In programming his computer to look at Burke’s equations, Rob had stumbled on his own equations of motion that penetrated, even if only slightly, into the unknown world of randomness. It was Claude Shannon who defined information as the amount of surprise you get on seeing something happen, and news spread fast around the physics department that Rob’s equations were generating a tremendous amount of information. By iterating these equations—repeating them over and over again—Rob could watch the process by which simple systems tip from order into chaos. While following this progression, he noticed two salient facts, which one might call the laws of chaos. The first law posits the “sensitive dependence” of all systems on their initial conditions. The second law states that whatever differences exist in systems will tend to increase over time. In the language of chaos theory, this second law predicts the “rapid divergence of nearby trajectories.” Given sensitive dependence on initial conditions and rapid divergence of nearby trajectories, one can expect small differences in systems to compound themselves over time into very large differences.
Without knowledge of chaotic solutions, and before the invention of computers capable of arriving at them, Poincaré described the essential insights of chaos theory as follows: “It may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon.”
As Doyne Farmer wrote in commenting on Poincaré’s observation: “Modern computer technology allows us to simulate dynamical systems that produce the ‘fortuitous phenomena’ and take them apart to study how, when, and under what conditions sensitive dependence on initial conditions occurs.” If the laws of chaos theory sound rather abstract, or of local interest to plumbers dealing with obstructions in their pipes, the larger implications of these laws should become evident when one mentions, for example, that sensitive dependence on initial conditions and rapid divergence of nearby trajectories could very well explain the development of species in Darwinian evolution.
Norman Packard said of his own surprise at finding strange attractors at work in the midst of chaos, “This idea of information generation is pretty powerful if you allow your imagination to wander a bit. We have dreams of generalizing the theory of information generation in chaotic systems to more general systems, like that of biological evolution. Two billion years ago there was a blob of pre-biotic chemicals on the earth. It mushed around to form a few strands of DNA and these reproduced and eventually formed life that got increasingly complex. And as it got increasingly complex, new information was generated. Each time evolution occurs, more information is being produced in these new, more complicated forms of life. Our hope is to quantify this kind of information generation in exactly the same way as we’re now doing for chaos. One of the things we try to do, for instance, is predict when you’re going to have chaotic behavior and how chaotic it will be. This chaos corresponds to the amount of information a system is generating. The more information it generates, the more chaotic it is. The appeal in studying them is that strange attractors produce information with ramifications in all sorts of fields, ranging from the theory of evolution to ecology, sociology, economics, and the workings of the human brain.”
Fascinated by chaos and the strange geometrical structures that govern it, Rob Shaw moved into his laboratory to work through the night on his computer. His advisers got worried. Having finished all the requirements for his doctorate, he was in the last stages of completing a dissertation on experimental superconductivity, a subject apparently unrelated to strange attractors. Rob had no more than two months of work remaining to finish the dissertation. His advisers came to talk to him. He could finish in one month, they said. When that didn’t interest him, they whittled it down to a couple of weeks. But Rob wasn’t listening. He was lost to chaos and there was no way anyone could bring him back to earth.
From the information generated by his computer, Shaw isolated different types of chaos and strange attractors. Many of his ideas were new, but others, he discovered later, had been arrived at independently by other scientists. Edward Lorenz, a meteorologist at MIT, had stumbled on the first strange attractor back in 1963. He was looking at models for weather prediction when he noticed something odd about convection currents. These display what might be called pockets of chaos—loops of information that demonstrate perfectly the laws of strange attraction: a sensitive dependence on initial conditions, and rapid divergence of nearby trajectories. The discovery of the Lorenz attractor, as this particular structure came to be known, had surprising ramifications in the everyday world, although its immediate effect was to explain why long-range weather prediction is impossible.
The Rössler attractor is named after another early explorer into chaos, Otto Rössler, who works as a theoretical chemist in Tubingen, Germany. Rössler is a gracious, soft-spoken man who lives surrounded by books, so that a conversation with him proceeds more in the nature of a colloquium. He pulls texts and citations off the shelf and piles them in front of him, while carrying on a kind of dialogue through the ages, with speaking parts given to Aristotle, Maxwell, Einstein, and anyone else who has something to contribute. He ascribes to Anaxagoras the earliest definition of chaos, but Rössler’s own discoveries in the field are pedestrian, rather than literary, in origin.
He was walking down the street one day when he saw a group of children standing in front of a window. He joined them in staring at what proved to be taffy puller—a machine with two arms that stretch and fold a sheet of taffy over and over again. Rössler stood and watched the taffy puller for half an hour. He was utterly transfixed. Not by taffy making, but by the rythmic motion of the machine, which produced what he recognized as a perfect example of strange attraction. Rössler imagined two raisins placed in close proximity on the surface of the taffy. While the arms of the puller stretched and folded the sticky mass, he followed his two imaginary raisins along a path of successive iterations. They traveled away from each other in an eloquent demonstration of sensitive dependence on initial conditions and rapid divergence of nearby trajectories. Still standing in front of the candy shop, Rössler scribbled the equations describing the strange attractor that bears his name, although he personally prefers to call it the taffy puller attractor.
