Chaos - James Gleick [64]
Ruelle had no experience with fluid flows, but that did not discourage him any more than it had discouraged his many unsuccessful predecessors. “Always nonspecialists find the new things,” he said. “There is not a natural deep theory of turbulence. All the questions you can ask about turbulence are of a more general nature, and therefore accessible to nonspecialists.” It was easy to see why turbulence resisted analysis. The equations of fluid flow are nonlinear partial differential equations, unsolvable except in special cases. Yet Ruelle worked out an abstract alternative to Landau’s picture, couched in the language of Smale, with images of space as a pliable material to be squeezed, stretched, and folded into shapes like horseshoes. He wrote a paper at his institute with a visiting Dutch mathematician, Floris Takens, and they published it in 1971. The style was unmistakably mathematics—physicists, beware!—meaning that paragraphs would begin with Definition or Proposition or Proof, followed by the inevitable thrust: Let….
“Proposition (5.2). Let Xµ be a one-parameter family of Ck vectorfields on a Hilbert space H such that…”
Yet the title claimed a connection with the real world: “On the Nature of Turbulence,” a deliberate echo of Landau’s famous title, “On the Problem of Turbulence.” The clear purpose of Ruelle and Takens’s argument went beyond mathematics; they meant to offer a substitute for the traditional view of the onset of turbulence. Instead of a piling up of frequencies, leading to an infinitude of independent overlapping motions, they proposed that just three independent motions would produce the full complexity of turbulence. Mathematically speaking, some of their logic turned out to be obscure, wrong, borrowed, or all three—opinions still varied fifteen years later.
But the insight, the commentary, the marginalia, and the physics woven into the paper made it a lasting gift. Most seductive of all was an image that the authors called a strange attractor. This phrase was psychoanalytically “suggestive,” Ruelle felt later. Its status in the study of chaos was such that he and Takens jousted below a polite surface for the honor of having chosen the words. The truth was that neither quite remembered, but Takens, a tall, ruddy, fiercely Nordic man, might say, “Did you ever ask God whether he created this damned universe?…I don’t remember anything…. I often create without remembering it,” while Ruelle, the paper’s senior author, would remark softly, “Takens happened to be visiting IHES. Different people work differently. Some people would try to write a paper all by themselves so they keep all the credit.”
The strange attractor lives in phase space, one of the most powerful inventions of modern science. Phase space gives a way of turning numbers into pictures, abstracting every bit of essential information from a system of moving parts, mechanical or fluid, and making a flexible road map to all its possibilities. Physicists already worked with two simpler kinds of “attractors”: fixed points and limit cycles, representing behavior that reached a steady state or repeated itself continuously.
In phase space the complete state of knowledge about a dynamical system at a single instant in time collapses to a point. That point is the dynamical system—at that instant. At the next instant, though, the system will have changed, ever so slightly, and so the point moves. The history of the system time can be charted by the moving point, tracing its orbit through phase space with the passage of time.
How can all the information about a complicated system be stored in a point? If the system has only two variables, the answer is simple. It is straight from the Cartesian geometry taught in high school—one variable on the horizontal axis, the other on the vertical. If the system is a swinging, frictionless pendulum, one variable is position and the other velocity, and they change continuously, making a line of points that traces a loop, repeating itself forever, around and around. The same system