approximate the equations actually governing the atmosphere by equations which are compatible with present computer capabilities. Generally two numerical solutions are compared. One of these is taken to simulate the actual weather, while the other simulates the weather which would have evolved from slightly different initial conditions, i.e., the weather which would have been predicted with a perfect forecasting technique but imperfect observations. The difference between the solutions therefore simulates the error in forecasting. New simulations are continually being performed as more powerful computers and improved knowledge of atmospheric dynamics become available. Although we cannot claim to have proven that the atmosphere is unstable, the evidence that it is so is overwhelming. The most significant results are the following. 1. Small errors in the coarser structure of the weather pattern-those

features which are readily resolved by conventional observing

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networks-tend to double in about three days. As the errors become larger the growth rate subsides. This limitation alone would allow us to extend the range of acceptable prediction by three days every time we cut the observation error in half, and would offer the hope of eventually making good forecasts several weeks in advance. 2. Small errors in the finer structure-e.g., the positions of individual clouds-tend to grow much more rapidly, doubling in hours or less. This limitation alone would not seriously reduce our hopes for extended-range forecasting, since ordinarily we do not forecast the finer structure at all. 3. Errors in the finer structure, having attained appreciable size, tend to induce errors in the coarser structure. This result, which is less firmly established than the previous ones, implies that after a day or so there will be appreciable errors in the coarser structure, which will thereafter grow just as if they had been present initially. Cutting the observation error in the finer structure in half-a formidable task-would extend the range of acceptable prediction of even the coarser structure only by hours or less. The hopes for predicting two weeks or more in advance are thus greatly diminished. 4. Certain special quantities such as weekly average temperatures and weekly total rainfall may be predictable at a range at which entire weather patterns are not. Regardless of what any theoretical study may imply, conclusive proof that good day-to-day forecasts can be made at a range of two weeks or more would be afforded by any valid demonstration that any particular forecasting scheme generally yields good results at that range. To the best of our knowledge, no such demonstration has ever been offered. Of course, even pure guesses will be correct a certain percentage of the time. Returning now to the question as originally posed, we notice some additional points not yet considered. First of all, the influence of a single butterfly is not only a fine detail-it is confined to a small volume. Some of the numerical methods which seem to be well adapted for examining the intensification of errors are not suitable for studying the dispersion of errors from restricted to unrestricted regions. One hypothesis, unconfirmed, is that the influence of a butterfly's wings will spread in turbulent air, but not in calm air. A second point is that Brazil and Texas lie in opposite hemispheres. The dynamical properties of the tropical atmosphere differ considerably from those of the atmosphere in temperate and polar latitudes. It is almost as if the

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tropical atmosphere were a different fluid. It seems entirely possible that an error might be able to spread many thousands of miles within the temperate latitudes of either hemisphere, while yet being unable to cross the equator. We must therefore leave our original question unanswered for a few more years, even while affirming our faith in the instability of the atmosphere. Meanwhile, today's errors in weather forecasting cannot be blamed entirely nor even primarily upon the finer structure of weather patterns. They arise mainly from our failure to observe even the coarser structure with near completeness, our somewhat incomplete knowledge of the governing physical principles, and the inevitable approximations which must be introduced in formulating these principles as procedures which the human brain or the computer can carry out. These shortcomings cannot be entirely eliminated, but they can be greatly reduced by an expanded observing system and intensive research. It is to the ultimate purpose of making not exact forecasts but the best forecasts which the atmosphere is willing to have us make that the Global Atmospheric Research Program is dedicated.

8. I. Gumowski and a Toulouse Research Group in the "Prehistoric" Times of Chaotic Dynamics Christian Mira Felix qui potuit rerum cognoscere causas. -Virgil Nihil est oppertum quod non revelabitur, et occulturn quod non scietur. -Matthieu

1. Introduction. Birth of the group. Approach of dynamic problems (processes). In the years 1957-1958, just after my engineering studies, I was beginning a research work on the behavior of the one-time constant parallel electronic integrator. For certain parameter choices this electronic circuit presents phenomena of complex, and very sensitive, oscillations. This work took place in the framework of the "Laboratoire de G6nie Electrique" directed by Prof. J. Lagasse, in the "Ecole Nationale SupCrieure d'Electrotechnique, d'Electronique et d'Hydraulique de Toulouse." At that time my knowledge in the nonlinear dynamics field (then more frequently called nonlinear mechanics) was at the very first step of embryonic state. During the last year of my engineering studies I only obtained a germinal information from two lectures, one given by a Russian scientist and another by Prof. Gilles, who wrote a book on nonlinear control systems. These two lectures began to make me more aware to the interest of nonlinear studies. Due to the quasiexclusive use of linear models in my earlier engineering teachings, this interest escaped to my attention. Nevertheless, in spite of this first 95

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information, it is likely that my research would have been oriented on purely technical aspects of electronics without an exceptional event. So I would surely have remained in a "linear world," as all the scientists of my local environment, considering that the most part of complex phenomena observed have only an academic interest, due to some parasitic effect without any direct interest. The exceptional event mentioned above was a first two years stay of Igor Gumowski from October 1958, in the laboratory of my first research steps. He arrived as Professor Assistant of "Ddpartement de GCnie Electrique" from "UniversitC Lava!" (QuCbec). Such encounter decided on the orientation of my research, i.e. my true entry in the "nonlinear world." From this encounter followed the creation of the group, which has ever since worked on nonlinear dynamics and its applications in Toulouse. This group operated under different names according to the administrative place of its activity: "Groupe The'orie et Simulation" in "Laboratoire de GCnie Electrique" and after "Laboratoire d'Automatique et Applications Spatiales" (CNRS) Toulouse, "Groupe Systbmes Dynamiques Non Liniaires" in "FacultC des Sciences of UniversitC Paul Sabatier of Toulouse," "Groupe d'Etudes des S y s t h e s Non Line'aires et Applications" in "Laboratoire &Etudes des Syst6mes Informatiques et Automatiques" of "Institut National des Sciences AppliquCes de Toulouse." For simplicity we shall always speak in this text of the Toulouse Group, the responsibility and the theses orientation of which I had until 1997. As far as I know, Igor Gumowski was at that time the western scientist who had the largest and the most profound information and understanding on the results of the schools on nonlinear dynamics in the former Soviet Union [I] [2] [4]. He also made a very strong impression on me by the wideness of his interdisciplinary scientific knowledge (the different fields of mathematics, engineering, physics, chemistry, biology, philosophy...). Thanks to him, I had access from 1958 to an exceptionally wide information, rather unknown in western countries at that time. This unawareness occurred in spite of a text due to J.P. La Salle and S . Lefschetz (J. of Math. Anal. and Appl., 2, pp. 467-499, 1961), who wrote in 1961 :

In USSR the study of differential equations has profound roots, and in this subject the USSR occupies incontestably the first place. One may also say that Soviet specialists, far from working in vacuum, are in intimate contact with applied mathematicians and front rank

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engineers. This has brought great benefits to the USSR and it is safe to say that USSR has no desire to relinquish these advantages. He also drew my attention to the high interest of the Japanese results of the "Hayashi' School" and their applications to electric and electronic circuits. This important contribution to nonlinear dynamics is recapitulated in the Hayashi's book [3] and in [65]. It must be noted that papers on what will be later called "chaotic dynamics" were published [66] [67] by Hayashi with his disciples Ueda and Kawakami, long before the appearance of this denomination (1976). More particularly, the chapter 7 of [66] gives the thrilling history of chaotic studies in Japan. This first meeting with I. Gumowski was followed by a long and friendly collaboration, source of fruitful results based on the qualitative methods, and the analytical methods of nonlinear dynamics. About the qualitative methods, it is reminded that their "strategy" is based on the characterization of the complex transcendental non tabulated solutions of continuous (resp. discrete) nonlinear models by their singularities. It is worth noting that the same method is used for characterizing the functions of the complex variable by poles, zeros and essential singularities. For continuous (resp. discrete) nonlinear models the singularities are equilibrium points (resp. fixed points), or periodical solutions (resp. cycles), phase trajectories (resp. invariant curves), stable and unstable manifolds, boundary (or separatrix) of the influence domain (domain of attraction, or basin) of a stable (attractive) stationary state, homoclinic, or heteroclinic singularities, more complex singularities of fractal, or non-fractal type. The qualitative methods concern the identification of two spaces associated with a dynamic system. The first one, called phase space, identifies these singularities (location, nature). The second one deals with the evolution of these singularities when the system parameters vary, or in the presence of a continuous structure modification of the system (study of bifurcation sets in aparameter space, or in a function space). Due to the absence of necessary and sufficient conditions of structural stability for m-dimensional dynamic systems, m > 2, the bifurcation studies were initially conducted with the conjecture that the conditions of m = 2 case are those of m > 2. This is not true in general. Sufficient conditions of structural stability were formulated by Smale in 1963, from which it resulted a new development of researches in Russia, in particular with the Shilnikov's results. The second approach corresponds to

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the analytical methods. Here, the above mentioned complex transcendental functions are defined by convergent, or at least asymptotically convergent series expansions, or in "the mean." The method of the Poincari's small parameter, the asymptotic methods of Krylov-Bogoliubov-Mitropolskiare analytical. So are the averaging method, and the method of harmonic linearisation in the theory of non linear oscillations. These approaches constitute two relatively independent branches of the nonlinear oscillations theory. They have the same aims: construction of mathematical tools for the solution of concrete problems, improvement and development of the dynamic systems theory. In the study of complex dynamics, more specially chaotic dynamics, analytical methods quickly attain limits, which restrict their use to some limited local behavior. It is the reason why the basic tools, at the origin of the results described in this text, are essentially those of the qualitative methods, which achieved their highest level of evolution in the framework of the Andronov' School [4]. This approach has been the background of theses preparation for many researchers of the group I managed in Toulouse, historical city of middle importance in the south-west of France. This town is famous for its aeronautical industry (Concorde, Airbus) and the epopee of the "Aeropostale" from 1920 (Saint-Exupdry, Mermoz, etc.. ..). I. Gumowski told me that it has been also known by specialists of nonlinear dynamics as the place of publication of the first translation (made by E. Davaux, a French Navy officer-engineer) of the most famous A.M. Ljapunov's text "Le ProblGme gknkral de la Stabilite' du Mouvement" (Annales de la Facultt! des Sciences de Toulouse, Series 2, vol. IX, 1907, 203-474). The original Russian text was published in 1892 by the Mathematical Society of Kharkow. The French translation was reviewed and corrected by Ljapunov (officer of the Russian Imperial Navy), who completed his text with an additional part. It has remained the basic text quoted in relation with stability problems, and it was reproduced in Annals of Mathematical studies (Vol. 17, Princeton University Press, 1949) long after. The important contribution (from I903 to 19 18) of S. Lattes, a Professor of the Facultk des Sciences de Toulouse, is less known in the field of discrete nonlinear dynamics. Lattes published fundamental texts on nonlinear recurrence relationships (or maps), in particular in relation with the stable and unstable manifolds of fixed points and cycles [83]-[86]. These results provided another set of fundamental bases for the Toulouse group studies in nonlinear discrete models, and their complex behaviors, with the Julia and Fatou publications.

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Guided by the Gumowski's experience, the Toulouse group has always chosen an inductive way to study nonlinear problems. It is a matter of understanding the dominating internal mechanisms of dynamic processes, and thus preserving a phenomenological transparency through the simplest possible "germinal" examples. So it was considered that the behavior analysis of complex models resulting directly from concrete systems (Physics, Engineering, ...) is facilitated by understanding more elementary relevant forms in a first step. Such forms are obtained by choosing the lowest dimension, and the simplest equation structure, in order to isolate in the t'puresttt configuration a "mathematical phenomenon," after elimination of "parasitic effects" of a more complicated structure. The bringing to light of the bifurcation phenomenon of period doubling cascades and their accumulations, from the simple one-dimensional quadratic map x' = x2-h, is of such a type. This result was discovered by Myrberg in 1963. It has had a fundamental importance for understanding the birth mechanism of complex subharmonic oscillations in continuous dynamical systems, and the road to chaos resulting from the bifurcation accumulations. Indeed, a rank ksubharmonic oscillation is related to a period k cycle by PoincarC section of a continuous model. The Smale's horseshoe is also of this type. Nevertheless it is important to note that such a model simplification is only a first qualitative step, i.e. the research of the "essential cause" of a phenomenon (Felix qui potuit rerum cugnuscere c a u s a ~-Virgil), Applied sciences requires a second step, which consists in a quantitative agreement with the observation, taking into account a given precision. So one has to ''converge" toward the observed system behavior from more and more refined models up to this quantitative agreement. Nevertheless, even at the level of this last step, one has to be aware of the fact that however refined a model may be, it only remains a mind construction which will never be the exact representation of a real object, and so cannot be confused with it. Indeed it is often forgotten that a model is simply the image of our relations with the ''naturett (the real world), rather than the image of the "nature" itself, as noted by Heisenberg. The present text relates the first period (i.e. the "prehistoric" one ending in 1976, emergence date of the word chaos due to May [63] and beginning of the "historic" times) of the Toulouse group activity in complex nonlinear dynamics and its collaboration with I. Gumowski. It tries to describe the research motivation, the work background, the problems approach, the nonclassical sources of information used, and the difficulties met at different steps. The most part of the related papers were published in French, or in not

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very widespread publications, during this period. This text tries to give the content of the corresponding results from a summarized presentation. Each of the following sections is devoted to a topic which led to the observation of complex dynamics: electronic circuit, basins and attractors of twodimensional noninvertible maps, resonant bifurcations, two-dimensional area-preserving map, one-dimensional noninvertible maps, and applications in engineering.

2. First Contact with Complex Dynamics (1958-1960) My research activity began with the study of the parallel electronic integrator (Fig. l), described at first approximation as a one-time constant circuit [5][6],as indicated above. The amplifier gain K is supposed close to 1 in the framework of a linear approximation. Let R i be input resistance, 1-K=6, FRC, a=R/Ri , u=t/z, t being the time. Then the circuit can be described by one of the two autonomous one-dimensional ordinary differential equations: (1)

dY/du=KE-V,(6+u),

dV,/du=E-V,(6+a)

-+E

I

I

Fig. 1. The parallel electronic integrator. The transistor TI plays the role of the switch 1 of the left scheme.

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(4

(el

Fig. 2. Response of the circuit for different polarizations of the amplifier and values of V,.

When 6+a= 0, V, increases linearly, the circuit behaves as an integrator, but the equation leads to a critical case in the Ljapunov sense, i.e. the eigenvalue of the stationary state is equal to zero. This means that the linear approximation is not sufficient to explain the circuit behavior, which may present phenomena of complex, and very sensitive, oscillations (Fig. 2b-e) for certain parameter choices. A first step toward the phenomena

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interpretation was made by introducing the amplifier nonlinearity in the form: (2)

y, = KV, + f(V,), f(V,) = -a,y3 + a,V,5 - a7y7, 25a: - 84a,a7 > 0,

This permitted to explain the nonlinear form of Ve(t)(Fig. 2a), but not the complex oscillations of Fig. 2b-e. A two-dimensional differential equation, by introduction of a parasitic capacitance (bounded bandwidth of the amplifier frequency response) is also inadequate for the interpretation of Fig. 2b-e [ 6 ] .In fact a dimension increase of the model, necessary to explain the complex oscillations observed, is introduced by a more realistic model of the transistor amplifier. Indeed I. Gumowski showed in 1959 [7]-[12] that a reliable description of the transistor amplifier dynamics implies a model containing not only a small parasitic capacitance, but also a time lug which has a nonlinear dependence with the input amplitude. Here the lag is due the transit time of the charge carriers, an increase of the electric field at the collector junction leading to a decrease of the diffusion distance. Then taking into account [7]-[121 the model of the amplifier (considered as isolated from the R, C, E circuit) is written in the following form: (3)

q d y ( t )/ dt + Y.(t)+ K y . [ t - h( v,)] = q d v , ( t ) / dt f f{y[t- h( y )]}

The qualitative relevance of this model was verified experimentally. Such a model, associated with the circuit of Fig. 1, permits a qualitative account of the observed oscillations at very high frequency from the small delay h, and their complexity from the nonlinear characteristic of this delay in addition to that of the function f. In this case the initial condition, necessary to render the problem unambiguously defined, must be a function V&)= vo(t) for - h l t
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3. Basin Boundaries of Two-Dimensional Noninvertible Maps (1963-1975)

3.1 Genesis of the results After two years of military service, I reintegrated the "Laboratoire de GBnie Electrique" at the end of 1962. Then my researches were oriented toward nonlinear sample data control systems, and some control and electronic systems using different types of modulations. The corresponding models were discrete in the form of recurrence relationship (equivalent designations: map, point mapping, iteration, substitution). More particularly, the studies were concentrated on the class of models described by twodimensional noninvertible maps (nonlinear by nature). The determination of the basin boundary of a stable stationary state is of prime importance for practical problems. Indeed it gives the domain of perturbations (classed as changes of initial conditions) that the system can undergo without changing its qualitative behavior. In 1963 the results on nonlinear maps (recurrences, iterations) were in an underdeveloped state with respect to those related to ordinary differential equations. I. Gumowski attended at a lecture dealing with the known results, given by Paul Monte1 on this topic in Quebec. This is the reason why in 1963 our first basic reference was the book "Leqons sur les rhrrences et leurs applications" of this mathematician [13], giving an account of the French school results on iteration. This school (Grdvy, Koenigs, Leau, LCmeray, Hadamard, Latths, Julia, Fatou) was the most active one in this field from the 19th century end to around 1930. With respect to the other authors of the French school, I. Gumowski and I were more particularly attracted by the "global" study dealt in the Julia & Fatou's papers [ 141 [ 151, in spite of the fact that they concern a restricted class of two-dimensional noninvertible maps. Indeed their results on onedimensional maps with a complex variable are directly related to the particular class of two-dimensional maps with real variables, defined by two functions satisfying the Cauchy-Riemann conditions. One of the basic tools used by these authors was the important Montel's theory of normal families. Julia and Fatou offered a starting point for studies of basin boundaries generated by general two-dimensional noninvertible maps with real variables. The results obtained by these authors can be summarized as follows. The critical points, and the set E of all the repulsive cycles, play a

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prime role for defining the map properties. The set E', derived from E , contains E, and E is perfect. In page 49 of [14] it is said that "la structure de E' est la mime duns toutes ses parties," which means that the E' structure is self similar, called fractal from I977 [87]. "Nonlinear analysts" were not interested by such questions when we began such researches. Now, it is well known that the set E' (called Julia set since the years 1980) is a basin boundary, when the map generates an attractor. Montel (Director of the "Institut PoincarC" in Paris) and Julia were still alive at that time. It was not the Fatou's case. So at first Gumowski and I met Montel in 1963 for information about the existence of eventual results on basin boundaries, when they are generated by two-dimensional real maps not satisfying the particular Cauchy-Riemann conditions. He told us the lack of such an extension, as far as he knew. Then I contacted Julia, approximately 70 years old at that time. About him it is less known that he was severely wounded at the face in 1915 during the First World War, where he has an heroic behavior. In particular this behavior was reported by newspapers, when he was elected to the French Academy of Sciences the fifth of March, 1934, becoming its youngest member. Julia told me that, due a health state partially related to his old wounds, he has left his mathematical activities since many years. Nevertheless after having confirmed Montel's opinion, he added that more general cases might present high difficulties associated with a lot of possible different qualitative situations. Such difficulties emerged gradually and laboriously long after in the Toulouse group researches. Some of them are described in [161. From this contact with Montel and Julia, 1. Gumowski and I decided to begin studies on basin boundaries of two-dimensional real noninvertible maps by collecting many very simple generic examples, illustrating different qualitative situations. They essentially concerned quadratic and cubic maps, or piecewise linear ones, a point having at most three real rank-one preimages. Acting in this way boils down to follow the old "Philosophia Perennis" rule, nihil in intellectu nisi prius in sensu, i.e. nothing in the intellect if not before in the senses. We imitated the botanists approach collecting piece by piece unknown plants, hoping to find relations between some of them in order to define classes of properties. We started with the property of invariance of a basin boundary by inverse mapping, given in the Julia and Fatou texts. A first basic tool was obtained with the introduction of a new singularity, the critical curve as locus of points having two merging rank-one preimages (inverses) [88]. Such a curve separates regions, the

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points of which have a different number of preimages. A second tool was given by the determination of the stable manifold of a saddle fixed point, or cycle, from the series expansions defined by Lattbs [83]. This permits to obtain a "germ," the inverse iterates of which lead to get the stable manifold of a saddle point. Following this way, step by step, after understanding of many "regular" examples, this method led from 1968 to three examples of basins with properties of self-similarity, generated by two-dimensional real maps not satisfying the Cauchy-Riemann conditions. Such a result was possible by using numerical programs, elaborated by A. Giraud [ 171. The first "Cyderec" permitted a direct obtaining of a large set of cycles from only one initial point. The second "Linva" gave the saddle manifolds, and their expansions in a parametric form up to relatively high degree (120 with the computers performances of that time) from algorithms based on Lattes and Picard results [17][19].

