Chaos - James Gleick [82]
Metaphorically—but only metaphorically—he knew that this region was like the mysterious boundary between smooth flow and turbulence in a fluid. It was the region that Robert May had called to the attention of population biologists who had previously failed to notice the possibility of any but orderly cycles in changing animal populations. En route to chaos in this region was a cascade of period-doublings, the splitting of two-cycles into four-cycles, four-cycles into eight-cycles, and so on. These splittings made a a fascinating pattern. They were the points at which a slight change in fecundity, for example, might lead a population of gypsy moths to change from a four-year cycle to an eight-year cycle. Feigenbaum decided to begin by calculating the exact parameter values that produced the splittings.
In the end, it was the slowness of the calculator that led him to a discovery that August. It took ages—minutes, in fact—to calculate the exact parameter value of each period-doubling. The higher up the chain he went, the longer it took. With a fast computer, and with a printout, Feigenbaum might have observed no pattern. But he had to write the numbers down by hand, and then he had to think about them while he was waiting, and then, to save time, he had to guess where the next answer would be.
Yet all in an instant he saw that he did not have to guess. There was an unexpected regularity hidden in this system: the numbers were converging geometrically, the way a line of identical telephone poles converges toward the horizon in a perspective drawing. If you know how big to make any two telephone poles, you know all the rest; the ratio of the second to the first will also be the ratio of the third to the second, and so on. The period-doublings were not just coming faster and faster, but they were coming faster and faster at a constant rate.
Why should this be so? Ordinarily, the presence of geometric convergence suggests that something, somewhere, is repeating itself on different scales. But if there was a scaling pattern inside this equation, no one had ever seen it. Feigenbaum calculated the ratio of convergence to the finest precision possible on his machine—three decimal places—and came up with a number, 4.669. Did this particular ratio mean anything? Feigenbaum did what anyone would do who cared about numbers. He spent the rest of the day trying to fit the number to all the standard constants—π, e, and so forth. It was a variant of none.
Oddly, Robert May realized later that he, too, had seen this geometric convergence. But he forgot it as quickly as he noted it. From May’s perspective in ecology, it was a numerical peculiarity and nothing more. In the real-world systems he was considering, systems of animal populations or even economic models, the inevitable noise would overwhelm any detail that precise. The very messiness that had led him so far stopped him at the crucial point. May was excited by the gross behavior of the equation. He never imagined that the numerical details would prove important.
Feigenbaum knew what he had, because geometric convergence meant that something in this equation was scaling, and he knew that scaling was important. All of renormalization theory depended on it. In an apparently unruly system, scaling meant that some quality was being preserved while everything else changed. Some regularity lay beneath the turbulent surface of the equation. But where? It was hard to see what to do next.
Summer turns rapidly to autumn in the rarefied Los Alamos air, and October had nearly ended when Feigenbaum was struck by an odd thought. He knew that Metropolis, Stein, and Stein had looked at other equations as well and had found that certain patterns carried over from one sort of function to another. The same combinations of R’s and L’s appeared,