Chaos - James Gleick [67]
Ruelle and Takens wondered whether some other kind of attractor could have the right set of properties. Stable—representing the final state of a dynamical system in a noisy world. Low-dimensional—an orbit in a phase space that might be a rectangle or a box, with just a few degrees of freedom. Nonperiodic—never repeating itself, and never falling into a steady grandfather-clock rhythm. Geometrically the question was a puzzle: What kind of orbit could be drawn in a limited space so that it would never repeat itself and never cross itself—because once a system returns to a state it has been in before, it thereafter must follow the same path. To produce every rhythm, the orbit would have to be an infinitely long line in a finite area. In other words—but the word had not been invented—it would have to be fractal.
By mathematical reasoning, Ruelle and Takens claimed that such a thing must exist. They had never seen one, and they did not draw one. But the claim was enough. Later, delivering a plenary address to the International Congress of Mathematicians in Warsaw, with the comfortable advantage of hindsight, Ruelle declared: “The reaction of the scientific public to our proposal was quite cold. In particular, the notion that continuous spectrum would be associated with a few degrees of freedom was viewed as heretical by many physicists.” But it was physicists—a handful, to be sure—who recognized the importance of the 1971 paper and went to work on its implications.
ACTUALLY, BY 1971 the scientific literature already contained one small line drawing of the unimaginable beast that Ruelle and Takens were trying to bring alive. Edward Lorenz had attached it to his 1963 paper on deterministic chaos, a picture with just two curves on the right, one inside the other, and five on the left. To plot just these seven loops required 500 successive calculations on the computer. A point moving along this trajectory in phase space, around the loops, illustrated the slow, chaotic rotation of a fluid as modeled by Lorenz’s three equations for convection. Because the system had three independent variables, this attractor lay in a three-dimensional phase space. Although Lorenz drew only a fragment of it, he could see more than he drew: a sort of double spiral, like a pair of butterfly wings interwoven with infinite dexterity. When the rising heat of his system pushed the fluid around in one direction, the trajectory stayed on the right wing; when the rolling motion stopped and reversed itself, the trajectory would swing across to the other wing.
The attractor was stable, low-dimensional, and nonperiodic. It could never intersect itself, because if it did, returning to a point already visited, from then on the motion would repeat itself in a periodic loop. That never happened—that was the beauty of the attractor. Those loops and spirals were infinitely deep, never quite joining, never intersecting. Yet they stayed inside a finite space, confined by a box. How could that be? How could infinitely many paths lie in a finite space?
In an era before Mandelbrot’s pictures of fractals had flooded the scientific marketplace, the details of constructing such a shape were hard to imagine, and Lorenz acknowledged an “apparent contradiction” in his tentative description. “It is difficult to reconcile the merging of two surfaces, one containing each spiral, with the inability of two trajectories to merge,” he wrote. But he saw an answer too delicate to appear in the few calculations within range of his computer. Where the spirals appear to join, the surfaces must divide, he realized, forming separate layers in the manner of a flaky mille-feuille. “We see that each surface is really a pair of surfaces, so that, where they appear to merge, there are really four surfaces. Continuing this process for another circuit, we see that there are really eight surfaces, etc., and we finally conclude that there is an infinite complex of surfaces, each extremely close to one or the other of two