3.2 First example: fractal organization of non connected basins (1968-1969) The starting point "fractal structure of non connected basins'' was the understanding of a very simple quadratic map generating a basin made up of only two non connected parts. Imbedding this map into a cubic one leads to a fractalization of the basin. This imbedding is a cubic "perturbation" of the initial map. The quadratic (non-perturbed) map T illustrates the bifurcation ''simply connected basin t) non-connected basin" [ 181 (also cf. p. 243-244 of [19]). It is given by: x' = y ,

(4)

y'=-ax-y-x,

2

a being a parameter, 0.04 < a < 0.31. The map T has two fixed points 0 (x = y = 0), and A (x = y = -a-2). 0 is a stable node point. A is an unstable node,

appearing as a "cape" (i.e. cusp point, its multipliers being S, > 1, S2< - 1 and ISZI>&) of boundary 300of the immediate basin DOof 0 (connected part containing the attractor, here 0).The boundary DO also contains a period two saddle cycle (&,I?,') with multipliers S, > 1, - 1 < S2< 0. The inverse map T I is given by:

{

x = -a f [la'

- 4(x'

y=k.

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Chaos Avant-Garde: Memories of the Early Days of Chaos Theory

The critical curve LC (ligne critique in Fig. 3) x+y= a2/4 separates the phase plane into two regions: Zo : x+y > a2/4, where the points have no real preimage, Z2 : x+y < a2/4, where the points have two rank-one real preimages.

Fig. 3. Map (4). (a,b) Basin boundary of the fixed point 0 (1969).

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Fig. 3. (Continued) Map (4). (c) Basin obtained by a scanning.

Such a map is called of (ZO-Z2)type. The curve LC.1, locus of merging for the rank-one preimages, is the line where the Jacobian of T cancels, i.e. x= -an,T(LCI)=LC. Denote R I , R2, the two open regions separated by LC-1. For every point ( x , ~ ) EZ2 let TI-', T2-' be the two determinations of the inverse map T-' :

T-'= TI-'uTi', with

TI-'(x,~)E RI, Ti'(x,y)~ Rz,

If ( x , ~ ) EL C , then Tl-'(x,y)=TZ.l(x,y)~ LC.1. When 0.031% a < ab= 0.236075, the basin of 0 is non-connected. It is made up of the immediate basin DO,and only one non connected part D1 = T-'(Ao), A o E Z ~being the domain bounded by the closed curve (T, A.1, S, 7') of the Fig. 3a. The non connected part DIE ZO,and so has no preimage. DI is bounded by the closed A1-2, T I ) : curve (KI, A.2, S-I,

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(7'-1,

A-2, S.1,

T.1) = Ti-'(Ao),

(T-I,A'-2, S.1, T-I)= Til(Ao).

This situation is represented in Fig. 3a, where a letter with a negative lower index n defines the rank-n preimage of a point designated by this letter, and (LC)naDo= P u Q u T u S . In Fig. 3b the basin of 0 is connected (a= 0.3). The bifurcation separating these two situations corresponds to a tangential contact of the critical curve LC (designated by l'ligne critique" in the figures) with 300 (belonging to the stable manifold of the saddle period two cycle). This occurs when T = Q for a = ab= 0.236075 (Fig. 3c). The fractal basin is obtained by imbedding (4) into a cubic map family T, from a "small" cubic term added to (4) [20] (also cf. p. 244-247 of [19]):

The critical curve of this map T is now made up of two straight lines bounding a region Z3 where every point has three first rank preimages. In the two complementary regions (Zl) a point has only one first rank preimage. Then the map T is said of ( Z l - Z3- 21) type. The fractalization of the basin is now possible with respect to the above example, because a non connected part of basin can have infinitely many preimages of increasing rank. Here the two fixed points 0 and A (-3, -3) have the same characteristics as for (4), but a new fixed point A1(-7, -7), a stable focus, appears, the total basin of which is also non-connected (Fig. 4a). Due the absence of a ZO region, and the presence of the Z3 region, the map gives rise to arborescentchains (sequences) [16][ 191 of infinitely many increasing rank preimages of the immediate basin of each of the two stable fixed points 0 and A l . These sequences belong to the total basin of 0 and A ] , made up of infinitely many non connected parts, the complementary part of the (xy) plane being the domain of divergence. In 1968 the existence of a limit set for such arborescent sequences was deduced [ 191 remarking that the increasing rank preimages must tend toward a repulsive set made up of infinitely many unstable cycles of increasing period, their increasing rank preimages, and their corresponding accumulation points [ 161. This limit set constitutes what will be called later a strange repeller [16]. It is a "jfulZydiscontinuous set'' in the Julia and Fatou's language ([14] p. 48). As far as I know, the map ( 5 ) has given the first example of a fractal non-connected basin for maps which do not satisfy the Cauchy-Rieman conditions. At that time the word "fLactal" did not exist, but the description given in [20], and the enfslarged parts of

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Fig. 4a have clearly shown the self similarity properties of the basin structure. Here this structure can be said "globally fractal" in the sense that it is the whole set of the basins non connected parts which has the self similarity properties. Locally the basin boundary of 0 has not this feature.

Fig. 4a. Map ( 5 ) . Basin boundaries of the fixed points 0 and A I (1969). The global structure of the basins is fractal.

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With modern personal computers using a method of scanning (pixel by pixel), the basins of the stable fixed points 0 and A1 are directly achieved in the form of Fig. 4b. This figure, now obtained in few seconds, can be compared with the 1969 result, based on the determination of the properties of the basin boundaries of the two stable fixed points, which necessitated more than two months of works. Nevertheless the "old" method conserves its interest because it gives a lot of information on the "microscopic" structure of a basin boundary from the knowledge of its cycles, the "germ" of the saddle cycles, the arborescent sequences of their successive preimages.

4.000

I

I,

-11.000I

-11.000

Fig. 4b. Map (5). Basins of the fixed points 0 and A , , obtained by a numerical scanning.

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3.3 Second example:fractalization of a basin boundary (1968-1969) This example concerns a simply connected basin the boundary of which has a fractal structure. It is generated by the non invertible map T of (ZO-22)type: (6)

x"y

,

y'= x/2 + x2/I0+ hxy

Fig. 5a. Map (6). Basin boundary of the fixed point 0 (1 969).

The origin 0 (x=y=O) is always a stable "star" node (multipliers Sl=-S2, 04'14). This map gives rise to an interesting non classical global bifurcation, by crossing through the parameter value h=O. 1, value for which the whole line x+y=5 is made up of points of unstable period 2 cycles. It results a sudden and important increase of the 0 basin area [21] [22] when b 0 . 1 . The fractalization of basin boundary (Fig. 5a) is described in the Giraud's thesis [17] for h= 0.1 1, i.e. after this global bifurcation. The corresponding study identified the mechanism of period doubling for the

Chaos Avant-Garde:Memories ofthe Early Days of Chaos Theory

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cycles located on this basin boundary, by variation of the parameter h (Fig. Sc). Such a basin boundary structure was called "halfjkzctall' later [16]. It is made up of smooth arcs of a saddle stable manifold bounded by points of a period four unstable node, and their arborescent sequence of increasing rank preimages. The limit of these preimages, when the rank tends toward infinity, consists of infinitely many cycles of increasing period and their increasing rank preimages. Some of such cycles appear in Fig. Sc from the vertical line h= 0.1 1, the other cycles obtained by period doubling being suggested for smaller h values by the interrupted horizontal lines. A method of scanning, now possible with modern computers, gives directly and very quickly the basin of 0 (Fig. Sb). It can be compared with the 1969 result laboriously obtained but giving a larger information, based on the determination of the basin boundary. h= .11000 8 30.000

.OOOOOf-

.OOOOO

I

,000 -25.000

5.000

Fig. 5b. Map (6). Basin of the fixed points 0, obtained by a numerical scanning.

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--

113

cycle col 16

a a

.

Nooud W

12

W

12

I

6 6

16

. " "

4 2 1

Noeud

1

Noeud

Col

Fig. 5c. Map (6). Period doubling of cycles (according to p. 43 of [17], 1969). Each number is a cycle period.

3.4 Third example:fractal organization with a multiply connected basin (1973) The third example is related to the bifurcation "simply connected basin t) multiply connected basin" [23] from the cubic map T: (7)

{

+ y' = p (xsincp+ ycoscp) + x2 + y3

x' = p (xcoscp - ysincp) xy

with (0 = 2n / 3, p= 1.05, for which the critical curve LC is made up of a curve presenting a cusp point. This curve limits a region Z3 with three rankone preimages (noted "3 antdcddents" in Fig. 6a), and a region ZI with only one preimage (noted "1 antdcddent" in Fig. 6a). Such a map is called of (Zl
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the region Z3. Using the methods of the last paragraph of sec. 3.1, the basin boundary was obtained from the stable manifold of the saddle fixed point A , with the unstable node B belonging to ( F ) . The holes of the multiply connected basin are generated by the arborescent sequences of increasing rank preimages of the domain containing the point P and bounded by two arcs, one belonging to LC, the other to (F).A strange repeller (in the sec. 3.2 sense) exists as limit set of the arborescent sequence of holes inside the domain limited by the boundary of the external basin (i.e. not containing the holes). The fractal organization of the holes does not appear directly at the Fig. 6a scale, because it is crushed along the external boundary of the basin. For a larger value of the parameter p (p= 1.07) this structure can appear as shown in Fig. 6b, obtained quickly now by a method of scanning, where the white holes belong to the divergence domain.

c

Fig. 6 . Map (7). (a) Basin boundary of the attractor. (1973).

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1.200

-1.600 -1.300

1.600

Fig. 6 (Continued).Map (7). (b) Basin obtained by a numerical scanning. A detailed explanation of the fundamental mechanisms leading to the

situations ofthis section 3 can be found in the relatively recent book [16].

4. Chaotic Attractors of Two-Dimensional Noninvertible Maps (1968-1975) 4.1 Mapsfar from the conservative case Consider the sec. 2 circuit, and V.@) given by Figs. 2b-e, and the plane [Vs(t), ciV,(t)4&], which can be regarded as a projection of the function space defined from the the differential equation (3) with the delayed argument h, and the initial condition function V,(t)= vo(t) for -hl t
1 I6

Chaos Avant-Garde: Memories ofthe Early Days o/Chaos Theory

in this plane would have had a chaotic aspect, but at that time this representation was not made. So the first publication of the Toulouse group, on what has been after called a chaotic attractor, or a strange attractor, was made in the "Proceedings of the International Pulse Symposium" (Budapest, 1968) [21]. This example was also given in a paper of the journal "Automatica" (1969) [22], and with more details, indicating the role of critical curves and bifurcations, in the Proceedings of the Colloquium "Point Mapping and Applications" (Toulouse, Sept. 10-14, 1973) [23]. The attractor is generated by the piecewise linear map:

It is a (Z,- Z3- Z,) map. When A= +E, Ab= 16/7.2= 2.222..., E>O being sufficiently small, the map gives rise to a period two area, without stable cycle, indicated as attractive limit set in [22], stochastic area in [23]. The Toulouse group also used "Pulkin'sphenomenon" associated with such behaviors. This designation was inspired from a Russian paper published in 1950 [25], showing that in an one-dimensional noninvertible map infinitely many unstable cycles and increasing classes of their limit points may lead to bounded complex iterated sequences. These fluctuations in the vocabulary choice reveal the group perplexity in presence of phenomena not well understood at that time. It is worth to note that Y . Ueda met equivalent behaviors with ordinary differential equations at the same period, proposing to call them "randomly transitional phenomena" (cf. p. 206 of [66]). The first simulation of the map behavior is reproduced in Fig. 7a (A= 2.3) [22], for which the two clouds of points are related to the limit set (chaotic attractor), the closed curve with a parallelogram shape represents the corresponding basin boundary, containing the period four saddle A , , i= 1,2,3,4, and the period four focus A,. A region designated by ''p antecedents," p= 1,3, is a 2, region in the sense of sec. 3.2. A segment of critical curve (designated by LC) is a part of the boundary of the limit set. This example permitted to see the important role of the critical curve in the definition of complex attractors generated by two-dimensional non-invertible maps (more

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details are given in the 1996 book [16]). In 1973 with a more recent computer, permitting more iterations, this map gave rise to the attractors of Figs. 7b (A= 2.23), 7c (A= 2.3), 7d (A= 2.4). It was numerically observed that the contact of the attractor boundary with the basin one leads to the destruction of this attractor. This bifurcation was studied more completely in 1978 [74]. Another interesting example was obtained by approximating the piecewise linear map by smooth one. This was made by replacing a neighborhood of the non smoothness points IXI= 1/2, with a polynomial of degree 2, which leads to the map [24]:

The chaotic attractor generated by this map, shown in Fig. 8, presents a different "internal structure" with respect to the Fig. 7c attractor produced by a nonsmooth map. Indeed arcs with a higher density of iterated points appear in the smooth case, whereas Fig. 7c shows areas with different densities of iterated points. Such "internal structures" were explained latter in 1977 [ 161 [19] [75] considering the rank-rn images of critical curves. Arcs of such images correspond to higher density of iterated points for the smooth case, and separation of areas with different densities of iterated points in the nonsmooth case. Another type of attractor (Fig. 9) is generated by the map (7). It was presented in Colloquium "Point Mapping and Applications" (Toulouse, Sept 10-14, 1973) [23], and also published in the Proceedings of IFIP (Stockholm, 1974) [24].

I18

Chaos Avant-Garde: Memories of the Early Days of Chaos Theory 2

A

Fig. 7. Map (8). (a) Chaotic attractor and its basin boundary, h= 2.3. According to [2 I ] (1 968).

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.

...

!.

.. .. : .- :. !* : . : , : I . : , . .: .: .I ;. : . , . . . .,. . ... . . . . .. . . .,. . . ... . . *... . . ... . . ... . . ... . ..... . ..... . ... ...,.......... ... .. 3

.

I

.

,

.

I

~

I

!

.

.

.

..r

(4 Fig. 7 (Continued). Map (8). (b) h= 2.23, period 2 chaotic attractor. (c) h= 2.3, period 2 chaotic attractor. According to [23] (1 973).

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(4 Fig. 7 (Continued). Map (8). (d) h=2.4, the chaotic attractor has now the period 1. According to [23] (1973).

Fig. 8. Map (9)-( 10). Period 2 chaotic attractor (1973).

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Fig. 9. Map(7). Chaotic attractor (1973).

4.2 Perturbation of conservutlvecases From 1970 two families of maps, obtained from a dissipative perturbation of a conservative map (issued from the studies described in sec. 6 below), were used for generating attractors called "chaotic" now. Before the adoption of the adjective chaotic, the term "stochastic" was finally adopted at that time, in order to describe the qualitative structure of phase trajectories generated by deterministic models. This choice was motivated by the so high complexity degree of trajectories, that from a casual point of view they appear to be random, without being really random. Now "sfochastic"is only reserved for processes defined as a one-parameter family of random variables, a random variable being defined as a measurable function [70]. The two-dimensional quasi-conservative maps, mentioned above, have the following forms:

(1 1)

i

+

+

= y - F(x) ay(1 ay2)

y' = --x

-t- Ffx' )

[ F ( x ) = p x +2(1- p))x2/(1+x2) x' = y - F ( x ) + ay(1-t-ay2) y*=-x+F(x')

i

F ( x ) = p x + ( l - l u ) x 2 e x p [ ( x 2 -1)/4]

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Fig. 10. Map(1 I), p= 1/8, w 5 . 1 0 3 ,a= -1. Chaotic attractor (1973)

Fig. 1 1 . Map(l l), p= 118, at5.103,a= -10. Chaotic attractor (1973)

Toulouse Research Group

Fig. 12. Map(1 I), p= 1/8, a=5.103, a= -29.48. Chaotic attractor (1973)

Fig. 13. Map(1 l), p= 1/8, w5.103,a= -29.58. Chaotic attractor (1973)

123

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where -1
5. Normal Forms for Resonant Bifurcations (1969-1974) These normal forms played a central role for the study of some chaotic behaviors by the Toulouse group researchers. In particular they were firstly applied to resonant Hamiltonian bifurcations generated by the areapreserving map obtained for a=O in the equations (10) and (11). The originality of these normal forms has been quoted in page 184 of [26]. Such normal forms were first formulated for a fixed point of a twodimensional map, the multipliers (eigen-values) of which are S I , ~ = This means that one has a critical case of stability in the Ljapunov' sense, i,e. the fixed point nature cannot be defined from the linear approximation, but by the nonlinear terms. For this purpose a variant of the Cigala's method was used. This method was described in [27] at the century beginning for solving the particular problem of stability in two-dimensional conservative diffeomorphisms. Extended to dissipative two-dimensional maps [28]-[30], having a sufficiently high degree of smoothness, the reduction to a normal form consists in identifying and isolating the "dominating terms'' of the nonlinear parts. This objective is achieved by means of a sequence of nonlinear transformations of almost-identity type, which successively remove the non dominating terms. $JQ.

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The critical case leads to two essentially different situations. The most complex one, called exceptional case, occurs when the angle cp is commensurable with 2n, cp= 2 h / q , and leads after a certain number of applications of Cigala's transformations to a problem of denominators which cancel, and small denominators near this angle value. Then at this level the normal form is obtained, because it is impossible to continue the process of successive transformations. The second situation, called non-exceptional case, is related to the angle cp, either non-commensurable with 2n, or if it is commensurable it does not lead to a problem of small denominators. Bifurcation by crossing through an exceptional case, or a n o n exceptional case, from a parameter variation is very often wrongly called "Hopf bifurcation." Since 1969 the Toulouse group has always called this qualitative change Nezmark's bifurcation. Indeed the first contribution is due to NeYmark when cp is non-commensurable with 2n, in the particular case of the crossing through a complex focus of multiplicity one, giving rise to only one invariant closed curve [3 11-[34]. Moreover the crossing through the exceptional cases cp= 2 d 3 , p= 2x14, was considered by this author in [34]. The Ne'lmark's results furnished some of the basic tools for the Toulouse group studies. In the non-exceptional case the Cigala's method permitted to extend the Ne'lmark's results to the crossing through the situation of a complex focus of multiplicity m, m21, when S,,2= efJQ.This bifurcation may give rise to s, 05 s-, invariant closed curves [35]-[37]. The exceptional cases lead to new singular points of different types, those defined locally by 2q asymptotes (complex saddle) giving the resonant situations [23] [29] [30]. Such studies of exceptional and non-exceptional cases were followed by their equivalent related to a four-dimensional map, a fixed point having two pairs of complex multipliers in this case. Due to the presence of two angles c p ~ and cpr the exceptional case gives rise to 8 basic situations ([38] and appendix C of [391)* Resonant cases also appear in the maps family [40]: x'= xcos@- y s i n @ + a p ( x , y , a )

y' = xsin@+ ycos@+ a q ( x , y , a ) where a and # are two parameters, # # 2 h , k 0, f l , f 2 , ..., p(0, 0, a)= q(0, 0,a)= 0, p and q are two smooth functions, each one having an

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expansion with respect to the real variables x, y , which can be limited to a polynomial of degree N . The functions p and q are also supposed sufficiently smooth with respect to the parameter a.

