Chaos - James Gleick [68]
THE FIRST STRANGE ATTRACTOR. In 1963 Edward Lorenz could compute only the first few strands of the attractor for his simple system of equations. But he could see that the interleaving of the two spiral wings must have an extraordinary structure on invisibly small scales.
The effort to pursue the hints put forward by Ruelle and Takens took two paths. One was the theoretical struggle to visualize strange attractors. Was the Lorenz attractor typical? What other sorts of shapes were possible? The other was a line of experimental work meant to confirm or refute the highly unmathematical leap of faith that suggested the applicability of strange attractors to chaos in nature.
In Japan the study of electrical circuits that imitated the behavior of mechanical springs—but much faster—led Yoshisuke Ueda to discover an extraordinarily beautiful set of strange attractors. (He met an Eastern version of the coolness that greeted Ruelle: “Your result is no more than an almost periodic oscillation. Don’t form a selfish concept of steady states.”) In Germany Otto Rössler, a nonpracticing medical doctor who came to chaos by way of chemistry and theoretical biology, began with an odd ability to see strange attractors as philosophical objects, letting the mathematics follow along behind. Rössler’s name became attached to a particularly simple attractor in the shape of a band of ribbon with a fold in it, much studied because it was easy to draw, but he also visualized attractors in higher dimensions—“a sausage in a sausage in a sausage in a sausage,” he would say, “take it out, fold it, squeeze it, put it back.” Indeed, the folding and squeezing of space was a key to constructing strange attractors, and perhaps a key to the dynamics of the real systems that gave rise to them. Rössler felt that these shapes embodied a self-organizing principle in the world. He would imagine something like a wind sock on an airfield, “an open hose with a hole in the end, and the wind forces its way in,” he said. “Then the wind is trapped. Against its will, energy is doing something productive, like the devil in medieval history. The principle is that nature does something against its own will and, by self-entanglement, produces beauty.”
Making pictures of strange attractors was not a trivial matter. Typically, orbits would wind their ever-more–complicated paths through three dimensions or more, creating a dark scribble in space with an internal structure that could not be seen from the outside. To convert these three-dimensional skeins into flat pictures, scientists first used the technique of projection, in which a drawing represented the shadow that an attractor would cast on a surface. But with complicated strange attractors, projection just smears the detail into an indecipherable mess. A more revelatory technique was to make a return map, or a Poincaré map, in effect, taking a slice from the tangled heart of the attractor, removing a two-dimensional section just as a pathologist prepares a section of tissue for a microscope slide.
The Poincaré map removes a dimension from an attractor and turns a continuous line into a collection of points. In reducing an attractor to its Poincaré map, a scientist implicitly assumes that he can preserve much of the essential movement. He can imagine, for example, a strange attractor buzzing around before his eyes, its orbits carrying up and down, left and right, and to and fro through his computer screen. Each time the orbit passes through the screen, it leaves a glowing point at the place of intersection, and the points either form a