Chaos - James Gleick [23]
The bad news arrived in the mail soon after Christmas 1959, when Smale was living temporarily in an apartment in Rio de Janeiro with his wife, two infant children, and a mass of diapers. His conjecture had defined a class of differential equations, all structurally stable. Any chaotic system, he claimed, could be approximated as closely as you liked by a system in his class. It was not so. A letter from a colleague informed him that many systems were not so well-behaved as he had imagined, and it described a counterexample, a system with chaos and stability, together. This system was robust. If you perturbed it slightly, as any natural system is constantly perturbed by noise, the strangeness would not go away. Robust and strange—Smale studied the letter with a disbelief that melted away slowly.
Chaos and instability, concepts only beginning to acquire formal definitions, were not the same at all. A chaotic system could be stable if its particular brand of irregularity persisted in the face of small disturbances. Lorenz’s system was an example, although years would pass before Smale heard about Lorenz. The chaos Lorenz discovered, with all its unpredictability, was as stable as a marble in a bowl. You could add noise to this system, jiggle it, stir it up, interfere with its motion, and then when everything settled down, the transients dying away like echoes in a canyon, the system would return to the same peculiar pattern of irregularity as before. It was locally unpredictable, globally stable. Real dynamical systems played by a more complicated set of rules than anyone had imagined. The example described in the letter from Smale’s colleague was another simple system, discovered more than a generation earlier and all but forgotten. As it happened, it was a pendulum in disguise: an oscillating electronic circuit. It was nonlinear and it was periodically forced, just like a child on a swing.
It was just a vacuum tube, really, investigated in the twenties by a Dutch electrical engineer named Balthasar van der Pol. A modern physics student would explore the behavior of such an oscillator by looking at the line traced on the screen of an oscilloscope. Van der Pol did not have an oscilloscope, so he had to monitor his circuit by listening to changing tones in a telephone handset. He was pleased to discover regularities in the behavior as he changed the current that fed it. The tone would leap from frequency to frequency as if climbing a staircase, leaving one frequency and then locking solidly onto the next. Yet once in a while van der Pol noted something strange. The behavior sounded irregular, in a way that he could not explain. Under the circumstances he was not worried. “Often an irregular noise is heard in the telephone receivers before the frequency jumps to the next lower value,” he wrote in a letter to Nature. “However, this is a subsidiary phenomenon.” He was one of many scientists who got a glimpse of chaos but had no language to understand it. For people trying to build vacuum tubes, the frequency-locking was important. But for people trying to understand the nature of complexity, the truly interesting behavior would turn out to be the “irregular noise” created by the conflicting pulls of a higher and lower frequency.
Wrong though it was, Smale’s conjecture put him directly on the track of a new way of conceiving the full complexity of dynamical systems. Several mathematicians had taken another look at the possibilities of the van der Pol oscillator, and Smale now took their work into a new realm. His only oscilloscope screen was his mind, but it was a mind shaped by his years of exploring the topological