, u-

.)..-3.636

0.332

(a) The invariant closed curve is attractive.

'f- y,,

t

-

1,5868rd

(r)

&

(b) Bifurcation situation merges into ( Y ) 1

-

*.-Q 0,132

'

rrO.02

(d) Conservative case

(r)

Fig. 14. Map(l3). Stable and unstable manifolds of the period 4 saddle, and invariant closed curve r.

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(9) The invariant closed curve is now repulsive.

t

-

*bl

(r)

0,132

#I.

(€1

second situation of bifurcation The clo8ed curve (y2) gives rise to a new invariant closed curve (r)

Fig. 14 ( C o ~ ~ Stable j ~ ~ and e ~unstable . ~anifoidsof the period 4 saddle, and invariant closed curve r.

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. .

4=~i2a , = 0.332

-

- .

I ,

(a) With such an increase of non-linearity homoclinic points appear.

(b) Fig. 15. Map(l3). Chaotic behaviors. (a) cp= d2, a= 0.332. (b) cp= 191"30', a=4.

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Resonances are now due to small denominators after having canceled secular terms in the polar form associated with the map. It is reminded that secular terms appear in the "Poincard's method of the small parameter," for the determination of periodic solutions generated by ordinary differential equations from series expansion. Cancellation of these terms ensures the periodicity of the solution. For the map (13) they play the same role. When a=O the map (13) is linear and conservative. The corresponding phase portrait is of the center type, made up of circles centered at the fixed point (0;O).These circles do not contain cycle points if @ is not commensurable with 27c. They are made up of period m cycles if @ is commensurable with 2n, @ = 2 h / m . When a= f ~ E>O , having a sufficiently small value, @ being different from certain values commensurable with 2n, the center phase portrait generally disappears. Then it gives rise to isolated invariant closed curves, which are alternately stable and unstable except for particular bifurcation cases. Some sets of the angle @, close to values commensurable with 27c, @ = 2k7dm, lead to small denominators by canceling secular terms in the polar @),the points of form of (13). It follows a domain of the parameter plane (a, which correspond to the existence of a period m pair of saddle and node (or focus) cycles. Inside this domain global bifurcations take place associated with an invariant closed curve, as shown by Figs. 14a-g, for 6 close to -n/2, p(x, y , a)=O q(x, y , a)=-0.3y+4y3. When E is sufficiently large homoclinic points, associated with a chaotic behavior, appear (Fig. 15). All papers of this section were written in French at that time, but their content was after reproduced in English in the book [39].

6. Two-Dimensional Conservative Maps (1970-1975) 6.1 Framework of the research From 1966 to 1976, I. Gumowski occupied a position of Senior Physicist in the "European Organization for Nuclear Research" in Geneva. In this framework he had particularly to study the problem of "stochastic" instability in accelerators and storage rings. Such instabilities appear to increase with the amount of nonlinearity. Physically this amount increases with the self-fields, and the latter increase with beam intensity. Here "stochastic," adjective also used in the Geneva research center, has the sense

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of sec. 4, i,e. it has been called "chao~~c" from 1975. Nevertheless here "stochastic" will be used in preference to "chaotic" in order to remain in the vocabulary context of that time. The new Gumowski's position led to a collaboration with the Toulouse group on a new topic: two-dimensional conservative (or Hamiltonian) maps [44]-[47]. Such maps are areapreserving. Considering the results obtained in the framework of this collaboration, it must be noted that the Gumowski's part was more important than the Toulouse group one. A map describing the transverse motion of a particle in an accelerator, or a storage ring, can be found in a straightforward manner if the assumption is made that the ring is composed of localized elements only. From an argumentation developed in [41] the map can be reduced to a twodimensjonal conservative one in the form: (14)

x' = y f F ( x ) ,

y' = --x

+ F( x'),

its Jacobian J being J= 1 . Two forms of the nonlinear term F(x) were considered called "bounded nonlinearities": (15)

F(x)= @+ 2( 1-p)x2/ ( I + x2),

or

For the longitudinal motion the map has the same form, but the noniinear term is an asymmetrical periodic function 1411 [42]: (17)

F(x)= x- (1 -p)[sin(bx+bo)-sinboll bcosbo.

In all these cases F(x) verifies the inequality -12 GI, and the "strength" of the non linearity increases when p decreases from +I. The parameter (1-p) is a sort of measure of the strength of the non linearity. The results given below were obtained from numerical programs (associated with adequate tests of precision) based on the fundamental features of bifurcation processes. This leads to mixed analytical numerical methods achieved in a computer algorithm. So lists of cycles were obtained, and in case of saddles their stable and unstable manifolds were drawn. An

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inductive reasoning, based on an ordering of cycles, permitted to find fundamental laws satisfied by the singularities generated by the map (14). More details on such an approach, and the correctness of its results, are given in [42]. The basic references used for these studies were the Birkhofs Collected Mathematical Papers [1968], which were published in three volumes. Birkhoff was intellectually the greatest PoincarC disciple, taking over the technics and problems of PoincarC and carrying on. So the results about fixed points, or periodic orbits, of saddle (hyperbolic) type, or center (elliptic) type, homoclinic and heteroclinic structures, unstable centers, instability rings, the concepts of signature, for the two-dimensional surface transformations, have given some of the greatest advances in this field. The Birkhoff s qualitative theory reaches its highest point with the long paper, written in French language, "Nouvelles recherches sur les syst2mes dynamiques" (Memoriae Pont. Acad. Sci. Novi Lyncaei, 1935, 53, vol. I , pp. 85-216) [43]. It was crowned by the Pontifical Academy of Sciences. This fine contribution resumes and extends his earlier results in dynamics. It is in this publication that he introduces the "signature,"two-dimensional symbol which displays the topology of an homoclinic, or heteroclinic, structure. Such a symbol is the ultimate in the qualitative description of a dynamic system. He also indicates in this paper, not well known, the formal analogies between the sets asymptotic to elliptic points (unstable centers) and the invariant curves passing through hyperbolic points (saddles). As far as I know the results described below for the map (14), and the particular nonlinear characteristics considered [ 151 [ 161 [ 171, were new at the time of their publication. Being remained relatively unknown, sec. 6 gives an outline of these results. More details can be found in [19] [41] [42]. The maps defined by (14) and one of the nonlinearities (1 5), or (16), or (1 7), induce particular difficulties due to their "bounded characteristics." Using the general approach adopted by the Toulouse group, i.e. by understanding more elementary relevant forms at once, so the simpler cases of "unbounded nonlinearities," defined by quadratic and cubic polynomials, were considered in a first step.

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6.2 Study of unbounded quadratic map

6.2.aAsymptotic properties In order to determine the characteristic features of the stochastic (i.e. chaotic) behavior of the map (14), in the phase plane (xy), as a first step it was studied in the simpler case of an unbounded quadratic non linearity in the form [41] 1421 [44]-[47]:

For such a purpose it was proceeded as follows: - (a) The properties and the distribution of the cycles (periodic points) were identified. - (b) Intersecting invariant curves (stable and unstable manifolds) passing through saddle fixed points, or cycles, were computed using an improved algorithm based on the iteration of a quadratic germ. This algorithm permits to obtain the slope of the tangent and the curvature at points of both invariant curves. This led to the determination of homoclinic and heteroclinic points. - (c) Varying the parameter p, typical qualitative changes (bifurcation) of the phase portrait were detected. Lists of cycles and the two manifolds passing through the saddle fixed point (1;O) show that cycles exist only inside the area enclosed by these two manifolds. A subset of cycles is such that the sequences of their points are bounded by intersection points of such manifolds (homoclinic points), when their period increases. Moreover these points are aligned on certain curves (Fig. 16a,b,c,d). Due to the symmetry of the map two of these privileged curves are y=O and y= x-F(x). Period two cycles are absent for this map.

Toulouse Research Group Main invariant curves and distribution of cycles. 0 centres, x saddles

Fig. 16a. Map (14) & (18). Stable-unstable manifolds of the fixed point (l;O), and distribution of cycles.

Fig. 16 (Continues). (b) Map (1 4) & (1 8).

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Fig. 16 (Continued). (c) Map (14) & (18).

Starting from the fixed point 0 (x=y=O), except for particular values of p, the classical Birkhoff structure of the phase plane (Fig. 17) was recognized in a neighborhood of 0. It is reminded that this structure consists of a center 0 surrounded by a set of regular closed ~nvar~ant curves (free of singularities), one or more island structures, and one or more instability rings [43] giving rise to homoclinic points. A period k island structure is made up of the stable and unstable manifolds of a period k saddle, intersecting only at the k cycles points. Inside this figure one has a companion period k center. In an instabi~ityring the saddle manifolds have

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infinitely many (homoclinic) intersections. With respect to the Birkhoff contribution it was shown that at least two new features appear out of a certain neighborhood of 0: a region of "diffusion" and a region of "stochasticinstabiliw.I'

Fig. 17. Phase plane picture according to Birkhoff.

The map (14) being area preserving the multipliers (eigen values) product related to a fixed point, or a cycle, is always hl hz= 1. Then except some bifurcation situations, the singularities of the phase plane are centers (hl,~=e*'?), saddles said of type 1 (Xpl), saddle said of type 3 (XI<-1). For a saddle the slopes of the corresponding eigen-vectors are designated by P I ,p2. An additional property of a cycle of period b2 is its rotation number r , equal to the number of turns made around an interior point of the polygon obtained following successive images (consequents) of one of its k points, until one of these images comes back to the initial state. For conciseness, a

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saddle (or center) cycle of period k, with the rotation number Y , is designated by saddle (or center) Wr or k(+ It is related to the PoincarC's rotation number N= r/k. For the quadratic map (14) (18), the fixed point (x= 1, y=O) is a saddle of type 1. In order to solve the step (a) defined above, an ordering of the infinite set of cycles generated by the map (14) is necessary, Tables of computed cycles were established until a maximum period equal to 97. An extension of the cycles list to higher periods encounters two essentially numerical constraints: increasing slow convergence of iteration methods near bifurcation values, and an increasing strong sensitivity to rounding and truncation errors when I hl I is very large. With computing refinements the extra gain is quite small, especially in view of the inescapable situation of infinitely many cycles generated by (14). Indeed, as stated by a well known theorem due to Birkhoff, the presence of homoclinic points ensures the existence of such an infinite set. This property was verified, accompanied by the discovery of other ones, from an inductive reasoning, based on an ordering of cycles with respect to an increasing period k for a constant rotation number r , and a classification in the order of decreasing Wr. Let Pm[k(,)]be the rn-th point of a period k cycle, l l m l k , with the rotation number Y , and hl[k(,Jthe associated eigenvalue. So new asymptotic relations appear for a set of saddle cycles k(r)of type 1 :

where x * , y*, andp* are finite constants. For another set of saddle cycles k(r)of type 1, one has:

The associated center cycles are such that:

where xa, y", and k, ,are finite constants. The position of some cycles with respect to the stable and unstable manifolds of the saddle fixed point (x= 1, y= 0) are given by Figs. 16a,b,c,d.

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It appears that the points P,[k(l)] approach asymptotically homoclinic points of the stable and unstable manifolds (invariant curves) of the saddle fixed point (x= 1, y=O), inside the area enclosed by these manifolds. A simple inspection of Figs. 16, and the cycles ordering, show this feature. Such a conclusion is also confirmed by a study of the limits of the sequences 031-p2) at Pm[k(l)]. Indeed the differences (p1-p~)approach asymptotically the difference of slopes of the saddle manifolds at the homoclinic points when k increases. Homoclinic points can therefore be characterized explicitly as accumulation points of a set of cycles having a suitably chosen order. So a set of saddle cycles hr)of type 1, and saddle k(r)of type 3, exists with such a behavior, which corresponds to asymptotic properties derived from (1 9):

where p1* and p ~ are * the slopes of the stable and the unstable manifolds of the saddle fixed point (1;O) at the homoclinic point B,. When converging toward Bm the points P m are aligned on the curves of Fig. 18. For each r a subset of cycles k(r)follows an arc defined by r, when k increases.

Homoclinic point point of rank 1

invariant curves

loci of cycler

Fig. 18. Loci of cycles 4r,

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Moreover, the map gives rise to the following behaviors: - (a) The distance between (0;O)and the point Pm[k(,)]is monotonically increasing as Wr increases. Considering a k(,) cycle, its distance from (0;O) increases when the non linearity strength (1-p)increases. - (b) The "amplitude" of a "loop" formed by one of the two manifolds of a saddle point, between two consecutive homoclinic points, increases rapidly as (1-p)increases, or as the distance from the fixed point (0;O) increases (k increases for the same r). In the same conditions the multiplier hl[k(rl]>l,and the angles formed by the two eigen-slopes p1 and p2 increases. - (c) The existence of islands structures and instability rings is limited to cycles up to a critical effective order Wr= a ] ,for a given parameter p. The appearance of heteroclinic points at the smallest possible distance from (0;O) may be used as an equivalent definition of al. In the region defined by Wr a2 the mean distance s,, grows exponentially, the iterated sequences of points being very complex. This corresponds to a region of stochastic instability. - (e) In [19] it is conjectured that the boundaries of a Birkhoffs instability ring, and the boundary separating the diffusion area from the one of these rings, correspond to irrational values of Wr.

6.2.b Bifurcations The center fixed point (0;O) undergoes the following sequence of bifurcations: (23)

center (0;O) -+ center (0;O) +island structure k(r)

An island structure k(,.)(or Wr) is released from (0;O)each time the angle cp of the multipliers e*Jsof (0;O) is (p=27cr/k, r and k being relative prime integers. This structure only exists for 'p> cp(r/k), i.e. p< cos(2xr/k). The same bifurcation also occurs for a center rllkl each point of which releases an island structure defined by r*/k2. The result is a "cyclic" island structure

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(r&/ ( k t k ~ }More . generally island structures (YI ...rm)/(kl...k,,,),contained inside island structures ( ~ ~ . . . ~ m (kt . l ~..&-I), can be defined. When p decreases from 1, the angle cp of the center cycle k(,),born from the bifurcation (23), increases from 0 to R, value giving rise to a flip bifurcation by crossing through V=R,or A,= A*= -1 : (24)

center k(p)+ saddle kr)of type 3 + center (2k)(2,).

Fig. 19. Map (14) C (18), p= 0.707. ~anifoldsC of a saddle 1711 of type 3, the loops of which give homoclinic points, and surround the center 34/2. CI are arcs of the manifolds of a saddle 17/1 of type 1.

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Chaos Avant-Garde: Memories ofthe Early Days of Chaos Theory

After the bifurcation the two manifolds of the saddle k(r) of type 3 surrounds two neighboring points belonging to the center (2k)(2,). These curves form an eight shaped figure with homoclinic points, as shown in Fig. 19. More generally equivalent bifurcations take place by doubling period:

(25)

center 2'k[d1)l+saddle 2'k[(l)1of type 3 + center 2'+'k[4,+,)~ ,

i= 0,1,2,... , k= 1,3,4,... Consider the period k cycles which do not belong to the set born from the bifurcation (23), or from equivalent bifurcation when a center cycle takes the place of the point (0;O). Such cycles can be generated from the following fold bifurcation:

(26)

cusp cycle (k) + saddle (k)of type 1 + center (k),

which occurs when the angle cp of the center cycle is cp= 0, the multipliers of the saddle being hl= hz= +I. These two cycles merge and give the cusp point at the bifurcation. Before the bifurcation the two cycles do not exist. For the quadratic map defined by (14) and (18) the cycles merging occurs for a pair of cycles 3(1) (or 311) when p= b=1-42, one of the cusp point being x= -0.207107..., y= 0. For p-p ~the , two cycles 3(1) no longer exist. By means of computer-programmed algebraic operations it was found that the two manifolds (invariant curves), passing through the point x= -0.207107...,y= 0, can be described by an at least asymptotically convergent series in fractional powers of x, the first term of which is:

y = +2[x3(1 - p,)/3]'. These manifolds therefore form a cusp locally. Longer segments are shown in Fig. 20. They do not join smoothly, giving rise to homoclinic points.

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Fig. 20. Map (14) & (181, p= 1-62. Manifolds ofthe double cycle 311, the loops of which give h o m ~ l i n i cpoints, and forming a stochastic cusp.

=. ir

7

U

Exceptional

CAI8

'p- 2 ~ 1 3a t ( 0 , O ) .

Singular invariaat curve traversial) the six-branch saddle ( 0 , O ) .

Fig. 21a. Nap (14) & (18), p=-112. One of the 6 branches of the complex saddle (0;O).

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Chaos Avant-Garde: Memories of the Early Days ofchaos Theory b

(c> Fig. 21b & c. Map (14) & (18). (b) p= -0.55. (c) . p= -0.45. Manifolds of the saddle 311.

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Due to the presence of the quadratic nonlinearity the normal form, deduced from applications of Cigala’s transformations (cf. sec 5), shows that one has an exceptional case for the fixed point (0;O) when p= -1/2 with cp= 2 d 3 . Then three points of the saddle cycle 311 of type 1 merge into the center (O:O), and their stable (unstable) manifolds give rise to a complex saddle with six branches presenting homoclinic points (Fig. 2 la). The situations for p< -1/2 is given by Fig 21b, and for p--112 by Fig. 21c. If p= -1/2 the instability immediately occurs from the fixed point (0;O). Equivalent bifurcation occurs for other center cycles. So the exceptional case cp= 2ni3 occurring for the center 311 is a bifurcation involving saddles 9/3 of type 1. For the quadratic map other values of cp commensurable with 2?c, p= 2Mq, do not lead to exceptional cases, that is to the presence of a canceling denominator by application of Cigala’s transformations for such cp-values. Nevertheless these transformations permit to have an idea of the form of invariant closed curves in a small neighborhood of the center (0;O). So for cp= 2k7d4 (i.e. p=O) and I x l > ~ ,l y l ~PO, being sufficiently small, arcs of invariant closed curves can be approximated by x2y2= const. (Fig. 22).

Fig. 22. Map (14) & (1 8). p= 0, (pp 2~14.Approximate closed invariant curve, or unsolved island structure Mr==4, k>l,r>> 1, surroundingthe center (0;O).

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6.3 Study of unbounded cubic map With an unbounded cubic non linearity the map equation is given by (14) with:

Properties, equivalent to those of sec. 6.2, are found in the case of the cubic non linearity, plus some others [19] [41] [42] [48].In particular three fixed points exist now: the center (O;O), and two saddles of type 1 (1;O) and (-1;O). Due to the absence of quadratic term the exceptional case neither occurs for cp= 2x13, nor for other values of cp commensurable with 27t, cp= 2hlq.

The bifurcation (23) occurs for an even period k2 4, but for an odd period k2 3 it becomes: (28)

center (0;O) + center (0;O) -t 2 island structures krl .

The two island structures are released from (0;O).They have a complementary symmetry and the same multipliers. For cp= 2x14 (i.e. p=O) , the invariant closed curves can be approximated by x4 +y4= constant. (Fig. 23). The deformation of the invariant curves with respect to an elliptical shape is therefore not the same as in the case of a quadratic nonlinearity. Asymptotic properties of sec. 6.2.a are also found to hold. Compared with the quadratic case, the major features of the phase portrait appear to differ only by the effect of the F(x)= - F(-x) symmetry, which leads to the doubling of cycles of an odd period k (the value k= 1 included), and the absence of instability for cp= 2h13. Another more fundamental difference with the quadratic case is the fact that the bifurcations related to q=x, h= -1, 9'0, h= 1 are not unique in the sense that they can occur more than once when the parameter p decreases. By crossing through the angle q=x (hi=h2= -1) one may have: (29)

center k(r) + center /qr)+ 2 saddles (2k)(z,, of type 1 + 2 centers (2k)(Zr),

By crossing through hl= h2=-1 :

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(30)

saddle k(r)of type 3

+ center lqr)+ saddle (2k)(2r)of type 1.

