Chaos - James Gleick [109]
This is deterministic pinball—no shaking the machine. Only one parameter controls the ball’s destination, and that is the initial position of the plunger. Imagine that the machine is laid out so that a short pull of the plunger always means that the ball will end up rolling out the right-hand ramp, while a long pull always means that the ball will finish in the left-hand ramp. In between, the behavior gets complex, with the ball bouncing from bumper to bumper in the usual energetic, noisy, and variably long-lived manner before finally choosing one exit or the other.
Now imagine making a graph of the result of each possible starting position of the plunger. The graph is just a line. If a position leads to a right-hand departure, plot a red point, and plot a green point for left. What can we expect to find about these attractors as a function of the initial position?
The boundary proves to be a fractal set, not necessarily self-similar, but infinitely detailed. Some regions of the line will be pure red or green, while others, when magnified, will show new regions of red within the green, or green within the red. For some plunger positions, that is, a tiny change makes no difference. But for others, even an arbitrarily small change will make the difference between red and green.
To add a second dimension meant adding a second parameter, a second degree of freedom. With a pinball machine, for example, one might consider the effect of changing the tilt of the playing slope. One would discover a kind of in-and–out complexity that would give nightmares to engineers responsible for controlling the stability of sensitive, energetic real systems with more than one parameter—electrical power grids, for example, and nuclear generating plants, both of which became targets of chaos-inspired research in the 1980s. For one value of parameter A, parameter B might produce a reassuring, orderly kind of behavior, with coherent regions of stability. Engineers could make studies and graphs of exactly the kind their linear-oriented training suggested. Yet lurking nearby might be another value of parameter A that transforms the importance of parameter B.
Yorke would rise at conferences to display pictures of fractal basin boundaries. Some pictures represented the behavior of forced pendulums that could end up in one of two final states—the forced pendulum being, as his audiences well knew, a fundamental oscillator with many guises in everyday life. “Nobody can say that I’ve rigged the system by choosing a pendulum,” Yorke would say jovially. “This is the kind of thing you see throughout nature. But the behavior is different from anything you see in the literature. It’s fractal behavior of a wild kind.” The pictures would be fantastic swirls of white and black, as if a kitchen mixing bowl had sputtered a few times in the course of incompletely folding together vanilla and chocolate pudding. To make such pictures, his computer had swept through a 1,000 by 1,000 grid of points, each representing a different starting position for the pendulum, and had plotted the outcome: black or white. These were basins of attraction, mixed and folded by the familiar equations of Newtonian motion, and the result was more boundary than anything else. Typically, more than three-quarters of the plotted