Chaos - James Gleick [69]
The process corresponds to sampling the state of a system every so often, instead of continuously. When to sample—where to take the slice from a strange attractor—is a question that gives an investigator some flexibility. The most informative interval might correspond to some physical feature of the dynamical system: for example, a Poincaré map could sample the velocity of a pendulum bob each time it passed through its lowest point. Or the investigator could choose some regular time interval, freezing successive states in the flash of an imaginary strobe light. Either way, such pictures finally began to reveal the fine fractal structure guessed at by Edward Lorenz.
EXPOSING AN ATTRACTOR’S STRUCTURE. The strange attractor above—first one orbit, then ten, then one hundred—depicts the chaotic behavior of a rotor, a pendulum swinging through a full circle, driven by an energetic kick at regular intervals. By the time 1,000 orbits have been drawn (below), the attractor has become an impenetrably tangled skein.
To see the structure within, a computer can take a slice through an attractor, a so-called Poincaré section. The technique reduces a three-dimensional picture to two dimensions. Each time the trajectory passes through a plane, it marks a point, and gradually a minutely detailed pattern emerges. This example has more than 8,000 points, each standing for a full orbit around the attractor. In effect, the system is “sampled” at regular intervals. One kind of information is lost; another is brought out in high relief.
THE MOST ILLUMINATING STRANGE ATTRACTOR, because it was the simplest, came from a man far removed from the mysteries of turbulence and fluid dynamics. He was an astronomer, Michel Hénon of the Nice Observatory on the southern coast of France. In one way, of course, astronomy gave dynamical systems its start, the clockwork motions of planets providing Newton with his triumph and Laplace with his inspiration. But celestial mechanics differed from most earthly systems in a crucial respect. Systems that lose energy to friction are dissipative. Astronomical systems are not: they are conservative, or Hamiltonian. Actually, on a nearly infinitesimal scale, even astronomical systems suffer a kind of drag, with stars radiating away energy and tidal friction draining some momentum from orbiting bodies, but for practical purposes, astronomers’ calculations could ignore dissipation. And without dissipation, the phase space would not fold and contract in the way needed to produce an infinite fractal layering. A strange attractor could never arise. Could chaos?
Many astronomers have long and happy careers without giving dynamical systems a thought, but Hénon was different. He was born in Paris in 1931, a few years younger than Lorenz but, like him, a scientist with a certain unfulfilled attraction to mathematics. Hénon liked small, concrete problems that could be attached to physical situations—“not like the kind of mathematics people do today,” he would say. When computers reached a size suitable for hobbyists, Hénon got one, a Heathkit that he soldered together and played with at home. Long before that, though, he took on a particularly baffling problem in dynamics. It concerned globular clusters—crowded balls of stars, sometimes a million in one place, that form the oldest and possibly the most breathtaking objects in the night sky. Globular clusters are amazingly dense with stars. The problem of how they stay together and how they evolve over time has perplexed astronomers throughout the twentieth century.
Dynamically speaking, a globular cluster is a big many-body problem. The two-body problem is easy. Newton solved it completely. Each body—the earth and the moon, for example—travels in a perfect ellipse around the system’s joint center of gravity. Add just one more gravitational object, however, and everything changes. The three-body problem is hard, and worse than hard. As Poincaré discovered, it is most often impossible. The orbits can be calculated numerically