By crossing through q=O :

(3 1)

center k(r)+ 2 centers k(r)+ saddle k(r) of type 1

In (29)-(31) the pair of saddle cycles, or the pair of center cycles, have complementary symmetry and the same multipliers.

Fig. 23. Map (14) & (27). p= 0, (pz 2d4.Approximate closed invariant curve, or unsolved island structure WF 4, k>> 1, r>>l, surrounding the center (0;O).

6.4 Study of m a p with bounded nonlinearity Among the different types of bounded nonlinearities studied such (1 5)-( 17), this section is limited to the map defined above by:

with -Iq,l<+l [19][41][42] [48] [64]. This form differs from the quadratic map by the presence of higher terms in the McMaurin development of F(x). The principal similarity is the existence of the main fixed points (0;O)and (1;O) which turn out to be respectively a center and a saddle of type 1. A

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particular feature is the existence of a third fixed point (a;O),which is a center, independently of the value of p. This point is defined by the coordinates: x=

a root of the transcendental equation x2exp[(1-2)/4]=1 ,

y= 0.

The fundamental difference with the quadratic nonlinearity is the fact that a cycles set of (32) exists for x>l. This new feature implies that it is necessary to classify cycles into at least four distinct families: (i) internal ones surrounding the main center (O;O), (ii) internal ones surrounding the secondary center (a;O), (iii) external ones surrounding both (0;O) and (a;O), (iv) mixed ones, characterized by the property that all their points are located inside the areas bounded by the intersections of the two manifolds of the saddle (1;O). Points of both internal and external cycles converge separately to the homoclinic points of the saddle (1 ;O) manifolds. The asymptotic relations of the form (19)-(21) remain valid in both the internal and external regions. Furthermore the positions of the higher-period cycles (k/r>> 1) appear to guide the higher-order loops of the two manifolds of the saddle (1;O) (Fig. 24). All the bifurcations identified in the case of the quadratic map are found to occur on internal cycles of (32). Nevertheless the bifurcation rate is higher; that is, the same degree of stochasticity is attained at a lower strength of nonlinearity with (32). Mixed cycles constitute a new qualitative feature, and can be differentiated into two kinds. Considering the cycles offirst kind, the manifolds of a mixed saddle do not form anything resembling an island structure, even if hl-l<
Toulousa Research Group

.

K

0

W

3

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The two manifolds of a mixed saddle of second kind have no tendency at all to follow the outline of the two manifolds of the saddle fixed point (1;O). Indeed the stable and the unstable manifolds of such a mixed cycle spiral instead around two companion mixed centers, with formation of homoclinic points. It appears that this configuration represents an unstable center (in the sense of Birkhoff [43]) approached simultaneously by spiraling and intersecting stable and unstable manifolds. Figure 26a represents this situation for k= 0.144, giving a mixed saddle 17/2 of type 1 and two companion mixed centers 17/2. Such centers admit homoclinic points in an arbitrary small neighborhood (Fig. 26b). Contrarily to internal or external centers, mixed centers only exist inside essentially stochastic phase plane cells, because their existence is inherently coupled to the existence of homoclinic points. Mixed centers appear to meet the validity conditions of a Zehnder's result [49] on the arbitrary small proximity of centers and homoclinic points. (cf. p. 159 of [42] a discussion on the Zehnder's theorem).

I

Fig. 25. Map (32), p= 0.13222. The Pecked arcs are manifolds of the mixed A , WF 1612. The "x" are other mixed saddles 16/2.

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..

t

2.23i3b

(a)

8

(b)

Fig. 26. Map (32), p= 0.144. (a) ~ ~ i f o lofd as mixed saddle 17f2 of type 1 (loops width strongly increased). (b) The Birkhoffs unstable center. The singularities introduced by a bounded nonlinearity induce a new behavior: the mean distance s,, between (0;O)and the successive images (or preimages) of an initial point ceases to grow exponentially, as this occurs for unbounded nonlinear characteristics. The region of stochastic instability disappears except for (p=2n/3at (0;O).The distance sn is bounded now. This situation is due to the presence of at least two close "simplerings," each one being constituted by a pair saddle and center cycles, such that the saddle manifolds of one "ring" intersect the saddle manifolds of the other ring. It results a creation of heteroclinic points, but such points are absent out of the region related to these rings including ail their saddle manifolds. This situation is possible by the non linearity limitation. The union of such simple rings was called region of ~ o ~ n d~e d~ ~ s i o n . Consider the sequences of points generated by the map now. Then an iterated sequence presents series of long stochastic transients and short stochastic ones which alternate around a center point. The long stochastic transients occur in the region of influence of one of the simple rings. They are separated by short stochastic transients, when the points sequence passes from the influence zone of a simple ring to the influence zone of another

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simple ring. This occurs as soon as a neighborhood of heteroclinic points is attained. More than two simple rings giving rise to a region of bounded diffusion appear in the peripheral part of Fig. 27, obtained from the map (14)-(15) with p=5/100. At least four simple rings can be identified by four sets of periodic closed curves. These curves are separated by regions materialized by clouds of iterated points related to the above transients.

Fig. 27. Map (14) & (15) with p=5/100.

Results related to other types of bounded non~inearit~es are given in [41] 1421. A dissipative perturbation of the ~orr~sponding maps may generate chaotic attractors as mentioned in sec. 4.2.

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6.5 Study of almost conservative maps (1973-1976) Section 4.2 considers the numerical simulations of almost conservative maps leading to chaotic attractors obtained until 1973. These simulations were followed by more complete studies using the sec. 6.2 approach by addition of a small dissipative term to a conservative map. Then the resulting map may be either invertible or non-invertible. The results obtained at that time were published later in the book [42] (cf. p. 184-21 1). They show the high complexity of this situation. A link with strongly non-conservative maps was sought [42] (cf. p.212-216).

7. Study of One-Dimensional Non-Invertible Maps (from 1972) 7.1 Basis of the study As indicated in sec. 1, the Toulouse group approach was based on understanding dominating internal mechanisms of dynamic processes, and thus preserving a phenomenological transparency through the simplest possible "germinal" examples (in particular the lowest dimension). When I began the map study 1 wrongly thought that a direct consideration of twodimensional maps gave the simplest "germinal" interesting situations. Progressively from 1968 the Pulkin's paper [25], published in 1950, weakened this conviction. Indeed Pulkin was the first to show in the more general case of piecewise continuous maps, or continuous ones, that the presence of unstable cycles may generate complex oscillating iterated sequences. His publication deals with one-dimensional maps generating infinitely many unstable cycles, giving rise to what is named completely invariant sets by this author. Such sets are related to the existence of limit sets of different classes. So infinitely many limit points of the unstable cycles set, when their period tends toward infinity, lead of class 1 limit set. The limit sets of class 1 generate limit points of class 2, and so on until limit points of class -. Such an organization of limit points leads to what has been called from 1976 a fractal structure. The Pulkin's results, associated with the Giraud's ones described in sec. 3.3, in particular the complex structure of cycles points on some basin boundaries and the appearance of the first ranks of period doubling (Fig. 5c), ultimately finished changing my mind.

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After long bibliographical researches, including studies of Herschel [ 18141, Pincherle [ 19201, Ulam and Von Neuman [ 19471, two other authors,

Myrberg and Sharkovskii, drew our attention by the prime importance of their contribution directly related to the Toulouse group preoccupations. The results of these authors are complementary. They gave from 1972 the reference points which permitted step by step to identify the fractal bifurcation organization generated by smooth maps defined by a function with only one extremum. It was called box-within-the-box bifurcation structure ("structure boftes-emboftdes" in French) [56]-[58] by the Toulouse group, and "embedded boxes'' by Guckheimer when he quoted this result [591-

From 1958 to 1963, Myrberg published a series of important papers [50]-[53] concerning the bifurcation properties of the one-dimensional quadratic map T (33)

2

-a

x being a real variable,

A a real parameter. They constitute one of the most

important basic contributions to the theory of dynamics. The map (33) is noninvertible, its inverse T'having zero, or two real determinations (rankone preimages), that is the map is of ZO- 2 2 type. The rank-one critical point separating these situations is x= -A.The "classical" singularities of (33) are constituted by two fixed points, the cycles points of period k, and their limit (fractal) sets when k + 00. The map (33) has at most one attractor. Myrberg has been the first to show a series of essential generic results for the theory of dynamic systems. - All the bifurcation values occur into a bounded interval, -1/4 < R < 2, for the map (33). - The number of all possible cycles having the same period k, and the number of bifurcation values giving rise to these cycles, increases very rapidly with k. - The cycles with the same period differ from one another by the cyclic transfer (shift) of one of their points, obtained by k successive iterations by T . These cyclic shifts were defined by Myrberg using a binary code constituted by a sequence of (k-2) signs [+, -1 (binary rotation sequence). More or less explicitly the Myrberg's papers give an extension of this notion to the case k + 00, and to general orbits (iterated sequences).

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For A< h ~=1,401155189 ).~ ..., the number of singularities is finite (Tis "Morse-Smale"). For h2 the number of singularities is infinite. A= 41)~ is an accumulation value of bifurcations by period doubling (Myrberg cascade called by Myrberg spectrum). The following cascades of bifurcations: "stable cycle k2' + unstable cycle k2' + stable cycle k2'+',occur when h increases, i = 1, 2, 3, ... ; k having a fixed given value, k = 1, 3,4, .... For i+ 00, the bifurcation values, for a given value of k, have a limit 2, which is an accumulation value by period point A= &)$, &),v<&).v< doubling from a cycle of period k. - It is possible to classify all the binary rotation sequences via an ordering law (Myberg's ordering law). - A binary rotation sequence can be associated with the d-value resulting from accumulation of bifurcations such that i + w , or k + 43. This rotation sequence satisfies the ordering law. All these fundamental results have been passed over in silence in most of contemporary papers dealing with this subject, which has a very large vogue since 1978. The most part of these results have been very often attributed to authors who rediscovered them after using another form of quadratic map such as the logistic map, or maps of the unit interval. It is the case of a characterization of a cycle (or a more general orbit) by a binary code "R,L" ("+,-" with Myrberg), the notions of invariant coordinate, kneading invariant which are directly related to properties of Myrberg's rotation sequences, the cascade by doubling period. In 1970 I had access to the Sharkovskij's prime rank contribution in discrete dynamics from the references of the M. Kuczma's book "Functional equations in a single variable" (PWN, Polish Scient. Publ., Warsaw, 1968), at a time when this contribution was quasi unknown in western countries. After I had a regular correspondence, and publications exchange with Sharkovskij, whom I met for the VZZ'h International Conference on Nonlinear Oscillations (ZCNO) Berlin, Sept. 1975. It is well known now that from 1962 this mathematician has produced important results (in particular [54] [SS]) related to the non-wandering sets generated by general forms of continuous one-dimensional non-invertible maps with several extrema. More particularly a famous ordering of the cycles, with respect to their period, was given by this author. It is less known that he also gave the first "extended" notion (with respect to the classical one) of homoclinic and heteroclinic

-

-

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points in m-dimensional non-invertible maps, r n l l , the case rn=l having a particular interest for the problem considered here [ 5 5 ] . From the Pulkin's results, the Sharkovskij's cycles ordering compared with the Myrberg's one was the germinal point of a thinking on cycles classification. Indeed the Sharkovskij's ordering is general in the sense that it concerns general forms of continuous one-dimensional maps, but cannot discern between the cycles having the same period, the number of which drastically increases with the period growth. For example, a quadratic map can generate (cf. [39] p. 96): 2 fixed points, 1 period-2 cycle, 2 period-3 cycles, 3 period-4 cycles, 6 period-5 cycles, 9 period-6 cycles, 18 period-7 cycles, 30 period-8 cycles, 99 period-10 cycles, 335 period-12 cycles, 1161 period-14 cycles, 52377 period-20 cycles, 35790267 period-30 cycles, 3714566310 period-37 cycles, etc. ....

The formula giving the total number of cycles having a fixed period k, and the corresponding number of their bifurcation can be found in [39]. The Myrberg's ordering is limited to the particular case of maps defined by a function with only one extremum (unimodal maps), but it permits to differentiate cycles of same period by their rotation sequence (permutation order of their points), which leads to a sharp classification. Considering these two basic results as a starting point the guide line, adopted to go further in the study of unimodal maps, was to introduce the set of the critical points of rank Y = 1, 2, 3, ...., (i.e. the sequence of images of the map extremum). Such a set is made up of "non-classical" singularities introduced by the map non-invertibility. This leads to identify the non-classical bifurcation corresponding to the merging of two singular points of different nature: a critical point and an unstable cycle for parameter values designated by h i . The introduction of this non-classical bifurcation permitted a new classification of the Myrberg' spectra, through the identification of the fractal box-within-the-box bifurcation structure (or embedded boxes). All the results of this section 7 were published in French before 1976, but they are presented in English in [16] and [39] with more details.

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7.2 The box-within-the-box bifurcation (or embedded boxes) structure Three basic bifurcations related to the two fixed points of the map T given by (33) must be noted at once. The first one is the fold bifurcation giving rise to the two fixed points (one ql with multiplier D1,the other q 2 with S-4) of T for A= -1/4,.when the curve x ' = x2 - A is tangent to the first bissectrix. The second bifurcation is related to the Myrberg spectrum by period doubling from the fixed point q2 in the parameter interval (designated by 01) jl(l)o IA< A(,)$,The third bifurcation is non classical in the above sense. It corresponds to a "final" bifurcation value A= A*l= 2, obtained for the merging of q1 with the rank two critical point x= A2-A. The parameter value ;I" gives rise to a completely invariant segment in the sense of Pulkin, -21x S2, containing all the cycles of T (which are unstable), their limit points of different classes, the arborescent chains of their increasing rank preimages. It was called in French "segment stochastic" or "segment S." For the period-k cycles, the bifurcation study is facilitated by drawing the curve representing the powers T k of the map (33) in the plane (x, x') from the coordinates of its extrema. These coordinates are easily obtained from successive images and preimages of the minimum of T (x=O). Such a process shows that k arcs of the oscillating curve T k , each one with only one extremum, reproduce the situation of T locally with respect to the first a fold bifurcation, giving rise bissectrix (x'= x). This means that for A= to two period-k cycles (one with multiplier S<1, the other with S l ) , corresponds to k arcs of the oscillating curve T k (each one with only one extremum) tangent to the first bissectrix. The same occurs for the Myrberg spectrum by period doubling from the period-k cycle characterized by S c l , in a parameter interval (designated by ok) &kb IA I &k)s and the equivalent of the non-classical bifurcation A= All. The interval 4 1 ,< )~ AI All was called < a box (the largest one) designated by Q l , the complementary interval AS A*1 was designated by A]. In this interval all the bifurcations related to cycles of period IF 3,4,5, ... take place and their related Myrberg' spectrum

WI WI.

If they are not produced by the same fold bifurcation, cycles having the same period differ by their rotation sequence, which implies a symbolism permitting to differentiate their bifurcations. More precisely a first set of cycles noted (kl , j l ) , kl = 3,4,...,jl = 3,4,...,PI ,PI+" if kl+w, is identified,

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Days ofchaos Theory

j l being an integer characterizing the cycle numbering in the Myrberg ordering of rotation sequences related to cycles having the same period kl,

\ 1

i

~

-1

an extremum. Xn,

Fig. 28. Map (33), ?. Behavior of the iterated sequences for h=h(j,o.e,E>O being small.

From the curve Tk' in the (x, x') plane, and for a parameter range of k A I , kl "parabolic shaped" arcs are situated as the parabola T with respect to x'=x for k Q .So when h increases the tangency of these kl "parabolic" arcs with x'=x defines a fold bifurcation giving rise to a pair of (kl , j ~cycles ) (one with multiplier S<1, the other with *I). This bifurcation value is designated by

From this value, for each of the kl arcs a parameter

increase leads to a situation equivalent to the A'],the kl points of the (kl ,j1) cycle with D1,merging with kl critical points of rank m= kl+l, ....,2kl. This "non-classical" bifurcation value is designated by A5

A>

becomes a box designated by

A*4

, the interval AJi

(kl )o

c A , . The interval A$kl)oSA5

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Aji( 4 ) s designated by contains the Myrberg spectrum generated from .c A5 A* is designated by the (kl ,j 1 ) cycle with %I. The interval Atkl).% 4 Ajl . This process shows that $2; is self similar to R1. Equivalently a kl “periodic segment S” having the period kl and the rotation sequence j l is obtained for

A> .

Fig. 29a. Map (33). Representation of T 3 .A=A*3.

) of Tk’.For a Considering now a cycle (k2 ,j2) and the power k2 ( parameter interval belonging to AJ’ each of the above kl “parabolic” arcs kl

’

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generates k2 "parabolic" arcs situated as the above kl arcs of T k l with respect to x'=x for LE

. For

a pair of cycles (klk2, jlj2) this leads to

. .

define the interval interval

A { ~ ~ k 2I ) oA I

atkj:k2)o I as

ajIj2 (klk2 1s

A*-jij2

klk2

as

as the box

QJk:;

c

equivalent to the Myrberg

d j 2 klk2

spectrum generated from the (kl ,j l ) cycle with %I, the interval AS A*jij2 as klk2

A$;*

. Then the interval

The parameter values

d 1 j 2 klk2

On the left

..

A$:k2kz), <

appears self similar to

A$k,)o on the left and A*4

subset of boxes Q{ when k+=.

Ail, the

A*4

.

on the right are limit of a is a limit of boxes

when k 2 j - 3 .

Fig. 29b. Map (33). Representation of T kfor A=

a?,

QGt

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3 Fig. 30. Map (33). Representation of 78 from a "parabolic"arc of f a

The continuation of this process of consideration of successive powers of the map leads to the de~nitionof boxes ~ ~ c A!$;:Ji;:,2 ,~A$ii'.'Jia, ; and intervals

M2'**.& ..,k containing the Myrberg' spectrum of the cycle 1 2

a

(ktkz ...k, , j 3 2 . , . j g ) with S4,a= 1,2, ,.,.-, Such boxes, contained in interval A and determined from a fold bifurcation and the merging of a critical point with an unstable cycle such as *I, are said embedded boxes offirst kind.

~

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Chaos Avant-Garde: Memories of the Early Days of Chaos Theory

t b

Fig. 31a. Map (33). Representation of T 2 for A=

Fig. 3 1b. Map (33). Representation of T 2 and T

A' 21

for d= A;l

,

-E,

00.

Toulouse Research Group

Similarly e~beddedboxes of second kind

141

QJ~J2*"Ju1

klk2 ...k, 2'

are defined with a

flip bifurcation (period doubling) as the lowest extremity, and the merging of a critical point with an unstable cycle such as S< -1, as the highest parameter value A= ,%*'1j2,**jut. Such boxes contain a subset of a Myrbergs spectr~m.

...

k,kz k, 2'

Fig. 32. Map (33). Representation of T 2and T4 for A=

.

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Figures 28, 29, 30, 31, 32 show the curves representing a power of T in different situations characterizing boundaries of boxes. Figures 33 and 34 represent the structure of embedded boxes of first and second kind. The

'

boundary ' (j1j2-'ja k 1 k 2 . . . k ~of ) ~ the Myrberg spectrum generated by period doubling of cycles (klk2...k02j ,j&.jO), i=0,1,2, ..., is on the right limit of parameter values A j i j 2 - j a ' when kO+w. (klk2 *-ka1s

Until 1980 these results were published in French [I91 [57] [58]. More details on these questions in English can be found in [16] [39].

