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close this bookUnited Nations University - Work in Progress Newsletter - Volume 14, Number 1, 1992 (UNU, 1992, 12 pages)
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The Origin of Chaos

By Yoshisuke Ueda

In structurally stable systems, mathematicians identify two typical "attractors" - one representing behaviour that reaches a steady state or fixed point (as, for example, the spot where a free-swinging pendulum eventually comes to rest), the other a cycle that repeats itself continuously. Some of the first footprints of chaos appeared to researchers in what came to be known as "strange attractors."

One of the first to focus on the problem of the strange attractor was Professor Yoshisuke Ueda of Kyoto University. In 1961, as a graduate student in electrical engineering at Kyoto, he encountered the strikingly beautiful image of a strange attractor while doing computer simulations of an oscillator. In the paper he presented to the April 1991 chaos symposium, Dr. Ueda discussed his own personal experience in helping to identify the role of strange attractors in the origin of chaos theory. - Editor

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.*

[* Edward Lorenz, MIT meteorologist, who in 1961 made his computer runs on weather patterns which played a seminal role in the development of chaos theory.]

At the 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 (that is to say, one where relationships are not strictly proportional)? And my paradigm has always been the phenomena themselves - not papers with their abstractions, but something we can actually observe or quantify.

As I remember that November day in 1961,1 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 Electrical Communication Engineers, to be held in December, and was carrying out some analog computer experiments in order to test the applicability of the approximate computation I quoted in my paper. At that time, I was a third-year graduate student at Kyoto University, working on the phenomenon of frequency entrainment under the guidance of Professor Chihiro Hayashi. Frequency entrainment is a phenomenon that occurs when a self-oscillating frequency is attracted to and synchronized with that of an external driving frequency.

The main purpose of my computer experiment was to simulate the nonlinear differential equation describing frequency entrainment. I also sought to examine the range of the frequency and amplitude of the driving external signals which resulted in synchronization, as well as the amplitude and the phase of its oscillation. (For the non-scientist, a linear equation is one in which the sum of two solutions provides another solution; nature, however is virtually all nonlinear, and the behaviour of linear equations is far from typical.) I made an approximate computation by rewriting the non-autonomous equation into an autonomous one using the averaging method - which had the result of suppressing any evidence of chaos.

As we now know, there are actually two kinds of asynchronous oscillations - quasi-periodic (represented by a limit cycle attractor in the averaged equation) and chaotic oscillation (represented by strange or chaotic attractors). In those days, however, only the equilibrium point and limit cycle were known to exist as steady states, or attractors, of an autonomous system. So it was understandable to have the preconception that an asynchronous condition meant quasi-periodicity.

The Broken Egg

On that November day when I changed the parameter - the 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 I had been seeing earlier, but was more like a broken egg with jagged edges. My first concern was that my computer had gone bad. But I soon realized that this was not the case. It did not take long for me to recognize the mystery of it all - the fact that during the 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.

One day in the fall or winter of 1962, Professor Hayashi asked me to do some computation involving third order simultaneous equations with four unknowns to draw amplitude curves of the Duffing equation. It was particularly difficult on the hand-cranked calculator, but I was somehow trusted by Professor Hayashi as someone having fast and accurate computation and careful drawing skills. As it was impossible to solve these equations with four unknowns by sheer brute force, I eliminated two variables, gave the amplitude ahead of time with a bit of "manipulation," and finally, by hand calculations, solved approximately 50 cases of third order equations. This crash project was completed within a few weeks. The damping coefficient was set at 0.2.

The amplitude characteristic curve drawn with only two frequency components was, of course, nothing but an approximation. The standard procedure, therefore, was to verify the results with the analog computer. During this process, I struggled with enough chaotic oscillations (the source of the Japanese attractor) to make me sick. But Professor 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 quasi-periodic oscillation appeared.

Only a fragmented 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 Professor 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 using a larger damping coefficient of 0.4 (Chaos does not occur for this large coefficient.) and done away with the problematical data.

I finished my doctorate in the spring of 1964 and was hired as a research assistant in the Department of Electrical Engineering at Kyoto University. During that year, Professor Hayashi's book, Nonlinear Oscillations in Physical Systems, was published by McGraw-Hill. The Hayashi Laboratory was overflowing with chaotic data produced by the analog simulations, and yet they were overlooked and thought to be quasi-periodic oscillations or transient states.

Enter the Digital Computer

The digital computer KDC-I became available to us in the laboratory at about that time. 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=0 to 2p, using the Runge-Kutta-Gill method that set the size of integration step at 2p/60. It wouldn't even make a toy today, but at the time I was deeply impressed by the fact that such a calculation -impossible to do by hand - was finally possible.

In our laboratory, mapping or the stroboscopic method had been used for analog computer experiments. In modern terms, hand-made automatic mapping is a device which uses the analog computer to create Poincaré maps.* (These maps, in effect, take a slice from the heart of the attractor, removing a two-dimensional section in much the same way as a pathologist prepares a section of tissue for the microscope. It enables the investigator to sample motion at given intervals - for example, the velocity of a pendulum bob each time it passes through its lowest point.) The device plots on recording paper a sampling of the analog signal at the same instant during each cycle of external force. Thanks to the automatic mapping device, drawing invariant curves became much easier. As a result, the application of discrete (stroboscopic) dynamical systems theory to nonlinear oscillation was speeded up considerably.

[* Named so after mathematician Henri Poincaré.]

During the latter half of the 1960s, work continued at the Hayashi Laboratory on nonlinear oscillations. Most of the research data from our group were published by Professor Hayashi at international meetings. Then during the late 1960s, I sent a paper of my own for the first time to the Journal of the Institute of Electronics and Communication Engineers. (This was a time when the university was in turmoil with student protest; the democratic atmosphere that prevailed on campus might have prompted me to take this action.)

Concept of Chaos Established

In the paper, I was able to demonstrate quasi-periodic oscillation and chaos in Rayleigh-van der Pol mixed type equation. Though it hadn't taken clear shape yet, the concept of chaos was already established in my mind at that time. This fact can be supported by an oral report made at a seminar on "Ordinary differential equations and nonlinear dynamics," held in December 1970 at the Research Institute for Mathematical Sciences at Kyoto University, under the leadership of Professor Minoru Urabe.

The record of the comments I made at the; and 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 either by small fluctuations in the oscillatory system or by external disturbances." These minimal sets are the unstable periodic motions in the attractor. The descriptions led to my proposed name, "randomly transitional phenomena," for the characteristics of chaos in a physical system.

I made the report in the Urabe seminar in the hope that the mathematicians might possibly support my interpretation of the random oscillations. But my gamble backfired, and Professor Urabe admonished me personally. "What you saw was simply the essence of quasi-periodic oscillations," he said. "You are too young to make conceptual observations." The existence of random oscillations - or chaos - was so obvious in my mind that this negative comment did not crush me. I was, of course, deeply disappointed that no one understood it no matter how hard I tried to explain -and from then on, I became even more careful in my research efforts.

Grasping the Concept

I sent off a second paper, based on my oral report, but it was rejected. I spent nearly a year rewriting it and sent it again in September 1972. It was finally accepted. The summary of the discrete dynamical system theory I included in the appendix was by far the most difficult work of the paper. 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 could 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.