Chaos - James Gleick [84]
Although the connection between numerics and physics was faint, Feigenbaum had found evidence that he needed to work out a new way of calculating complex nonlinear problems. So far, all available techniques had depended on the details of the functions. If the function was a sine function, Feigenbaum’s carefully worked-out calculations were sine calculations. His discovery of universality meant that all those techniques would have to be thrown out. The regularity had nothing to do with sines. It had nothing to do with parabolas. It had nothing to do with any particular function. But why? It was frustrating. Nature had pulled back a curtain for an instant and offered a glimpse of unexpected order. What else was behind that curtain?
WHEN INSPIRATION CAME, it was in the form of a picture, a mental image of two small wavy forms and one big one. That was all—a bright, sharp image etched in his mind, no more, perhaps, than the visible top of a vast iceberg of mental processing that had taken place below the waterline of consciousness. It had to do with scaling, and it gave Feigenbaum the path he needed.
He was studying attractors. The steady equilibrium reached by his mappings is a fixed point that attracts all others—no matter what the starting “population,” it will bounce steadily in toward the attractor. Then, with the first period-doubling, the attractor splits in two, like a dividing cell. At first, these two points are practically together; then, as the parameter rises, they float apart. Then another period-doubling: each point of the attractor divides again, at the same moment. Feigenbaum’s number let him predict when the period-doublings would occur. Now he discovered that he could also predict the precise values of each point on this ever-more–complicated attractor—two points, four points, eight points…He could predict the actual populations reached in the year-to–year oscillations. There was yet another geometric convergence. These numbers, too, obeyed a law of scaling.
Feigenbaum was exploring a forgotten middle ground be tween mathematics and physics. His work was hard to classify. It was not mathematics; he was not proving anything. He was studying numbers, yes, but numbers are to a mathematician what bags of coins are to an investment banker: nominally the stuff of his profession, but actually too gritty and particular to waste time on. Ideas are the real currency of mathematicians. Feigenbaum was carrying out a program in physics, and, strange as it seemed, it was almost a kind of experimental physics.
ZEROING IN ON CHAOS. A simple equation, repeated many times over: Mitchell Feigenbaum focused on straightforward functions, taking one number as input and producing another as output. For animal populations, a function might express the relationship between this year’s population and next year’s.
One way to visualize such functions is to make a graph, plotting input on the horizontal axis and output on the vertical axis. For each possible input, x, there is just one output, y, and these form a shape represented by the heavy line.
Then, to represent the long-term behavior of the system, Feigenbaum drew a trajectory that started with some arbitrary x. Because each y was then fed back into the same function as new input, he could use a