7.3 Consequences A first consequence of the above results for the map ( 3 3 ) concerns the existence of infinitely many unstable cycles with limit points of different classes (in the Pulkin sense), when A> &I).$. Then consider such a A value giving a stable cycle (kj)of T. Its basin is the open segment D bounded by the unstable fixed point q1 ( P 1 ) and its rank-one preimage 41'' different from 41, deprived of the set F' made up of all the unstable cycles, the limit points (class 1) of such cycles when the period tends toward infinity, the limit points of increasing classes in the Pulkin sense, and the increasing rank preimages of all these points. The set F' is the intersection of the Julia set of the complex map z'= z2-Awith the real axis. The map p has k stable fixed points each one inside its proper immediate basin Do(j),j= 1,2, ...,k. Each stable fixed point has a total basin 00')made up of infinitely many nonconnected segments, given by the arborescent sequence of the increasing rank preimages of the immediate basin D&). These segments are intertwined with those of the basins of the other fixed points of p. The accumulation points of all these segments are constituted by the set F'. The set of such accumulation points has a fractal structure, due to the properties of the above arborescent sequences of preimages. The publications [56]-[58] and [60] in

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French deal with this question. They also quote the case of the basin of the two stable cycles of cubic maps, and the case of two-dimensional maps with homoclinic points. Such basin boundaries were called 'yronti8resfloues," or fuzzy boundaries, and after 1980 fractal basin boundaries [61]. They play the role of strange repellers. The above consideration leads to the following remark. Consider the stable cycle (k$), with respect to T (and not giving k stable fixed points). The set F' is also a strange repeller in the sense that it induces the existence of domains corresponding to what can be called chaotic transients, located inside D n[uDoG)].It is also a fuzzy basin boundary, which can be viewed as related to unstable chaos, a stable chaos being obtained for the parameter * '

values A,l . It is worth noting that the well-known Li & Yorke's paper "Period three implies chaos" (1975) [79] implicitly deals with a situation which leads to stable or (and) unstable chaos. This means that period three generally implies unstable chaos for a smooth one-dimensional map. The Sharkovskij's cycles ordering (1964) indicates that a period three cycle implies the existence of infinitely many cycles of any period. In general they are unstable, except some of them which may be stable. Likewise this situation is implicitly related to a case which leads to stable or (and) unstable chaos. The Pulkin's paper [25], published in 1950, was the first to show in the more general case of piecewise continuous maps, or continuous ones, that the presence of unstable cycles can generate complex oscillating iterated sequences (i.e. chaos). The increasing rank classes of limit points, introduced by this author from the limit of unstable cycles when their period tends toward infinity (class I ) , leads to a fractal structure. These considerations about the Pulkin' classes (1 950) give a link with the Sharkovskij (1964) and Li & Yorke' results (1975), these authors independently starting from different points of view and approaches. I discussed with both Sharkovskij and Yorke about the question "Period three implies chaos" and related aspects in the framework of the YZfh International Conference on Nonlinear Oscillations (ZCNO Berlin, Sept. 1975), just after the Yorke's talk. Another opportunity of discussion with a specialist of dynamics, about the problems induced by the box-within-a-box bifurcation structure, occurred during my two years stay in the postgraduation center COPPE of the Federal University of Rio de Janeiro ( 1 9741976). Indeed Smale made a lecture on the May's results in this center at that time (I have forgotten the exact date). It was a question of the chaos

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generated by the logistic map, when the parameter value crosses through the accumulation of period doubling bifurcations from the fixed point with multiplier S4.For the Myrberg's map this accumulation value is &I),~, the The parameter interval (the chaos one) after chaos occurring for ;1> such a crossing was presented as an unknown domain in this lecture. A posttalk exchange of points of view permitted to inform Smale on the relatively recent results of the box-within-a-box bifurcation structure, corresponding to what was considered as an "unknown domain'' before. The second consequence concerns the interpretation of the iterated sequences behavior for ilvalues very close to the bifurcation values and

A>

A"( 410

defined above, which are limit of infinitely many self similar

A$k,)o-E, E>O being boxes. The publication [62] has called the situation il= sufficiently small, "cycle en valeur moyenne" (cf. Fig. 28). It was rediscovered in 1980, and is known as intermittency phenomenon now. The paper [62] has called the situation il= A* I

ki

+E,

e 0 being sufficiently small,

''segment stochastique cyclique en valeur moyenne." Rediscovered in 1982 this phenomenon is called chaotic attractor in crisis or boundary crisis now. The problem of random number generation was also considered by I. Gumowski in [77].

8. Applications A first application (1 967- 197 1 ) concerns rectifiers "alternating current-direct current" using thyristors with voltage feedback, or current feedback. Figure 35a gives the corresponding circuit with current feedback and two phases. Figure 35b represents the circuit without representation of the primary winding of the transformer, e being the input voltage. Connection (AB) corresponds to a voltage feedback, and (BC) to a current feedback. In order to have a correct switching time from a classical periodic switching curve (sinusoid, or piecewise linear), either an anti-saturation device [I71 [71], or a limiting one [68] [69] [71] was used. A discrete model in the form of a recurrence relationship was constructed without any approximation, the discrete states of this system being determined at two consecutive switching

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times of the thyristors. The analytical form of this model is very complex and bulky, but can be programmed in a computer without any difficulty. It is of implicit and parametric type, including several relations [17] [19][68] [69][71].If it might be analytically solved, the explicit form corresponding to the determination having a physical sense would be:

where X is the state vector, t,+l is the smallest root, larger than t, (switching time), of the relation g(X,, t)= 0. For a voltage feedback X is a one-dimensional vector. With a current feedback X is a two-dimensional vector. Then a stable fixed point of (34) corresponds to the specifications related to the periodic behavior (specified residual ripple) of the rectifier output, the mean value of which is associated with the input e. Its frequency is Norm, N being the phases number of the secondary transformer winding, o the angular frequency at the primary winding. A period4 cycle corresponds to a rank-k subharmonic with respect to the specified frequency Nol2n. The corresponding amplitude of residual ripple increases with the subharmonic rank, due to the low-pass filter properties of the load (inductance + resistance).

TH2

Fig. 35a. The rectifier circuit with current feedback considered in [17].

Figure 36 (K=6, 2= 10, E=0.3) is extracted from the Giraud's thesis (April 1969) [17].It represents the output behavior in the phase plane ( X J ) in the case of a current feedback, and a piecewise linear periodic switching

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curve. The state variables x,y are respectively reduced forms of the load current i and the output voltage u, K is the "amplifier-integrator" gain (transfer function U p ) , z is the load time constant, E is the reduced form of the input e. The points A , B i , Cj, i= 1, 2, 3, represent the cycle points (discrete phase trajectory), the curves are related to the continuous output of the rectifier. This figure shows two attractors: the stable fixed point A ("cycle 1'I) coexisting (multi-stability) with a "cycle stable d'ordre dlevt" ("stable cycle with a high period").

$

Y

C-

I

I

J

Fig. 35b. The rectifier circuit with current, or voltage feedback considered in [68].

It is important to note that the Giraud's thesis (page 190) says that the expression "cycle stable d'ordre dleve" is used for convenience sake in order to qualify a steady state without any periodicity, called ''pseudo-periodique" by this researcher. Such a denomination does not belong to the classical classification of dynamic motions. With respect to the maps of sec. 4, we have another example of the fluctuations in the vocabulary choice, revealing the group perplexity in presence of phenomena identified as chaotic after [71]. Figure 36 also shows two saddle ("colt') cycles of period 3 and 6, represented by their points and the corresponding continuous phase trajectories passing through them.

168

n

4

*‘

9%

16

48

3

8

Chaos Avant-Garde: Memories of the Early Days of

d

”-p . c

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Figure 37 [17] represents the non-connected (i.e. the map (34) is noninvertible) basin boundary (F) of the stable fixed point A. The immediate basin contains A and has a "Y-shape." Its boundary is made up of the stable manifold of the period 3 saddle Bi, i= 1,2,3. A non-connected part of (F)is represented in the lower right corner of Fig. 37. Out of the basin of A, basins of several stable steady states coexist, with among them the basin of the

Fig. 37. Non-connected basin boundary (F)of the stable fixed point A (accordingto [ 171, 1969).

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I70

above chaotic behavior. The transient toward A from the initial state M I , with the continuous phase trajectory passing through the discrete states Mi i= 1,2,..., 5 , ...., is given by Fig. 38 [17]. The one toward the chaotic behavior by Fig. 39. Figure 40 represents some bifurcation curves in the parameter plane (Kyz) for E= 0.3. Q, is a flip curve leading to a period doubling. Intersections of bifurcation curves are related to the multi-stability property. The regions I and IV correspond to a global stability for the fixed point A ; in this case it is the unique attractor for the corresponding parameter choice. A more complete study of the parameter plane (K,z) [80] was made for a limiting device [68] [69] [71] instead an anti-saturation device [I71 [71] in 1985. The boundary of the region leading to the existence of the fixed point (specified residual ripple) as a unique attractor appears as the limit of a set of fold bifurcation curves associated with cycles having an increasing period (p. 386-387 of [71]).

Figure 5.4.2.b

k.6

7-40

6.0.8

-t

Fig. 38. Transient toward the stable fixed point A (according to [ 171, 1969).

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Before the Toulouse group studies, made in the years 1967-1971, the manufacturers of rectifiers thought that subharmonic and complex behaviors in such systems were essentially due to asymmetries of the phases of the secondary winding, and "antenna effects." The results described in [ 171 [19] [68] [69] [71] have shown that, even with a perfect phases symmetry, the thyristors nonlinearities generate these non-wanted behaviors. Moreover, a correct choice of the gain K for a given z can lead to the global stability of the wanted residual oscillation at the frequency NwL2.n. A second application related to the longitudinal motion of particles in an accelerator is due to 1. Gumowski [41] [42]. The model is the area preserving map (14), with an asymmetrical periodic function F(x) as the nonlinear term: JJ'=-x+ F(x'), x' = yt F(x), F(x)= x- ( 1-p)[sin(bx+bo)-sinboll bcosbo ,O< bo
I

The parameter bo plays the role of a scale factor, and for a special particle accelerator, called microtron, bo was fixed as bo = 7c/2 -arctg(l-p)h. I. Gumowski considered also the longitudinal motion in an accelerator with the assumption that the ring is composed of discrete elements only. One formulation without space charge, taking into account one RF-cavity gap, is:

where p is the particle momentum, Ap the momentum deviation, a the momentum compaction factor, @ the phase, n the number of cavity gap traversals, V the amplitude of the sinusoidal accelerating voltage, h the harmonic number, e the particle charge, and p, 3/, c the usual relativistic quantities. The subscript s refers to synchronous values.

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Chaos Avant-Garde:Memories of the Early Days of Chaos Theory

fisurr

5&&5

Kr6

Z.tD

C~0,ll

@ -J h

Other appiications giving rise to complex dynamics have been d ~ s c r i b e ~ for systems of satellite attitude controlled by a frequency modulator of second kind (model having the form (34)) [72], and in a satellite with inertia are periodic function of time [73].

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9, The International Colloquium ''~ r ~ n s f o r ~ ~ t ~ o n s Ponctuelles et leurs Applications" (Point mapping and Applications): Two Exhibitions of Chaotic Images (1973 and 1975) The InternationaI Colloquium " ~ ~ a n s f ~ r ~Ponctuelles a i ~ o ~ s et leurs Applications" (Point mapping and Applications) was held in the framework of "Laboratoire d' Automatique et d'Analyse des SystBmes" (LAAS), in TouIouse, from Sept. 10 to 14, 1973. Prof. Lagasse, director of the LAAS, was its chairman. I. Gumowski led the scientific committee. I was vicechairman. Eleven papers were devoted to "stochastic" (chaotic) ~ e h a ~ iino ~ , the midst of the 33 ones of the Proceedings (Colloque International du CNRS n"229, Editions du CNRS, Paris 1976). Among the authors o f papers we can quote: J. Ford, J.H. Bartlett, M. Kuczma, I. Gumowski, M. Urabe, M. Henon, B.V. Chirikov, V.K. Melnikov.

Fig. 41. Conservativ~map 114) & (15) with p=-0.2.

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The colloquium was accompanied by an exhibition of chaotic images generated by equations studied by the Toulouse group, a part having been obtained with the J. Bernussou's collaboration. The above Figs. 7d, 8-1 1, 27 were exhibited with some others among which Figs. 41-53. I devoted the introductory presentation of the colloquium to a short history of nonlinear dynamics (pages 19 -27 of [Sl]). It is in this framework that I announced this exhibition of "stochastic" images, quoting the Birkhoff s papers dealing with the laws of aesthetic (vo1.3 of [82], p. 320-364), and a PoincarC text extracted from a "Notice SUT Hulphen" in the "Journal de I'Ecole cahier, also see [78], page LXV). In this text PoincarC Polytechnique" (60kme deals with the aesthetic emotion which can be communicated by mathematics in the following terms:

Fig. 42. Conservative map (14) & (15) with p=0.05.

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Fig. 43. Conservative map (14) & (15) with p=0.3.

Le savant digne de ce nom, le ge'om8tre surtout, kprouve en face de son ceuvre la mdme impression que l'artiste; sa jouissance est a u ~ s ~ grande et de mdme nature. Si j e n'e'crivais pour un public amourew de la Science, j e n'oserais pas m'exprimer ainsi; j e redouterais l'incre'dulite' des profanes. Mais i d , j e puis dire toute ma pensbe. Si nous travaillons, c'est moins pour obtenir ces r~sultatsp o s ~ ~ ~ s auxquels le vulgaire nous croit uniquement attache's, que pour ressentir cette &motionesthe'tique et la communiquer 2 ceux qui sont capables de l'e'prouver.

I took the liberty of saying that these images had begun to manifest such an emotion in a form opened not only to specialists as Poincard said, but also to a general public (page 27 of [Sl]), due to the new possibilities offered by numerical simulations. The same ~xhibition,entitled ' ' ~ o r ~ ~ o g et ~n~se

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Muthkmatiques," was organized by Mr. Marcel Barthes, Director of the "Alliance Franqaise Rio de Janeiro-Centre" in the "Centre Culture1 de la Maison de France," from May 8 to 30, 1975.

Fig. 44. Conservative map (14) & (15) with ~ ~ 0 . 2 5 .

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x'= y - F(x)+asl(l+uy2) - 1 . h -I-Xx'Il5-kF(x')

F(x) = flx+2(1-/i)x2

/(1+x2)

Fig. 45. Non i n ~ e ~ i bmap, i e a= 0.025, a= 10, p==0.525,

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Fig. 46

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12

- 10

- 10

14

Fig. 50

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7

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Fig. 51

185

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mpc:

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(x)-az Ax (x'-*-x+xJ ( x ) a45 a W-xJ ( x ) - 2 A(x+x,- 1) +ys

powo< x < x , +11,4(x-xJ+y,

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Chaos Avant-Garde: Memories of the Eurb Days of Chaos Theory

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10. Conclusion. The "history" of chaotic dynamics begins from around 1976, when the word "Chaos" was initially introduced in dynamics, with the well-known May's paper [63] which quotes publications [56] [57]. The Toulouse group continued a fruitful collaboration with I. Gumowski, who occupied a new Professor position in the Faculty of Sciences of Paul Sabatier University, until his retirement in 1987. The results obtained during the "prehistoric" times were the starting point of many developments of the problems considered in this text. Among such developments, the fractal box-withinthe-box bifurcation structure (or embedded boxes) was extended to onedimensional maps defined by functions with several extrema, to twodimensional diffeomorphisms in a foliated parameter plane (i.e. considered as made up of sheets, each one being associated with a well defined cycle). Complex communications (spring urea, saddle urea and crossroad area) between these sheets were identified (cf. [39] and papers published from 1991 in "International Journal of Bifurcation and Chaos"). Using a PoincarC section, various types of models in the form of ordinary differential equations were studied from the point of view of the foliated parameter plane and its communication areas. Such studies have led to strong links of collaboration with H. Kawakami (University of Tokushima). Two-dimensional noninvertible maps have been the object of extended studies [16], using as powerful tool the notion of critical set, natural extension of the critical point of the one-dimensional case. This topic has given the opportunity of a fruitful collaboration with Laura Gardini (University of Urbino), A. Barugola, and J.C. Cathala (Marseille) the results of which gave rise to a book [16]. This collaboration has also included H. Kawakami, and Y . Kevrekidis (University of Princeton) who organized the first meeting devoted to noninvertible maps and applications ("Noninvertible Dynamical Systems: Theory, Computation, Applications Workshop," Minneapolis, 14-18 Mars 1995) with McGehee. Maps with canceling denominators constitute a favored subject of researches with L. Gardini and G.I. Bischi since 1994. This choice is due to the richness of new singularites and bifurcations generated by such models often met in economical processes.

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A point is worth to be mentioned about the local background of these researches. The complex dynamics studies of the Toulouse group has never occurred in a favorable environment. So my projects of developing this topic and applications were systematically refused during the "prehistori~"times of "chaos," with the argument "nobody is interested by such a matter," which I frequently heard, Disheartened, I left the "Laboratoire d' Automatique et Applications Spatiales" (CNRS) and its excellent conditions of equipment, associated with large financial means. I stayed two years in Brazil (19741976). hen I returned in Toulouse I. Gumowski obtained a professor position in Toulouse. Then our research took place in a new framework: "Faculty of Sciences of the University Paul Sabatier," and "Institut National des Sciences AppliquCes de Toulouse." Even if the grants were at a level of "vital minimum," a neutral (but far from enthusiastic) environment permitted a research liberty, and the results mentioned in the first paragraph of this conclusion. Such a resistance (it would be better to say reluctance) to non classical studies, and interpreta~ionof phenomena observed, coming from the direct work surroundings, is not exceptional. Y. Ueda in chapter 7 of [66] describes the numerous difficulties he met, and his discouragement periods, when he tried to convince of the interest of his interpretation of complex dynamic phenomena that he observed in electrical and electronic circuits. The natural milieu always resists what it feels new. Indeed an unusual idea is frequently rejected a priori by the majority. It is not seriously considered because the milieu confers a sacred nature to what is well established by force of habit. After having suffered from this situation, I consider now that it is a common law, which also confirms the truth of the evangelic assertion, "Nobody is prophet in his own ~ o ~ n t ~ . ' ' Consider the situation out of the above local aspect now. It is interesting to mention that the post-I976 developments of the Toulouse group have occurred in a new international scientific environment for researches in dynamics. Indeed nonljnear dynamics was not a favored choice for pure mathematicians in western countries before 1976. This in spite of the highly significant Smale's contribution of prime interest, leading in particular to the "horseshoe" map properties (1 963), and other rare interesting publications. The most part of publications out of SSSR and before 1977 concern authors coming from engineering, physics, and applied mathematics. Therefore the Toulouse group publications were not written according to the standards of what can be called an "abstract theory of nonlinear dynamics," implicitly defined by certain pure mathematicians after 1976. The purpose of the

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Toulouse group has been to facilitate the communication with researchers in applied sciences. Indeed the simplest possible termjnoIogy has been adopted, in order not to add artificial vocabulary-hurdles to the intrinsic difficulty of the subject. Ars-pro-artis generalizations have been deliberately sacrificed in order not to obscure the dominating internal mechanisms of dynamic processes and thus preserve a phenomenological transparency (cf. p. 1 of [42] and preface of [393). From this point of view the Toulouse group approach has been in agreement with Birkhoff (quoted by M. Morse in Bull. Amer. ~ a t hSoc., . May 1946,52(5), 1,357-391), when he considered that:

The systematic organization, or exposition, of a mathematical theory is always secondary in importance to its discovery, , . . some of the current mathematical theories being no more than relatively obvious ~ l a ~ o r a t i of o nconcrete ~ examples. Moreover a relatively large part of the results obtained at those "prehistoric" times of "chaotic dynamics" were essentially attained via a numerical way (with classical checks of precision), from programs guided by the fundamental considerations of the qualitative and analytical theories of nonlinear dynamics, In particular it is the case of the section 6 results. The resulting formulation of problems has therefore appeared primitive, even simplistic, the terminology old-fashioned, and the results non relevant for many abstractly inclined researchers. Some of them found occasions to feel irritated by such a "modus operand!," for example through rejection of manuscripts. More precisely, it is possible to say that onl linear Dynamics has been the action field of two sets of scientists since some 25 years (the "historic" times of chaotic dynamics). The first set is made up of pure mathematicians. The second one is formed with some applied mathematicians, and more essentially with scientists coming from Engineering, or Applied Sciences. If we except the Poincare and Birkhoff cases and very rare others, the second set was historically the first which considered such questions since the century beginning, in order to solve practical problems of oscillations in their technical fields 111 [2]. The researchers of this set implicitly adopt as their own this J. Fourier' statement (quoted in [78] page XXXV):

LWude approfondie de la nature est la source la plus fkconde des dkcouvertes mathkmatiques. Non seuiement, cette dtude, en offrant

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aux recherches un but de'termine', a l'avantage d'exclure les questions vagues et les calculs sans issue; elle est encore un moyen assurk de former I'Analyse elle-mime et d'en ddcouvrir les dle'ments qu'il importe le plus de connaitre et de conserver. Ces dle'ments fondamentaux sont ceux qui se reproduisent dans tous les esfets naturels.

