Chaos - James Gleick [71]
Hénon had to plot these points by hand, but eventually the many scientists using this technique would watch them appear on a computer screen, like distant street lamps coming on one by one at nightfall. A typical orbit might begin with a point toward the lower left of the page. Then, on the next go-round, a point would appear a few inches to the right. Then another, more to the right and up a little—and so on. At first no pattern would be obvious, but after ten or twenty points an egg-shaped curve would take shape. The successive points actually make a circuit around the curve, but since they do not come around to exactly the same place, eventually, after hundreds or thousands of points, the curve is solidly outlined.
Such orbits are not completely regular, since they never exactly repeat themselves, but they are certainly predictable, and they are far from chaotic. Points never arrive inside the curve or outside it. Translated back to the full three-dimensional picture, the orbits were outlining a torus, or doughnut shape, and Hénon’s mapping was a cross-section of the torus. So far, he was merely illustrating what all his predecessors had taken for granted. Orbits were periodic. At the observatory in Copenhagen, from 1910 to 1930, a generation of astronomers painstakingly observed and calculated hundreds of such orbits—but they were only interested in the ones that proved periodic. “I, too, was convinced, like everyone else at that time, that all orbits should be regular like this,” Hénon said. But he and his graduate student at Princeton, Carl Heiles, continued computing different orbits, steadily increasing the level of energy in their abstract system. Soon they saw something utterly new.
First the egg-shaped curve twisted into something more complicated, crossing itself in figure eights and splitting apart into separate loops. Still, every orbit fell on some loop. Then, at even higher levels, another change occurred, quite abruptly. “Here comes the surprise,” Hénon and Heiles wrote. Some orbits became so unstable that the points would scatter randomly across the paper. In some places, curves could still be drawn; in others, no curve fit the points. The picture became quite dramatic: evidence of complete disorder mixed with the clear remnants of order, forming shapes that suggested “islands” and “chains of islands” to these astronomers. They tried two different computers and two different methods of integration, but the results were the same. They could only explore and speculate. Based solely on their numerical experimentation, they made a guess about the deep structure of such pictures. With greater magnification, they suggested, more islands would appear on smaller and smaller scales, perhaps all the way to infinity. Mathematical proof was needed—“but the mathematical approach to the problem does not seem too easy.”
ORBITS AROUND THE GALACTIC CENTER. To understand the trajectories of the stars through a galaxy, Michel Hénon computed the intersections of an orbit with a plane. The resulting patterns depended on the system’s total energy. The points from a stable orbit gradually produced a continuous, connected curve (left). Other energy levels, however, produced complicated mixtures of stability and chaos, represented by regions of scattered points.
Hénon went on to other problems, but fourteen years later, when finally he heard about the strange attractors of David Ruelle and Edward Lorenz, he was prepared to listen. By 1976 he had moved to the Observatory of Nice, perched high above the Mediterranean Sea on the Grande Corniche, and he heard a talk by a visiting physicist about the Lorenz attractor. The physicist had been trying different techniques to illuminate the fine “micro-structure” of the attractor, with little success. Hénon, though dissipative systems were not his field (“sometimes astronomers are fearful of dissipative systems—they’re untidy”), thought he had an idea.
Once again, he decided to throw out all reference