Rob Shaw’s own discoveries in the realm of strange attraction were made public in equally quixotic fashion. Norman Packard was leafing through a copy of Scientific American when he found an advertisement announcing a prize being given by Louis Jacot, a French businessman, for the most original essay on the origin of the universe. Norman wrote for details about the competition, and then he convinced Rob to submit an essay on chaos. His paper, called “Strange Attractors, Chaotic Behavior, and Information Flow,” was accompanied by a cover letter explaining the relevance of strange attractors to the theory of evolution. Shaw won honor-able mention for the Prix Louis Jacot and an award of two thousand francs—about five hundred dollars at the time—which he spent on flying to Paris to pick up his prize.
This marked the first public awareness of work being done by the Chaos Cabal. But soon there would be a flood of interest in the nighttime discoveries made by this group of hackers and would-be gamblers. When reporters from Newsweek and the Los Angeles Times came to inquire about what the Dynamical Systems Collective was finding out in the phosphorescent green sea of chaos, they discovered researchers holed up in a laboratory that looked like a submarine conning tower full of computers, terminals, plotters, printers, monitors, dials, gauges, and other paraphernalia needed for tracking strange attractors through the murky reaches of turbulence.
The formation of the Chaos Cabal in 1977 was itself a nice example of sensitive dependence on initial conditions. Like Shaw, the other cabalists had jettisoned careers in one of the more established branches of physics to join him in programming chaos into computers. Jim Crutchfield, who had been Rob’s assistant when he was st
udying superconductivity, had no problem picking up the new language of strange attractors.
Norman’s conversion was more complex. Phlegmatic and easygoing as he might be, he was the star of the physics department. He had zipped through the curriculum and passed his qualifying exams all in the first year—an unusual feat—and everyone expected him to do his doctoral research in statistical mechanics, which is one of the more gentlemanly pursuits in classical physics. The faculty was dismayed when he announced he was joining the Chaos Cabal, whose research looked to them like a kinky hybrid between philosophy and computer programming.
“I had made a name for myself in the department,” said Norman. “They didn’t see my truly lazy and sluggardly ways until the following summer when I went out to Las Vegas to play roulette. That gave them second thoughts about me, and then when I began working on chaos they had third and fourth thoughts. I fell into utter disrepute, from which I was only rehabilitated when we won our National Science Foundation grant.”
Doyne, having dropped out of school for a year and a half to work on roulette, had long since given up the thought of becoming an astrophysicist. By the time he dropped back into school he was already a connoisseur of chaos. As he wrote in the acknowledgements to “Order Within Chaos,” his doctoral dissertation: “Had Rob never heard of the Lorenz attractor, none of this [the formation of the Chaos Cabal] would ever have occurred. I would quite likely have gotten bored with physics and dropped out, and I would now be happily playing my harmonica for the hippies down on the mall. Instead, Rob planted the seeds of chaos in my brain, and here I am trying to be a respectable scientist. Such is sensitive dependence on initial conditions.”
Except for its reliance on computers, the formal study of chaos has little to do with beating roulette. In fact, the “Santa Cruz school of nonpredictable physics,” as Norman called it, was doing its best to subvert the classical Laplacian assumptions on which predictability in roulette is based. “In his deterministic, classical dynamics, Laplace said that if you give me the position and velocity of every particle in the universe, I can tell you exactly what it’s going to do a million years from now. He was wrong,” said Norman, in his calm, almost laconic tone of voice, “and the extent to which he was wrong has only been realized very recently.”
The Chaos Cabal remarked the fallacy in Laplace’s assumption when they found, as one would never expect in his classical physics, that very simple systems can evolve from order into chaos. “Originally,” said Norman, “people thought the reason behavior looked complicated was because only a complicated set of equations could describe it, equations involving many different interactions. It was natural for Laplace to think this. But it turns out that you can have complicated behavior even for very simple systems, systems with very few interacting components.
“In the case of water flowing down a pipe periodically, if you take your fist and knock the pipe, the water will jiggle around. It changes its motion as it jiggles. But then in time it settles down again. That’s what it means to be an attractor. You perturb the system by giving it a knock. Its behavior is altered. And then it settles back to what it was doing before. But then again—and this is the philosophically interesting part—you may knock the pipe and the water, rather than resuming its periodic flow, may go off and do something completely different and never come back to what it was doing before. If that’s the case, its previous state wasn’t caused by an attractor; it was unstable.
“These arguments against the Laplacian world view are disconcerting to a lot of people who thought the world was predictable in principle. And the fact that it isn’t predictable has all sorts of philosophical implications that have yet to be worked out. For example, take the ongoing debate about determinism and free will. The determinists use the Laplacian world view to argue that the motion of physical systems—humans included—is determined in advance by the laws of physics. If this is the case, then there is no sense talking about free will. Your motion for the rest of time is already programmed. How strange attractors affect this argument is not trivial, and they do affect it, because strange attractors allow for the possibility of spontaneous change, so that your behavior for the rest of time can in no way be determined.”