Indeed this text says that the study of the Nature (that is the real world) is the most productive source of mathematical discoveries, providing the advantage of excluding vague problems and unwieldy calculations. It is also a means to isolate the most important aspects to be known in the Mathematical Analysis, these fundamental aspects being those which appear in all natural effects. The approaches of each of the two sets work with different standards, have different purposes, but each approach has its own merits and interest in its proper field. This depends on the study context. Nevertheless this situation is the source of high difficulties o f communication between the two sets. They lie in the fact that what is important for one set is not for the other, and inversely. For the first set only results in a sufficiently general abstract form are signijkant. They consider that analyses performed in a particular study, or without using their patterns, are without any relevance. Some members of the first set do not accept the disuse of their standards for papers writing, which increases the above difficulties of communication. For the second set, only papers written in the least abstract form, using the least sophisticated mathematical tools, can transmit an information permitting to solve well defined practical problems. This last set considers that the following "proposition": the more a result is formulated in an abstract form, the less it can be used for solving a practical problem, is a result of experience and dictated by the common sense. My contacts with many researchers of this second set have shown that some of them, as "impure1'mathematicians daring to put a foot in the reserved field of a noble caste, have felt a kind of contempt from certain (fortunately not all) pure mathematicians, considering that only their approach is significant and meaningful. This text gives me the opportunity to pay a tribute to all the researchers of the Toulouse group since the beginning of its activity. At the prehistoric times of chaotic dynamics when this denomination was not introduced (i.e. before 1976), they permitted me a better understanding of this matter. The quality of their results, obtained in the framework of a thesis preparation,

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contributed a great deal to the development of chaos studies in Toulouse. Unfortunately the scientific management was not favorable at that time to this type of research, regarded as having only an academic interest. So these promising young researchers, subject to the surrounding pressure after their thesis, were led to change their activity domain, this some years before the increasing popularity of what will be called "chaotic dynamics." This account gives me the opportunity to quote the Pal Fischer (now Professor in Guelph University, Canada) participation to the group studies at that time during two years. I am greatly indebted to Igor Gumowski, now retired of scientific activities since more 12 years, without whom the Toulouse group would not exist. I dedicate this text to this very high level multidisciplinary scientist, who has been a model for me. Unfortunately his modesty let his contribution be relatively unknown in the nonlinear dynamics world. It is in particular the case of his results on nonlinear ordinary differential equations with time lag, conservative and almost dynamic systems. I would like to express my gratitude to the eminent specialists of nonlinear dynamics, Professors Abraham and Ueda, thanks to whom the publication of the story of the first times of the Toulouse group has been possible. This text gives me the pleasant opportunity to mention the constant and precious encouragement I have received from Professor Chua, Editor of "International Journal of Bifurcation & Chaos." I am also grateful to Professors L. Escande, A. BlancLapierre, and R. Thom for having accepted to present a part of the Toulouse group publications in "Comptes Rendus de 1'AcadCmie des Sciences de Paris." When J. Descusse (Ecole Centrale de Nantes) occupied a responsible position in the "Ministbre de I'Education Nationale," he obtained a special grant to the Toulouse group, in a time of the lean kine for this group. This text gives me the opportunity to thank him. Remarks from J. Gallas (Federal University of Port0 Alegre) permitted an improvement of the text.

References [I] C. Mira, Some historical aspects of nonlinear dynamics: Possible trends for the future. Double publication: (1) Intern. Journal of Bijiurcation and Chaos, 7(9& 10) ( 1 997), 2 145-2 174; (2) The Journal of the Franklin Institute, 3348(5/6)(1997), 1075-1 113. [2] C. Mira, Chua's circuit and the qualitative theory of dynamical systems. Double publication: (1) Intern. Journal of Bijiurcation and Chaos, 7(9&10) (1997),

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I9 1 1- 1 9 16; (2) The Journal of the Franklin institute, 334B(5/6) ( I 997), 737744. (31 Ch. Hayashi, Nonlinear Oscillations in Physical Systems. McGraw-Hill, New York, 1964. [4] E.S. BoTko, The Academician A . A . Andronov' School (in Russian). Ed. Nauka, Moscow, 1983. [5] J. Lagasse and C. Mira, Etude non IinCaire de l'integrateur parallele a I'aide d'une approximation du premier ordre algdbrique, Comptes Rendus Acad. Sc. Paris, 251 (1960), 2314-2316. [6] I. Gumowski, J. Lagasse and C. Mira, Etude d'un intdgrateur parallde, L'Onde Elecfrique No. 412-413 (1961), 1-9. [7] I. Gumowski, Sur un effet non lineaire dam les amplificateurs & transistor avec rdaction, Comptes Rendus Acad. Sc. Paris, 249 (1 959), 25 14-25 17, and 250 (1960), 822-825. [8] I. Gumowski, J. Lagasse & Y.Sevely, Mise en &quationd'un amplificateur A transistor non IinCaire, Compfes Rendus Acad. Sc. Paris, 250 (1960), 19951998. f9] I. Gumowski, Sur te comportement d'un amplificateur B transistor non M a i r e au voisinage de sa limite de stabilite. Comptes Rendus Acad. Sc. Paris, 250 (I960), 3 142-3145. [lo] I. Gumowski, Calcul de la rCponse en frCquence d'un amplificateur ii transistor non lindaire au voisinage de sa limite de stabilitd, Comptes Rendus Acad. Sc. Paris, 250 (1960), 4322-4325. [I 11 I. Gumowski, Sur la rdponse transitoire des amplificateurs a transistor avec rkaction, Comptes Rendus Acad. Sc. Paris, 253 (1 96 l), 167 1-1 674. [I23 I. Gumowski, Calcul de la rdponse transitoire a une onde carde d'un ampli~cateur transistor non lineaire, C o ~ p ~ Rendus es Acad. Sc. Paris, 253 (1961), 2207-2210. [13] P. Monte], Leqons sur les ricurrences et leurs applications, Ed. GauthierVillars, Paris, 1957. [I41 G. Julia, Memoire sur I'itdration des fonctions rationnelles, J.Nath. Pures & Appl., 4(1), 76mesCrie (1918),47-245. 1151 P. Fatou, Mdmoire sur les Cquations fonctionnetles, Bull. SOC.Math. France, 47 (1919), 161-271; 48 (1920), 33-94 and 208-314. [I61 C. Mira, L. Gardini, A. Barugola and J.C. Cathala, Chaotic Dynamics in Twodimensional Noninvertible Maps. World Scientific, "Series A on Nonlinear Sciences" (Editor L, Chua), vol. 20, 1996. [17] A. Giraud, Application des recurrences a 1'Ctude des systemes de commande, Thdse de Docteur-Ingthieur, No. 205 (April 1969), FacultC des Sciences de Toulouse. [I81 C. Mira & F. Roubellat, Cas ob Ie domaine de stabilitd d'un ensemble limite attractif d'une recurrence du deuxieme ordre n'est pas s~mp~ement connexe, Comptes Rendus Acad. Sc. Paris, SCrie A , 268 ( 1969), 1657- 1660.

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[ 191 I. Gumowski and C. Mira, Dynamique Chaotique: Transition Ordre-Dksordre, Ed. C6paduds Toulouse, 1980. [20] F. Roubellat, Contribution d I'ktude des solutions des rkcurrences non liniaires et applications am syst2mes ri donnkes e'chantillonnkes. Thkse de Doctorat kssciences Physiques No. 364, FacultB des Sciences de I'Universitd de Toulouse, 1969. 1211 C. Mira, Etude de la frontikre de stablitd d'un point double stable d'une rdcurrence non lindaire autonome du deuxikme ordre. Proceedings of International Pulse Symposium, Budapest, 1968, D43-7/11, 1-28. [22] I. Gumowski and C. Mira, Sensitivity problems related to certain bifurcations in nonlinear recurrences relations, Automatica, 5 (1969), 303-3 17. [23] J. Bernussou Liu Hsu and C. Mira, Quelques exemples de solutions stochastiques bornees dans les rdcurrences autonomes du second ordre. Collected preprints of Colloque International du CNRS No. 229 "Transformations Ponctuelles et Applications," Toulouse, Sept. 1973. Proceedings Editions du CNRS Paris, 1976, 195-226. [24] I. Gumowski and C. Mira, "Point sequences generated by two-dimensional recurrences, Proceedings of Information Processing 74 (IFIP), Stockholm 1974. North-Holland Publishing Company, 1974,851-855. [25] C.P. Pulkin, Oscillating iterated sequences (in Russian). Dokl. Akad. Nauk SSSR, 76(6) (1950), 1129-1132. [26] Ph. Holmes & R.F. Williams, Knotted periodic orbits in suspensions of Smale's horseshoe: Torus knots and bifurcation sequences, Archive for Rational Mechanics and Analysis, 90(2) (1 985), 1 15-1 94. [27] A.R. Cigala, Sopra un criterio di instabilita, Annuli di matematica, Ser. 3, 11 (1905), 67-75. [28] C. Mira, Etude d'un premier cas d'exception pour une rdcurrence, ou transformation ponctuelle, du deuxieme ordre, Comptes Rendus Acad. Sc. Paris,. Sdrie A , 269 (1969), 1006-1009. [29] C. Mira, Etude d'un second cas d'exception pour une rbcurrence, ou transformation ponctuelle, du deuxikme ordre, Comptes Rendus Acad. Sc. Paris,. Sdrie A , 270 (1970), 332-335. [30] C. Mira, Sur les cas d'exception d'une rdcurrence, ou transformation ponctuelle, du deuxikme ordre, Comptes Rendus Acad. Sc. Paris,. Serie A , 2 70 (1 970), 466469. [31] Yu. I. Ne'imark, Method of maps in the nonlinear oscillations theory (in Russian), Inr. Cont on Nonlinear Oscillations (iCN0). Kiev, I96 1 . Proceedings Ukr. Acad. Nauk., 1963, vol. 2,285-298. [32] Yu. I. Neimark, Stability of a fixed point of a map for a critical case (in Russian), Radiofsika, 3(2) (1960), 342-343. [33] Yu. I. NeTmark, Method of Maps in rhe Nonlinear Oscillations Theory (in Russian. Ed. Nauka, Moscow, 1972.

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[34] L.O. Barsuk, N.M. Belosludstiev, Yu. I. Ne'imark and N.M. Salganskaja, Stability of a fixed point in a critical case: Bifurcations (in Russian), Radiojsika, Il(l1) (1968), 1632-1641. [35] J.P. Babary and C. Mira, Sur un cas critique pour une recurrence autonome du deuxikme ordre, Comptes Rendus Acad. Sc. Paris,. Sdrie A , 268 (1969), 129132. [36] C. Mira, Traversee d'un cas critique pour une recurrence autonome du deuxikme ordre, sous I'effet d'une variation de paramktre, Comptes Rendus Acad. Sc. Paris, Serie A , 268 ( I 969), 62 1-624. [37] I. Gumowski and C. Mira, Bifurcation pour une recurrence du deuxibme ordre par traversee d'un cas critique avec deux multiplicateurs complexes conjuguds, Cornptes Rendus Acad. Sc. Paris, SBrie A , 278 (1974), 1591-1594. [38] C. Mira, Cas critique d'une recurrence, ou transformation ponctuelle, du quatrikme ordre avec multiplicateurs complexes, Comptes Rendus Acad. Sc. Paris, SBrie A , 272 (1971), 1727-1730. [39] C. Mira, Chaotic Dynamics: From the One-dimensional Endomorphism to the Two-dimensionalDiSfeomorphism. World Scientific, Singapore, 1987. 1401 C . Mira, Sur les courbes invariantes fermees des recurrences non linhaires voisines d'une recurrence lineaire conservative du deuxikme ordre. Collected Preprints o f Colloque International du CNRS No. 229 "Transformations Ponctuelles et Applications, I' Toulouse, Sept. 1973. Proceedings, Editions du CNRS Paris, 1976, 177-194. [41] I. Gumowski, Some properties of large amplitude solutions of conservative dynamic systems. Part 1 : "Quadratic and cubic non-linearities." Part 2: "Bounded nonlinearities." Rapport CERN/SI/Int. BW72-1, Genkve, 1972. [42] I. Gumowski and C. Mira, Recurrences and Discrete Dynamic Systems: An Introduction. 250 pages. Lecture notes in mathematics No. 809, Springer, 1980. [43] G.D. Birkhoff, Nouvelles recherches sur les systzmes dynamiques. Memoriae Pont. Acad. Sci. Novi Lyncaei, 1935, 53, vol. 1 , pp. 85-216. [44] 1. Gumowski and C. Mira, Boundaries of stochasticity domains in Hamiltonian systems. Proceedings 8th Con$ on High-Energy Accelerators. CERN Genkve, 1971,374-376. [45] I. Gumowski and C. Mira, Sur la distribution des cycles d' une rdcurrence, ou transformation ponctuelle, conservative du deuxikme ordre, Comptes Rendus Acad Sc. Paris,. SBrie A , 274 (1972), 1271-1274. [46] I. Gumowski and C. Mira, Sur la structure des lignes invariantes d' une recurrence, ou transformation ponctuelle, conservative du deuxikme ordre, Comptes Rendus Acad. Sc. Paris, SBrie A , 275 (1972), 869-872. [47] I. Gumowski and C. Mira, Stochastic solutions in a conservative dynamic system. Proceedings of 4th Int. Con$ on Nonlinear Oscillations (ICNO) Poznan 1972. PWN- Warzawa (1 973), 4 17-438. [48] I. Gumowski, Solution structure of a conservative second order recurrence with an unbounded nonlinearity. Colloque International CNRS, No. 229,

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"Transformations Ponctuelles et leurs Applications." Toulouse, 10- 14 Sept. 1973.Proceedings: Editions du CNRS, Paris 1976,79-92. [49]E. Zehnder, Homoclinic points near elliptic fixed points, Com. Pure and Appl. Math., 26 (1973),131-182. [50]P.J. Myrberg, Iteration von Quadratwurzeloperationen. I., Ann. Acad. Sci.. Fenn., Ser. A, 256(1958), 1-10. [51]P.J. Myrberg, Iteration von Quadratwurzeloperationen. 11. Ann. Acad. Sci.. Fenn., Ser. A, 268 (1959),1-10. [52]P.J. Myrberg, Sur I'itdration des polynemes reels quadratiques. J. Math. Pures Appl. 1J(9) (1 962), 339-351. [53]P.J. Myrberg, 1963. "Iteration von Quadratwurzeloperationen. 111. Ann. Acad. Sci.. Fenn., Ser. A, 336 (1963),1-10. [54]A.N.Sharkovskij, Coexistence of cycles of a continous map of a line into itself. Ukrain. Mat. Journal, 16 (1) (1964),61-71. [55] A.N. Sharkovskij, Problem of isomorphism of dynamical systems. Proceeding 5th. International Conference on Nonlinear Oscillations, Kiev, 1969,vol. 2, 541-544. [56] I. Gumowski and C.Mira, Sur les rbcurrences, ou transformations ponctuelles du premier ordre avec inverse non unique, Comptes Rendus Acad. Sc. Paris,. S&ie A , 280 (1975),905-908. [57]I. Gumowski and C. Mira, Accumulation de bifurcations dans une recurrence, Comptes Rendus Acad. Sc, Paris,. SBrie A, 281 (1975),45-48. [58]C.Mira, Accumulation de bifurcations et structures boites emboitdes dans les rdcurrences et transformations ponctuelles. 7th Int. Con$ on Nonlinear Oscillations (ICNO) Berlin, Sept. 1975.Proceedings: Akademic Verlag, Berlin, 1977. [59]J. Guckenheimer, The bifurcation of quadratic functions, New York Acad. Sc. 75(1) (1980),343-347. [60]C. Mira, Sur la notion de frontiere floue de stabilite, Proceedings of the third Brazilian Congress of Mechanical Engineering," Rio de Janeiro, Dec. 1975, D-4,905-918. [61] G. Grebogi, E. Kostelich, E. Ott and J.A. Yorke, Multi-dimensioned intertwined basin boundaries; basin stricture of the kicked double rotor, Plasma preprint UMLPF 86-033(Feb. 1986),Physics Publication, 86-174. [62]C. Mira, Sur la double interpretation, deterministe et statistique, de certaines bifurcations complexes, Comptes Rendus Acad. Sc. Paris, Sdrie A , 283 (1976), 91 1-914. [63]R.M. May, Simple mathematical models with complicated dynamics, Nature 261 (1976),459-470. [64]I. Gumowski, Some properties of a conservative second order recurrence with a bounded nonlinearity, Colloque International CNRS (No. 229), "Transformations Ponctuelles et leurs Applications," Toulouse, 10-14 Sept. 1973.Proceedings: Editions du CNRS, Paris 1976,143-153.

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[65] Ch. Hayashi, Selected Papers on Nonlinear Oscillations. Nippon Printing and Publishing Company, 1991. [66] Y. Ueda, The Road to Chaos. Aerial Press, Santa Cruz, Calif., 1992. [67] Ch. Hayashi, Y. Ueda and H. Kawakami, Periodic solutions of Dufing's equation with reference to doubly asymptotic solutions, Proceedings. of 5th Int. Conference on Non-Linear Oscillations, Kiev, 1969, 235-255. [68] R. Valette, Etude du comportement 6 fort signal de systdmes de commande comprenant un redresseur polyphase'. These de Docteur-Ingenieur, Facult6 des Sciences de Toulouse, No. 295, 1971. [69] R. Prajoux, Etude des ge'ne'rateurs utilisant un redresseur polyphase' et unpltre dynamique en tant que systBmes de commande. These de Docteur Bs Sciences Physiques, Universitb Paul Sabatier de Toulouse, No. 462, 1971. [70] R. Brown and L.O. Chua, Clarifying chaos 111. Int. J. Bifurcation & Chaos, 9 (5) (1999), 785-803. [71] C. Mira, Systdmes Asservis Non Line'aires. Ed. Hermes, Paris, 1990. [72] J. Bernussou, Contribution d! I'e'tude des solutions des re'currences non line'aires. Application ci I'ktude de certains syst2mes Li modulation. ThLe de Doctorat &-Sciences Physiques de I'Universitd Paul Sabatier de Toulouse, No. 596,1974. [73] Liu Hsu, Contribution ci I'e'tude des solutions des re'currences non linbaires. Application aux systBmes dynamiques conservatifs. These de Doctorat tsSciences Physiques de I'UniversitB Paul Sabatier de Toulouse, No. 597, 1974. [74] I. Gumowski & C. Mira, Bifurcation ddstabilisant une solution chaotique d'un endomorphisme du second ordre, Comptes Rendus Acad. Sc. Paris, Sdrie A, 286 (1 978), 427-43 I . [75] I. Gumowski and C. Mira, Solutions chaotiques bornees d'une recurrence ou transformation ponctuelle du second ordre h inverse non unique, Comptes Rendus Acad. Sc. Paris, Sdrie A, 285 (1977), 477-480. [76] I. Gumowski, Stochastic effects in longitudinal phase space. Paper contributed to the 9th International Conference on High Energy Particle Accelerators. Stanford, Calif., 2-7 May 1974. [77] Z. Grossman and I. Gumowski, Random number generation and complexity ine certain dynamic systems. Proceedings of Informatica 75, Bled, October 1974, Publication 3-4,4 pages. [78] H. Poincard, "Oeuvres" publides sous les auspices du Ministere de I'Instruction Publique par G. Darboux. Tome 11, Gauthier Villars et C", Paris 1916, "Eloge historique d'Henri Poincard." [79] T.Y. Li and J.A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992. [80] S. Nasri, Contribution d! I'analyse et a la synthBse des redresseurs bouclbs d thyristors. These de Docteur de 3""' Cycle, No. 35, INSA de Toulouse, 1985. [81] C. Mira, Expose d'Introduction. Colloque International du CNRS No. 229, "Transformations Ponctuelles el Applications," Toulouse, Sept. 1973. Proceedings: Editions du CNRS Paris, 1976, 19-27.

