Chaos - James Gleick [22]
Both subjects, topology and dynamical systems, went back to Henri Poincaré, who saw them as two sides of one coin. Poincaré, at the turn of the century, had been the last great mathematician to bring a geometric imagination to bear on the laws of motion in the physical world. He was the first to understand the possibility of chaos; his writings hinted at a sort of unpredictability almost as severe as the sort Lorenz discovered. But after Poincaré’s death, while topology flourished, dynamical systems atrophied. Even the name fell into disuse; the subject to which Smale nominally turned was differential equations. Differential equations describe the way systems change continuously over time. The tradition was to look at such things locally, meaning that engineers or physicists would consider one set of possibilities at a time. Like Poincaré, Smale wanted to understand them globally, meaning that he wanted to understand the entire realm of possibilities at once.
Any set of equations describing a dynamical system—Lorenz’s, for example—allows certain parameters to be set at the start. In the case of thermal convection, one parameter concerns the viscosity of the fluid. Large changes in parameters can make large differences in a system—for example, the difference between arriving at a steady state and oscillating periodically. But physicists assumed that very small changes would cause only very small differences in the numbers, not qualitative changes in behavior.
Linking topology and dynamical systems is the possibility of using a shape to help visualize the whole range of behaviors of a system. For a simple system, the shape might be some kind of curved surface; for a complicated system, a manifold of many dimensions. A single point on such a surface represents the state of a system at an instant frozen in time. As a system progresses through time, the point moves, tracing an orbit across this surface. Bending the shape a little corresponds to changing the system’s parameters, making a fluid more viscous or driving a pendulum a little harder. Shapes that look roughly the same give roughly the same kinds of behavior. If you can visualize the shape, you can understand the system.
When Smale turned to dynamical systems, topology, like most pure mathematics, was carried out with an explicit disdain for real-world applications. Topology’s origins had been close to physics, but for mathematicians the physical origins were forgotten and shapes were studied for their own sake. Smale fully believed in that ethos—he was the purest of the pure—yet he had an idea that the abstract, esoteric development of topology might now have something to contribute to physics, just as Poincaré had intended at the turn of the century.
One of Smale’s first contributions, as it happened, was his faulty conjecture. In physical terms, he was proposing a law of nature something like this: A system can behave erratically, but the erratic behavior cannot be stable. Stability—“stability in the sense of Smale,” as mathematicians would sometimes say—was a crucial property. Stable behavior in a system was behavior that would not disappear just because some number was changed a tiny bit. Any system could have both stable and unstable behaviors within it. The equations governing a pencil standing on its point have a good mathematical solution with the center of gravity directly above the point—but you cannot stand a pencil on its point because the solution is unstable. The slightest perturbation draws the system away from that solution. On the other hand, a marble lying at the bottom of a bowl stays there, because if the marble