The revolution currently taking place in the midst of physics is one whose battles are being fought on the insides of computers. They alone—never tiring in their electronic compulsion to repeat—are capable of the iterations required for tipping simple systems from order into chaos. Only through the doggedness of silicon can one look at sensitive dependence on initial conditions and project this dependence far enough into the future to watch for divergence of nearby trajectories. Taffy pullers have been put to good use, but the new physics owes its insights and methods to the electronic computer.
“It’s only with the advent of modern digital and analog computers,” said Tom Ingerson, “that people have been able to solve complex mathematical equations in all their glorious horror. Instead of studying the real forms that equations happen to have, physicists for years tried to beat them into forms which were familiar. Some curves had names and others didn’t. But Mother Nature didn’t give a damn for these distinctions, and she went on being as complex as ever. Computers don’t give a damn either. They can treat equations however they happen to behave. This field of strange attractors and chaos, or nonlinear dynamics, or whatever you want to call it, has arisen because people are attempting to understand the mathematics associated with more general forms of equations, which don’t necessarily describe particularly complicated phenomena. It’s just that Mother Nature doesn’t happen to be linear all the time. In fact, she seldom is. One of the fascinating things that’s come out of the study of chaos is the fact that a large class of nature is unpredictable. This still blows some people away when you explain it to them. But there are certain equations that indicate the intrinsic unpredictability of systems, and that’s truly an amazing fact.”
The Chaos Cabalists, in their published papers, and later when lecturing at scientific conferences, became known as artists in the realm of silicon. Out of their laboratory full of computers and printers came some of the earliest pictures of these strange worlds that displayed, as Doyne put it, a “peaceful coexistence between order and chaos.” As preparation for launching themselves into the forefront of theoretical physics, Doyne and Norman had spent a year and a half at home building roulette computers from scratch. Rob Shaw had resurrected an analog computer from the junkheap of outmoded technology, and Jim Crutchfield had elevated the profession of computer hacker into a way of life. Their exaltation had humble origins, but the Chaos Cabal had been in the right place at the right time, with the right technology.
“These guys are real lucky,” said Ingerson. “They got in on the ground floor, with the perfect combination of factors for success in science: native intelligence, opportunity, luck, and resources.”
When its theories captured the attention of Scientific American, Douglas Hofstadter wrote in a review article on the field of strange attractors and chaos that “the simplicity of the underlying ideas gives them an elegance that in my opinion rivals that of some of the best of classical mathematics. Indeed, there is an 18th- or 19th-century flavor to some of this work that is refreshingly concrete in this era of staggering abstraction.
“Probably the main reason these ideas are only now being discovered is that the style of exploration is entirely modern: it is a kind of experimental mathematics, in which the digital computer plays the role of Magellan’s ship, the astronomer’s telescope and the physicist’s accelerator. Just as ships, telescopes and accelerators must be ever larger, more powerful and more expensive in order to probe ever more hidden regions of nature, so one would need computers of ever greater size, speed and accuracy in order to explore the remoter regions of mathematical space. By the same token, just as there was a golden era of exploration by ship and of discoveries made with telescopes and accelerators, characterized by a peak in the ratio of new discoveries
uncovered to money spent, so one would expect there to be a golden era in the experimental mathematics of these models of chaos. Perhaps this era has already occurred, or perhaps it is occurring right now.”
While sailing their computers through the golden age of chaotic exploration, the Cabal never forgot the port from whence they had embarked. Doyne had “chaos in my brain,” as he put it. But he was equally obsessed by roulette. New fields in physics do not remain new for long. One either seizes the moment of discovery or watches it slip into more enterprising hands. But the same was true of beating roulette. For all the Projectors knew, there might be dozens of engineers in the Silicon Valley tinkering in their garages with roulette computers. The Chaos Cabal had flags to plant on unexplored terrain. Eudaemonic Enterprises had a Pie to fill with roulette winnings. Their only choice was to do everything at once: roulette by day and chaos by night. Or vice versa.
“The Project gave us a weird perspective on the academic experience,” said Norman. “Whenever we approached the end of the quarter or vacations, there was not only the pressure of finishing courses and writing papers and giving lectures, but there was also the pressure of getting the computers ready to go for one of the trips scheduled during vacations. We tried, but we never really had time to do everything.”
The Projectors developed a manic schedule of work and more work. Every break from school they crammed computers and roulette gear into the Blue Bus and shot over the mountains to Tahoe or Reno or farther across the desert to Las Vegas. They became quick-change artists adept at transforming themselves from graduate students into gamblers. They learned fast how to wipe every sign of intelligence off their faces. One day they’d be programming a PDP 11/45 mainframe computer with Belousov-Zhabotinsky reactions or Lyapunov exponents, and the next day they’d be checked into a flophouse off the Strip practicing betting patterns with computers strapped to their chests.
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