Toulouse Research Group

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[82] Georges David Birkhoff, Collected Mathematical Papers. Dover Publications, Inc., New York, 1968. [83] S. Lattes, "Sur les dquations fonctionnelles qui ddfinissent une courbe ou une surface invariante par une transformation." Ann. di Matematica 13(3) (1906), 1137. [84] S. Lattes, Sur la convergence des relations de recurrence surface invariante par une transformation, Comptes Rendus Acad. Sc. Paris, 150 ( 19lo), 1 106- 1 109. [85] S. Lattes, Sur les formes rdduites des transformations ponctuelles dans le domaine d'un point double, Bull. Soc. Math. France, 39 (191 I), 309-345. [86] S. Lattes, "Sur les suites rdcurrentes non lindaires et sur les fonctions gdndratrices de ces suites." Annales Facultk des Sciences de Toulouse 3 (3), (1912), 73-124. [87] B.B. Mandelbrot, Fractals: Form, Chance and Dimension. Freeman, San Francisco, 1977. [88] C. Mira, Ddtermination pratique du domaine de stabilitd d'un point d'dquilibre

d'une recurrence non lindaire du deuxieme ordre I variables rdelles, Comptes Rendus Acad. Sc. Paris, SBrie A, groupe 2,261 (19--), 5314-5317.

9.

The Turbulence Paper of D. Ruelle and F. Takens Floris Takens Gronningen University

In the Spring of 1969 I finished my Ph.D. at the University of Amsterdam under the direction of Professor N. H. Kuiper. This thesis was in differential topology, and was published in Inv. Math. 6 (1968), pp. 197-244. In that academic year I had started to read the literature in dynamical systems. In the academic year 1969-1970 I was at IHES (with a grant from the Netherlands Organisation for Pure Research), where other visitors were Smale, Pugh, Shub, Zeeman, etc., and followed Shub's course which was later published in Asterisque. At one point Thom ask me to speak in his seminar on the Hopf bifurcation. At that time this material was not so well known, at least I did not know what a Hopf bifurcation was. Thom gave me a copy of the original paper (published by the academy in Leipzig in 1942, hence not very widely available) and asked me to give a more geometric treatment. So it happened. After the lecture Ruelle informed me that Hopf considered this bifurcation, and repetitions of it leading to quasi-periodic motions with many frequencies, in order to explain turbulence (with sufficiently many frequencies one can obtain a dynamics which is arbitrarily erratic). By that time I was already sufficiently introduced to the dynamical systems a la Smale, and its ideology to give as a first reaction that quasi-periodic motion is not generic, but that on the other hand there are many examples of hyperbolic attractors, which are persistent, and which can explain a dynamic which is also erratic. This was essentially how the paper started. Though it was not much more than the remark that chaotic motion (as we now call it) is possible also in the case of a small number of degrees of freedom, and that this might be 199

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Chaos Avant-Garde: Memories ofthe Early Days of Chaos Theory

applicable to turbulence. Ruelle convinced me that it should be published since in the physical community the Landau-Hopf theory was still generally believed. The final publication was rejected by the Arch. Rational Mech. Then Ruelle used his position as editor of Comm. Math. Phys. to publish it in that journal [Vol. 20, 197 1 , pp. 167- 192 (received Nov. 2, 197 1 )]. After publication we got a reaction from Arnold in which he complained that we duplicated much of the work of the Russians. Also the nonpersistence of quasi-periodic motion was not as convincing as it first seemed to us: these motions are non persistent in the topological sence, but still they survive perturbations "with positive probability." The physical community was rather sceptical about the whole idea; I think it was mainly Ruelle who tried to convince them. The main experiments which convinced the physicists of the relevance of these ideas were, I think, the Henon attractor as a computer experiment, Swinney's laserDoppler experiment on the Couette-Taylor system, and the detection of chaos in chemical reactors.

10. Exploring Chaos on an Interval T. Y.Li Michigan State University

James A. Yorke University of M ~ ~ ~ a n d In the late 1970s, many scientists learned of chaos. They found through computer studies that their favorite systems of equations behaved chaotically for some range of parameters. Until then they thought the simple deterministic physical systems they studied exhibited only two kinds of behavior (if we ignore transient motions), namely, steady state and quasiperiodic (or periodic). This third type was characterized by "sensitivity to initial data," a superb phrase that is apparently due to David Ruelle. Much of our work can be characterized as trying to discover the pure mathematics the scientist needs; we have often aimed at reformulating ideas so they would be of interest and value to scientists. When chaos theory seemed to suddenly spring to life full grown, scientists were unaware that the ideas had actually been developed by mathematicians over a period of many decades. Ideas established before 1900 include sensitivity to initial data (Maxwell's investigation of gas laws), Cantor sets, concepts of countable and uncountable infinities (Cantor), and the dynamic co~plexityof in~nitelymany coexisting periodic orbits of different periods (Poincard, who also invented stable and unstable manifolds and investigated period doubling bifurcations). By 1920 the following ideas were growing: fractal dimensions (beginning with Caratheodory and H a u sdorf~, symbolic dynamics (Hadamard), and there were topological investigations of connectedness, compactness, and continuity, and numerical methods including efficient differential equation solvers (Runge-Kutta). 20 1

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The twenties and thirties brought topologically strange invariant attractors of Birkhoff, embedding (Whitney), and ergodicity theorems (Birkhoff and von Neumann-based on the new measure theory of Lebesgue). The generic dynamics theory of Smale and his collaborators had been prepared by Sard in the 1940s. The ideas of the Kolmogorov-AmoldMoser theory had been prepared by Siegel's studies of "Siege1 disks," also in the '40s. In the 1940s and '50s Cartwright and Littlewood explored the complicated topology of dynamics, topology that had been developed over a long period, including PoincarC's ideas that eventually became homology theory, including the Lefschetz Fixed Point theorem that is particularly useful for nonlinear dynamics. The more one tries to touch upon the basic roots, the more one realizes how hopeless the task is, because nonlinear dynamics is also a description of how the world works, and almost every mathematical tool is at least occasionally needed, from probability theory and numerical analysis, to ordinary and partial differential equations. For each name mentioned above, several are unfairly omitted. Our goal here is to describe the environment and personal interactions that drew us into this realm. We have often thought of our work as attempts to make connections between the mathematical theories and the needs of the scientists, showing the scientists how useful the mathematics can be at identifying concepts as well as at proving theorems. One of our efforts began with a colleague, a fluid dynamicist Alan Faller, who was fascinated by several of Edward Lorenz's papers [2-51, at a time when few if any mathematical dynamicists knew of his work. In the early ' ~ O Sthe , faculty of the Institute of Fluid Dynamics and Applied Mathematics (IFDAM) at the University of Maryland, currently known as the Institute for Physical Science and Technology (IPST), researched diverse areas such as plasma physics, applied mathematics, meteorology and others. One of its faculty members, Prof. Alan Faller, an experimental fluid dynamicist and oceanographer, brought to our attention the four papers of Lorenz on fluid dynamics and weather prediction models. Faller had made many copies (this was before Xerox machines) and gave them to anyone he could interest. He confessed that those works were somewhat too theoretical for generic practical meteorologists even though they were all published in meteorology journals. They were beautifully written and Lorenz clearly described the structure of what is now called the "Lorenz Attractor."

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Exploring Chaos on an Interval

What attracted us the most was the highly irregular asymptotic behavior of the trajectories of the set of equations: x' = o ( y - x )

y' = px - y - xz

(L)

for certain parameter values 0, p, and b . According to the PoincarCBendixon theory, the limit sets of bounded trajectories of two-dimensional smooth differential equations in the phase space are quite regular. While no one had been able to extend the theory in a useful way to higher dimensional equations, most mathematicians and scientists tended to believe that the asymptotic trajectories in higher dimensions were, more or less, quite regular. Lorenz discovered a useful way of thinking of the solution of his system of equations. As he solved his system numerically he stored the local maxima of z(t) as it oscillated. When he plotted a huge number of pairs of consecutive maxima they seemed to lie along a curve that was smooth except at one point; that is, they seemed to satisfy a relationship Zn+l = F(ZJ The dynamics of this process seemed to us much easier to explain than the original system. The function was quite similar in spirit to the tent-map 1 for0 Ix I ax 2 < ( x ) := 1 a(1-x) or-IxIl f 2 where 1 < a < 2. One-dimensional Topology. We knew from Smale's horseshoe, the stretching property of a map can have infinitely many periodic points of different periods. Let f be a continuous map f : R' + R' and write f * = f(f)and f 3= f(f'). Iffhas the property that for some point xo

i

204

Chaos ~ v a n ~ - G ~Memories r ~ e : of the Ear& Days of Chaos Theory

f3(xO2 ) xo < f ( x o )< f 2 ( x o )

(wf2(Xo)

c f ( x o )< xo 5 f3((no))>

t h e n ~ h a speriodic points of all periods when it is iterated. And, it happens that a continuous one-dimensional map having a periodic point of period three constitutes a speciat case of these properties. When f k ( a )= cx, white f ( a )# a for 1 -< j c k, then a is called a periodic point of period k. To explain Lorenz's results we discovered a theorem in early 1973 on sensitivity to initial data. For f :R1 -+ R' ,write f " = f ( f " - ' ) . Theorem 1 ("Period Three Implies Chaos"): If a continuous map f :J 3 J has a point of period three, then (a) for each positive integer n, there exists a point of period n; (b) there exists an uncountable set S c J that is "scrambled," that is for any two points x ~ t iny S, we have lim in++" ( x ) - fa(yj( = o "4-

-

lim suplf"(x) f"(y)J> o

"4-

We completed the proof in April 1973, and submitted the first version of the paper [S] to MAA (Mathematic Association of America) Monthly. The first four References of that draft were [2-51. Shortly after submission, we learned that the editorial opinion on the pager was not too favorable. The major criticism was the strong research tendency of the article, which might not be suitable for the readership of the magazine. However, if we insisted on resubmittjng the paper back to the M o ~ t we ~ must ~ ~ revise , the paper massively to make the paper more readable. This was a result we felt a lot of people should know about, so we persisted. Our target for our simple ideas was the large readership of the magazine, so we never considered submitting it elsewhere. Nevertheless, occupied by other research projects, we put off the revisions for almost a year, Yorke had once submitted a paper to Science and got referee reports including assertions like, "This paper contains an identity. Identities are always true, so nothing can be learned from them." So going to an even larger readership journal seemed out of the question. Each academic year, the Math ~epartmentof the Univers~tyof ~ a ~ l a n d routinely organized a special year program. The topic of the program for the academic year of 1973-4 was mathematical biology. Robert May, who was trained as a physicist but had become a professor of biology at Princeton ~ n ~ v e r s i t was y , one of the dist~nguishedinvited speakers of the program.

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Exploring Chaos on an Interval

During his visit in the first week of May, 1974, Professor May delivered five lectures, one per day. The subject of his fifth talk was the Logistic Model, commonly known as the iterations of the quadratic map.

T, - ( x ) = ux(1-

x),

x

E [0,1], a E [0,4].

He described his discovery, the now well-known doubling period bijkcations of the map as a varies. He did not know what happened in the chaotic region, the region beyond the main cascade of doubling period bifurcations, and we had not known of period doubling! Since he addressed the topic only at the end of his visit, we only had a chance to discuss the topic with him in the airport that afternoon. We revised our paper the very next week. It was accepted by the Monthly in August, 1974. In the summer of 1974, Professor R. May was invited to give talks by many institutions in different countries in Europe. He adopted our use of "chaos" as a mathematical term, and Period three implies chaos therefore began to attract considerable worldwide attention by his strong advocacy in talks and papers [5,6]. A byproduct was the emerging popularity of the strange attractor of Lorenz. As use of the word "chaos" spread, it became a word people loved to hate: they didn't have a better word but didn't like chaos. We were quite surprised when use of the word spread to the Soviet Union, where there already was an established term: "stochasticity." Worldwide attention led to our awareness of the existence of the Sharkovskii ordering:

here, if "p< q" then the existence of a period p point implies the existence of a period q point. Actually, when we were preparing the original version of the paper, a natural question was automatically raised: what about period five? The counter-example we constructed that period five does not imply period three essentially terminated our desire to pursue it any further. In the fall of 1975, Yorke attended a large conference on nonlinear oscillations in East Berlin, meeting Sharkovskii on a tour boat on the river Elbe. Sharkovskii did not speak English, but A. Lasota and C. Mira translated. He said he had a better theorem. Despite the translator's efforts,

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Chaos Avant-Garde: Memories of the Early Days of Chaos Theory

he wouldn't say what it was, but promised to mail the paper to us. Mailing scientific papers out of the Soviet Union was discouraged and required permissions and explanations. We were quite amazed by the Shar~ovskii Theorem in 177 when we received it from him in 1976, and disappointed that his result was so much sharper than part (a) of our theorem. Of course it turned out that May was not the first discoverer of cascades but his papers were so well written that the idea of period doubling (see [9]>, became so well known that he was the last discoverer of period doubling cascades!!

Chaos Implies Statistic~lRegularity. Yorke's first chaos paper was actually extremely different in approach. The distinguished Polish mathematician Andrezej (Andy) Lasota was a frequent visitor to IFDAM. On one visit he announced that he had an approach to understand~ngthe theory of certain one-dimensional maps like the tent map He had a beautiful technique but could only apply it in very special cases, Yorke, who knew nothing of ergodic theory, was able to push the method through in considerable generality, yielding a result that we might calf "chaos implies statistical regularity," [lo). The assumption used in [lo] is C: For some interval J , assume F : J 3 J is piecewise cont~nuous (continuous except at a finite n u ~ b e rof points), and piecewise C' and C2, and for some constant B > 1

c.

-(dj

2B

where

dF

- is defined. dx

This is a chaos assumption because it guarantees that every trajectory is unstable. ~ e we felt papers [I] and [lo] were so differe~tthat we It is r e ~ a r k a b that did not have them refer to each other. One was based on one-dimensional topology [I] and the other [lo] was based on linear operators in a Banach space, but both applied to at least some of the tent maps 5 . This paper established that hypothesis C implied that there i s an absolutely continuous invariant density g for F, a density that would be called today a Sinai-RuelleBowen density or measure, even though it preceded the Bowen-Ruelle paper [ 1 I] by 2 years. Sinai had introduced this topic (See for example his survey [IS]) by studying cases where the attractor is the entire space. Our goal was to be able to "choose a point x at random'' and describe how its trajectory points were distributed on J. "Choosing x at random'I

207

Exploring Chaos on an Interval

might mean choosing it from a uniform distribution or more generally from a probability distribution with some density g, so that the probability of choosing a point in an interval (x, x + Ax) is approximately &)Ax for small Ax. The goal was to tell people what they would typically see in a trajectory. Hence, it was important to study initial distributions that were absolutely continuous with respect to Lebesgue measure. Since the maps were piecewise expanding, there was a guarantee of sensitivity to initial data. There was no assumption that the map takes an interval onto itself nor of continuity. Hence sensitivity to initial data was the key hypothesis. Our approach was to choose an initial probability density g and see how it evolved in time. So if a point x is distributed with density g, we can ask how F(x) is distributed. The point F(x) will have a density g&) which we write as P g where P is called the Frobenius-Perron operator. The second iterate 2 ( x ) would have a distribution P’g. If you compute the average up to time n of these densities, the resulting densities converge

and the resulting limit density h*, is invariant. That is Ph* = h*. We can summarize [lo] by the following statement in which h* is constructed as shown above. Theorem 2: Condition C implies F has an invariant density h*. Lasota, who was on an extended visit to the United States visiting the University of Maryland, had seen the entire theoretical structure of this problem. He realized that densities of bounded variation were the key. What makes it possible to prove a theorem is the fact that if one starts with g which has bounded variation, then P g does also, and there is a uniform bound B > 0 on the variation of all the densities { P g } .Hence the averages

1 -cn

J=l

P’g

will also have the same bound on their variations, and the set of densities GB with variation bounded by B is compact in the metric d ( g , , g , ) = j l g l ( x )- g,(x)ldx. Hence some limit point will exist. Li wanted to make this approach more explicit to the scientist. He asked how you would find h*. He felt it could be computed. So he began to write a Ph.D. thesis on this topic and discovered how to compute it [ 121, and in the

208

Chaos Avant-Garde: ~ e ~ o r iofthe e s Eurt'y Days of Chaos Theory

process he solved a long-standing problem of Ulam. We followed up with [13,14). It was a very exciting time for us since we were learning a great deal of established mathematics as we worked, from topology to operator theory to numerical methods!

References [ l ] T.Y. t i and J.A. Yorke, Period three implies chaos, Amer. Math. ~ o n j 82~ ~ , (1975), 985-992. [Z]E.N, Lorenz, The problem of deducing the climate from the governing equations, Teiius, 16 (I 964), 1-1 1. [3] E.N. Lorenz, D e t e r ~ ~ ~ inonper~od~c st~c flow, J . ~ ~ ~ o s pSci., ~ e 20 r ~(1963), c 130-141. [4]E.N. Lorenz, The mechanics of vacillation, J. Atmospheric Sci., 20 (1963), 448464. [5f E.N. Lorenz, The predictability of ~ydrodynamicflow, Trans. N. Y. Acad Sci., Ser. I], 25 (1963), 409-432. 161 R.M. May, Simple mathematical models with very complicated dynamics, Nahzre, 261 (1976), 459-467. [7] R.M.May, Biological populations with non overlapping generations, stable points, and chaos, Science, 186 (1974), 645-647. [ S ] A.N. Sarkovskii, Coexistence of cycles of a continuous map of a line into itself, ~ k r a ~Math. ~ j aJ.,~16(1964), 61-71. [9] P.J. Myrberg, Sur i'iteration des poIyn8mes rCels quadratiques, J, Math. Pures Appl., 41 (1962), 339-351. [ 101 A. Lasota, and J.A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488. [ 1 I] R. Bowen and D, Ruelle, The Ergodic Theory of Axiom-A Flows, Inv. Math, 29 (1979, I8 1-202. [I21 T.Y.Li, Finite app~oximati~n for the Frobenius-Perron operator: A solution to Ulam's conjecture,J. Appr0.x. Theory, I 7 (1 976), 177-186. [I31 T.Y. Li and J.A. Yorke, Ergodic transformations from an interval into itself, Trans. Amer. Math. Soc., 235 (1978), 183-192, [I41 T.Y. t i and J.A. Yorke, Ergodic maps on [&I] and nonlinear pseudo-random number generators, Nonlinear Anal., 2 (1 978), 473-48 1. [IS] Sinai, Ya. Gibbs measure in ergodic theory, Russ. Math. Surveys, 166 (1972). 2 1-69.

