Chaos - James Gleick [110]
FRACTAL BASIN BOUNDARIES. Even when a dynamical system’s long-term behavior is not chaotic, chaos can appear at the boundary between one kind of steady behavior and another. Often a dynamical system has more than one equilibrium state, like a pendulum that can come to a halt at either of two magnets placed at its base. Each equilibrium is an attractor, and the boundary between two attractors can be complicated but smooth (left). Or the boundary can be complicated but not smooth. The highly fractal interspersing of white and black (right) is a phase-space diagram of a pendulum. The system is sure to reach one of two possible steady states. For some starting conditions, the outcome is quite predictable—black is black and white is white. But near the boundary, prediction becomes impossible.
To researchers and engineers, there was a lesson in these pictures—a lesson and a warning. Too often, the potential range of behavior of complex systems had to be guessed from a small set of data. When a system worked normally, staying within a narrow range of parameters, engineers made their observations and hoped that they could extrapolate more or less linearly to less usual behavior. But scientists studying fractal basin boundaries showed that the border between calm and catastrophe could be far more complex than anyone had dreamed. “The whole electrical power grid of the East Coast is an oscillatory system, most of the time stable, and you’d like to know what happens when you perturb it,” Yorke said. “You need to know what the boundary is. The fact is, they have no idea what the boundary looks like.”
Fractal basin boundaries addressed deep issues in theoretical physics. Phase transitions were matters of thresholds, and Peitgen and Richter looked at one of the best-studied kinds of phase transitions, magnetization and nonmagnetization in materials. Their pictures of such boundaries displayed the peculiarly beautiful complexity that was coming to seem so natural, cauliflower shapes with progressively more tangled knobs and furrows. As they varied the parameters and increased their magnification of details, one picture seemed more and more random, until suddenly, unexpectedly, deep in the heart of a bewildering region, appeared a familiar oblate form, studded with buds: the Mandelbrot set, every tendril and every atom in place. It was another signpost of universality. “Perhaps we should believe in magic,” they wrote.
MICHAEL BARNSLEY TOOK a different road. He thought about nature’s own images, particularly the patterns generated by living organisms. He experimented with Julia sets and tried other processes, always looking for ways of generating even greater variability. Finally, he turned to randomness as the basis for a new technique of modeling natural shapes. When he wrote about his technique, he called it “the global construction of fractals by means of iterated function systems.” When he talked about it, however, he called it “the chaos game.”
To play the chaos game quickly, you need a computer with a graphics screen and a random number generator, but in principle a sheet of paper and a coin work just as well. You choose a starting point somewhere on the paper. It does not matter where. You invent two rules, a heads rule and a tails rule. A rule tells you how to take one point to another: “Move two inches to the northeast,” or “Move 25 percent closer to the center.” Now you start flipping the coin and marking points, using the heads rule when the coin comes up heads and the tails rule when it comes up tails. If you throw away the first fifty points, like a blackjack dealer burying the first few cards in a new deal, you will find the chaos game producing not a random field of dots but a shape, revealed with greater and greater sharpness as the game goes on.
Barnsley’s central insight was this: Julia sets and other fractal shapes, though properly viewed as the outcome of a deterministic process, had a second, equally valid existence as the limit of a random process. By analogy, he suggested, one could