Chaos - James Gleick [73]
THE ATTRACTOR OF HÉNON. A simple combination of folding and stretching produced an attractor that easy to compute yet still poorly understood by mathematicians. As thousands, the millions of points appear, more and more detail emerges. What appear to be single lines prove, on magnification, to be pairs, then pairs of pairs. Yet whether any two successive points appear nearby or far apart is unpredictable.
As an element in the world revealed by computer exploration, the strange attractor began as a mere possibility, marking a place where many great imaginations in the twentieth century had failed to go. Soon, when scientists saw what computers had to show, it seemed like a face they had been seeing everywhere, in the music of turbulent flows or in clouds scattered like veils across the sky. Nature was constrained. Disorder was channeled, it seemed, into patterns with some common underlying theme.
Later, the recognition of strange attractors fed the revolution in chaos by giving numerical explorers a clear program to carry out. They looked for strange attractors everywhere, wherever nature seemed to be behaving randomly. Many argued that the earth’s weather might lie on a strange attractor. Others assembled millions of pieces of stock market data and began searching for a strange attractor there, peering at randomness through the adjustable lens of a computer.
In the middle 1970s these discoveries lay in the future. No one had actually seen a strange attractor in an experiment, and it was far from clear how to go about looking for one. In theory the strange attractor could give mathematical substance to fundamental new properties of chaos. Sensitive dependence on initial conditions was one. “Mixing” was another, in a sense that would be meaningful to a jet engine designer, for example, concerned about the efficient combination of fuel and oxygen. But no one knew how to measure these properties, how to attach numbers to them. Strange attractors seemed fractal, implying that their true dimension was fractional, but no one knew how to measure the dimension or how to apply such a measurement in the context of engineering problems.
Most important, no one knew whether strange attractors would say anything about the deepest problem with nonlinear systems. Unlike linear systems, easily calculated and easily classified, nonlinear systems still seemed, in their essence, beyond classification—each different from every other. Scientists might begin to suspect that they shared common properties, but when it came time to make measurements and perform calculations, each nonlinear system was a world unto itself. Understanding one seemed to offer no help in understanding the next. An attractor like Lorenz’s illustrated the stability and the hidden structure of a system that otherwise seemed patternless, but how did this peculiar double spiral help researchers exploring unrelated systems? No one knew.
For now, the excitement went beyond pure science. Scientists who saw these shapes allowed themselves to forget momentarily the rules of scientific discourse. Ruelle, for example: “I have not spoken of the esthetic appeal of strange attractors. These systems of curves, these clouds of points suggest sometimes fireworks or galaxies, sometimes strange and disquieting vegetal proliferations. A realm lies there of forms to explore, and harmonies to discover.”
Universality
The iterating of these lines brings gold;
The framing of this circle on the ground
Brings whirlwinds, tempests, thunder and lightning.
—MARLOWE, Dr. Faustus
A FEW DOZEN YARDS upstream from a waterfall, a smooth flowing stream seems to intuit the coming drop. The water begins