11.

Chaos, Hyperchaos and the Double-Perspective Otto E. Ri)ssler Tubingen University

Chaos and Hyperchaos "Unknotting" is a favorite activity of the mind. When I told Art Winfree of my aim to design a "knotted'l limit cycle in 3-space, he immediately sent me his precious collection of reprints and preprints on chaos, including the by then still unpublished "period 3 implies chaos" founding paper [ 11. Chaos is a locally combable tangle of knots in 3-space, if you so wish [2]. Hairs and noodles and spaghettis and dough and taffy form an irresistible, disentanglable mess. The world of causality is thereby caricatured and, paradoxically, faithfully represented. An expanding spiral produces chaos if re-injected [3] between two 1dimensional crossing planes: Fig. 1. And an expanding screw produces hyperchaos [4] if reinjected between two 2-dimensional crossing planes: Fig. 2. The principle carries through to arbitrary dimensions. Always, but a single nonlinearity (a quadratic term, say) is needed in a single variable, all other variables being linear, to obtain maximal hyperchaos, as Gerold Baier and Sven Sahle recently found in an explicit example [5]. Waves in a linear chain with a single reflecting variable of nonlinear type on the one end, generate a beautiful carpet [5]. At the same time, they make the connection to wavy media with obstacles [ 6 ] . And they contain an uncountable number of periodic solutions of arbitrarily complex shapes in nspace. Chaos control (by Pyragas' feedback method [7]) thus ought to allow

209

210

Chaos Avant-Garde: Memories ofthe Early Days of Chaos Theory

I

Figure 1

Figure 2

Chaos, Hyperchaos and the Double-Perspective

21 1

one to extract arbitrary messages (like a speech of Einstein's) from an initial segment, fed into such a system in a custom-made computer. Is this science or fiction? Most recently, vowels were discovered to represent beautiful tangles embedded in low-dimensional chaotic attractors, by Sebastian Fischer. The hoarse voice and the stuttering car engine [8] and baby cries (H. Herzel) thus are no longer alone. The world of sounds becomes disentanglable.

Superfat Attractors Let me start all over anew by looking at Georg Cantor's basic map: Fig. 3. There is a full interval on the left, and there is an empty interval (except for a Cantor set) at the bottom. In between, every submap contains the same structure as the original map (insets). What is so beautiful about this? The sunshine reveals it.

---+

Figure 3

Days of Chaos Theory Chaos Avunr-Garde: ~ e ~ o ~of ithe e Early s

212

If we elongate everything toward the back, we have the "clotheslines phenomenon," as Jack Hudson calls it: Fig. 4.

=3

Figure 4

A set of one-dimensional clotheslines is suspended in space in such a manner that, when they are viewed from the top, all light can pass through. But if the same clotheslines are viewed from the side, their shadows make for a continuous dark surface. This Cantorian phenomenon is robust and occurs in many chaotic attractors, in arbitrary dimensions [9]. Nowhere differentiable attractors (on a Cantor set) are implicit and the paradoxical Kaplan-Yorke p h e n o m ~ ~ ~ n [ I 01 is thereby explained: "Hyperfat attractors" with topological dimensionality unity but fractal dimensionality maximal (potentially filling almost the whole phase space) become a matter of course.

Chaos, Hyperchaos and the Double-Perspective

213

A special case are the "flare attractors," in which the autocatalytic growth of one variable is multiplicatively dependent on a chaotic input according to the Milnor principle [ 113. Economic firms (like in the king of Wienerwald's story "From waiter to millionaire and back" [12]) fit in here [131. The stock market comes to mind as a testing ground. Another fractal is that of the galaxies in the sky, as Cantor's timedisplaced younger friend Benoit Mandelbrot first saw [ 141. Strangely, in such a fractal, gravity is distributed with a bias: Relative to an average point, "most" others have a lower gravitational potential (are redshifted), and apparently the more so the farther they are. Thus, "bang-like events'' may stop to exist for a universe that is looked at with a loving (transfinitely exact) eye.

Owls and Cats We at last come to the two-perspectives problem raised by chaos. It was invented in our time by Ed Lorenz, the second butterfly man of the 20th century after Erwin Schrodinger (as we shall see). His sharp eye saw what no one else had seen so clearly: the most important message of chaos for everybody to take home [IS]. Of course, Lorenz had his predecessors, like Maxwell and PoincarC and Julia and Birkhoff and Hopf and Cartwright and Levinson and Ueda and Arnold and Smale and Abraham (cf. [16, 171). I remember being told by a mathematical friend in the Fall of 1975 to stop running my homespun ODES on a computer; after, he had found out that their solutions depended on the stepsize chosen for the very first step, all later steps of the Runge-Kutta algorithm having the same step size. This was against the very grain of science and reproducibility: "pathologies should not be looked at." So what is the meaning of Lorenz's butterfly effect, which to the lay public and the philosophers alike has become the epitome of chaos? Is it a misunderstanding? Ed Lorenz is like an owl. He can turn his head 180 degrees (or owls can, as he pointed out to my wife and me when he acted as our tour guide through the Boston Science Museum in late 1976). No one else can. With the one position of the head, you look at the universe in which the butterfly did not flap its wings; with the other position of the neck, you look at that other

214

Chaos Avant-Garde: Memories of the Early Days of Chaos Theoty

universe in which it did. Both universes diverge exponentially from each other. The late Joe Ford and I independently invented the "Schradinger butterfly'' in homage to Ed, cf. [18]. In that case, the two universes do not diverge (when the butterfly's wings are controlled by a quantum decision between two equiprobable outcomes of a superposition [ 191). Everybody sees that something goes awfully awry here. Quantum and chaos become irretrievably entangled, one feels. The owl pierces deeper with his glance. The self-similar fractal never stops, said Anaxagoras [20] in ancient Greece (if one reads him after having met Mandelbrot). Is the world infinitely exact or transfinitely exact? Anaxagoras, who had invented the technical term ''recurrence'' (perichoresis [21]), sided with the transfinite camp which came into existence only 2 millennia later with the notion of the long line, cf. [22]. Most recently, Dean Driebe made the distinction clear: The description of the world is qualitatively different dependent on whether you choose the state point infinitely small or transfinitely small [23]. Only in the former case can you replace everything by a time-symmetry breaking distribution, Is chaos thus metaphysics? It certainly is not. It is the most Platonically idealistic way of looking at the world, the epitome of rationalism. It is Newton's exactness shining back and forth across the centuries and millennia. But who believes in ODES, in the song of the spheres, today? Some cohorts of people can be mentioned besides the age-wise and the artists. A 2year-old once asked me to remove the milk again which I had inadvertently poured into his rosehip tea, and declared in perfect self-confidence that I was "thinking wrongly" when I told him that I was unable to do so. Everybody has grown accustomed to this impotency in our own world. The demons of Laplace and Maxwell teach us otherwise: They represent Newtonian-chaotic transfinitely-exact myths. Are they perhaps to be revived from the dustbin?

The Interface We come to a synthesis. Synthesis with the mind is, after all, what chaos is all about. Konton (chaos) means, you need not even open your eyes to see exactly, as Yoshi Kuramoto told me in 1978.

Chaos, Hypedtaos and the Double-Perspective

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The piercing glance of the owl returns. There is an outside and an inside perspective. Lorenz has invented a demon, too, the demons of Laplace and Maxwell are not alone. Boscovich [24] had struggled with the same problem in the 18th century, and so did Bohr and Einstein. The potentially best-~tting word is "micro relativity." The world is "microscopically relative" to minute changes, as the butterfly effect shows. The sensitive dependence of the world is even more far-reaching, however. The "effective forcing function" that impinges on you and me is dependent on our own internal micro dynamics [25]. This time, the alternatives are not generated by an external butterfly but, so to speak, by the butterfly within. If this is the ultimate message of chaos, we are all "bubble boys" and "bubble girls," as Peter Weibel used to say. The observer is co-responsible for the world [26]. Microrelativity tries to describe the interface. Blood-stained cut or tender interface? The "red line"-the fractal attractor in a walking-stick map [27]--was perhaps implicitly referred to already in the ~ r i m mbrothers' tales, as a fine scar remaining after a head had been cut off and healed on in the wrong position (much like in that of an owl's). This squinting glance which no one else can follow was called "relativity" by the young Einstein 2281. Microrelativity-the logical extension-was discovered two years after Einstein's passing away by equally young Hugh Everett 111 in I957 [29]. His famous "many worlds theory" is indeed no more a many worlds theory than Minkowski's theory (with its many cuts) is. For "world" and 'kut" is the same thing. What we suddenly feel is a tinge of that ominous feeling that is said to occur shortly before the consistency of the world breaks down: Could we possibly have such a trans~nitelyexact red line cutting right through our neck and brain? Forgive me for positing that this ultimate fear is the penultimate teaching of chaos. The world always tries to tell us to either wake up or emerge smiling serenely. Only we do not listen to it. Chaos is the koan. Ueda's attrator looks like a smile. If the present memoir stopped here, it could perhaps still have passed as a philosophical essay. But we live between morning and evening in a coldblooded world that asks for "tangible" implications-applications, that is. Is there such a deeper meaning to the beauty of the chaotic metaphor as everybody knows it to date? Let me close on a few words on Everett in this context. He not only discovered the cut that presupposes the accuracy of chaos in a quantum context, but also the "world bomb." That would be an application of sorts:

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the ultimate icecream bomb. A technology of an unprecedented power, but so at a high cost. Suppose it were true that there is an invisible red interface running right through the universe and your head at this moment, the socalled now-world generator. How could one prove that this is not metaphysics? " D i b solo basta," said the rabbi's favorite granddaughter Santa Teresa in medieval Spain, with a similar reasoning in her mind. The technology at stake is not levitation. Everett disappeared not in a cloister but in the Pentagon, for the remaining 26 years of his short life. He had only published a single paper in the open literature (titled "relative state formulation" as we saw). If a "world change technology" were implicit in chaos theory, then both relativity and quantum mechanics would have to become explicable as "interface realities" as a first step so to speak. The late Kazuhisa Tomita [30], as well as Joe Ford [31], thought along similar lines. Is the objective world as tight to our skins as the Now is? The argument of "counterfactuality'' [32] strongly speaks against the provability of any such dream. Note that the butterfly effect by definition is counterfactual. Nevertheless, some recent developments in physics (like the "relativistic Bell experiment" [33] and the "gravitation-induced size increase" in proportion to the local clock slow-down [34]) could be taken to speak in favor of this idea. The observer-centered light bubble which unites relativity and quantum mechanics would then no longer be only a prison to fear: Any better-understood bondage provides the seeds for its own undoing. A "technology of liberation'' would join the theology of liberation. Thus, ''chaos'' oscillates today between discouragement and encouragement in much the same manner as it did through past millennia. Who dares say what the next turn of the mixture will bring to the surface?

Acknowledgments Thank you, Ralph and Yoshisuke. I would also like to thank Ilya Prigogine, Roland Wais, Gonsalv Mainberger, John Nicolis, Okan and Demet Gurel, Hans Degn, Evgeny Selkov, Rutherford Aris, John Ross, Bob May, John Tyson, Peter Ortoleva, Frank Hoppensteadt, Fred Marotto, Phil Holmes, Norman Packard, Rob Shaw, Doyne Farmer, Jim Crutchfield, Gottfried Mayer-Kress, Paul Rapp, Colin Sparrow, Bruce Clarke, ,Bill Smith, Ray Kapral, Jim Gleick, Kuni Kaneko, Bruce Stewart, John Casti, Hans Primas,

Erik Mosekilde, Vladimir Gontar, Mohamed El Naschie, Rent5 Thorn, Barkley Rosser, Keisuke Ito, the late Eberhard Hopf, GySrgy Targonski and Masaya Yamaguti for discussions. For J.O.R.

References [l] T.Y. Li and J.A. Yorke, Period three implies chaos, Amer. ~ a t h~. o n t h l y82 , (1975), 985-992. [2] I. Gumowski and C. Mira, Dynamique Chaotique. Cepadues Editions, Toulouse, 1980, [3] 0.E. RGssler, Chaotic behavior in simple reaction systems, 2. ~aturforsch.,3 f a (1976), 259-264. [4] O.E.R6ssler, An equation for hyperchaos, Phys. Lett. A, 72 (1979), 155-157. [5] G. Baier and S. Sahle, Design of hyperchaotic flows, Phys. Rev. E, 51 (1 99.9, 2712-2714. [6] O.E.RClssler, J.J. Kozak and D. Hoffmann, Linear chaos (abstract), Physics I il) (1984), 416. [7] Pyragas, K . Phys. Lett. A, I70 (1992), 42 1. [S] O.E. ROssler, R. R6ssler and T. Wiehr, Chaos. Anafog-computer-generated

movie with a stereo display to accompany the real-time "sound of chaos" (1 976). 191 O.E. Rassler, C. Knudsen, J.L. Hudson, and I. Tsuda, Nowhere-differentiable attractors, Inf.J. Infell.Systems, I0 (1995), 15-23,

[I 01 J.L. Kaplan and J.A. Yorke, Chaotic behavior of multidimensional difference equations. In: Functional Diferential Equations and Approximation of Fixed Poinrs (H.4. Peitgen and H.O. Walther, eds.); Berlin: ~pringerVerlag Lecture Notes in Mathematics, 730 ( 1979), 204-227. Ell] O.E. RUssler and G.C.Hartmann, Attractors with flares, Fractals, 3 (1995), 285-296. (121 F. Jahn, A Life for Wienerwald: From Waiter to Millionaire and Back (in German).Wienerwald Publications, Munich, 1993. [131 G.C. Hartmann and O.E.RUssler, Coupled flare attractors, Discrete Dynamics in Nature dG Society, 2 (1998), 153-159, [14] B.B. Mandelbrot, The Fractal Geometry of Nature. W.H. Freeman, San Francisco, 1982. [IS] E.N. Lorenz, Does the flap of a butterfly's wings in Brazil set off a tornado in Texas? (1972). Reprinted in: E.N.Lorenz, The Essence of Chaos. University of Washington Press, Seattle, 1993. [I61 R. Abraham and J.E. Marsden, Foundations of Mechanics, 2nd ed. London: Benjamin/Cummings, 1978; Abraham, R.H. and C.D. Shaw, Dynamics: The Geometry of Behav~or,Part 2: Chaotic B e h ~ ~Aerial ~ r . Press, Santa Cruz, Calif,, 1983.

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[I71 Y. Ueda, Some problems in the theory of nonlinear oscillations. Ph.D. Thesis, Kyoto University, 1965. Reprinted in: Y. Ueda, The Road to Chaos (R.H. Abraham and H.B. Stewart, eds.), pp. 1-71. Aerial Press, Santa Cruz, Calif., 1992. [IS] O.E. Rbssler, Chaos and endophysics (in German). In: Quanten, Chaos und Ddmonen (K. Mainzer and W. Schirmacher, eds.), pp. 220-235. BIWissenschafisverlag, Mannheim, 1994. [19] E. Schrodinger, The present situation in quantum mechanics: A translation of Schrodinger's 'cat paradox' paper, Proc. Amer. Phil. SOC.,124 (1980), 323-338; first published in German in 1935 in Naturwissenschajlen, 23. [20] O.E. Rossler, Anaxagoras' idea of the infinitely exact chaos. In: Chaos in Education, Teaching Nonlinear Phenomena, Vol. I1 (G. Marx, ed.), pp. 99-1 13. Veszprem, Hungary: National Center of Educational Technology Publications, 1987. For a new translation of Fragment # 12, see [21]. [21] O.E. Rossler, Endophysics: The World as an Interface. World Scientific, Singapore, 1998. [22] O.E. Rossler, Long-line attractors. In: Iteration Theory and Its Functional Equations (R. Liedl, L. Reich and G. Targonski, eds.). Berlin: Springer Lecture Notes in Mathematics, 1163 (1985), 149-160. [23] D. Driebe, Fully Chaotic Maps and Broken Time Symmetry. Kluwer Academic Publishers, Lancaster, 1999. [24] O.E. Rossler, Boscovich covariance. In: Beyond Belief: Randomness, Prediction and Explanation in Science ( J . Casti and A. Karlqvist, eds.), pp. 2546. CRC Press, Boca Raton, FL, 1991; for a new translation of Boscovich's seminal 1755-paper, see [21]. [25] O.E. Rossler, H.H. Diebner, and W. Pabst, Micro relativity, Z. Naturforsch., 52a (l997), 593-599. [26] P. Weibel, The world from within: Endo & Nano. In: Ars Electronica, 92, pp. 8-12. PVS-Verleger, Linz, 1992. [27] O.E. Rossler, Different types of chaos in two simple differential equations, 2. Naturforsch., 31a (1976), 1664-1670. Paper accompanying the movie of ref. [8]. [28] A. Einstein, On the electrodynamics of moving bodies. In: Einstein's Miraculous Year, Five Papers that Changed the Face of Physics (J. Stachel, ed.), pp. 123-160. Princeton Univ. Press, Princeton, NJ, 1998. First published in German in 1905 in Annalen der Physik, 17. [29] H . Everett 111, "Relative state formulation'' of quantum mechanics, Rev. Mod. PhyS., 29 (1957), 454-462. [30] K. Tomita, Conjugate pair of representations in chaos and quantum mechanics. In: A Chaotic Hierarchy (G. Baier and M. Klein, eds.), pp. 353-363. World Scientific, Singapore, 1991. [31] J. Ford, Directions in classical chaos. In: Directions in Chaos, Val. I (Ha0 Bailin, ed.), pp. 1-16. World Scientific, Singapore, 1987. [32] H. Stapp, Amer. J. Phys., 40 (l972), 1098.

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[33] O.E.Rbssler, Einstein completion of quantum mechanics made falsifiable. In: Compfexity, Entropy and the Physics of Information (W.H. Zurek, ed.), pp. 367373; A Proceedings Volume in the Santa Fe Institute Studies in the Sciences of Complexity. Addison Wesley, Redwoood City, Calif., 1990. [34] O.E.Rbssler, H. Kuypers and J. Parisi, Gravitational slowing-down of clocks implies proportional size increase. In: A Perspective Look at Nonlinear Media (J. Parisi, S.M. Mtlller and W. Zimmermann, eds.). Berlin: Springer Lecture Notes in Physics, 503 (1998), 370-372.

Sources Ch. 1. Steve Smale, On how I got started in dynamical systems. Taken from The Mathematics of Time by Steve Smale (Springer-Verlag, 1980, pp. 14715 1). Reprinted by permission of Steve Smale and Springer-Verlag. Ch. 2. Steve Smale, Finding a horseshoe on the beaches of Rio. Taken from The Mathematical Intelligencer 20 (Springer-Verlag, 1998, pp. 39-44). Reprinted by permission of Steve Smale and Springer-Verlag. Ch. 3. Yoshisuke Ueda, Strange attractors and the origin of chaos. Taken from The Road to Chaos, by Yoshisuke Ueda (Aerial, 1992). Also published in Nonlinear Science Today, Vol. 2, No. 2, pp. 1-16, Springer-Verlag, 1992, and as Chapter 17 in The Impact of Chaos on Science and Society, edited by Celso Grebogi and James A. Yorke (United Nations University Press, 1997). Reprinted by permission of Yoshisuke Ueda, Aerial Press, Springer-Verlag, and the United Nations University Press. Ch. 4. Yoshisuke Ueda, My encounter with chaos. Originally published in Japanese in Transactions of the Institute of Electronics, Information and Communication Engineers of Japan, 1994 (Vol. 77, pp.48 1-484). English translation by Mrs. Masako Ohnuki and Dr. H. Bruce Stewart, published here for the first time. Ch. 7. Edward Lorenz, The butterfly effect. This is Appendix One from The Essence of Chaos, by Edward N. Lorenz (Univ. of Washington, 1993). Reprinted by permission of the University of Washington Press. The other six chapters were written especially for this volume, and were received by February 29,2000.

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