Chaos - James Gleick [85]
For an ecologist, the most obvious sort of function for population growth is linear—the Malthusian scenario of steady, limitless growth by a fixed percentage each year (left). More realistic functions formed an arch, sending the population back downward when it became too high. Illustrated is the “logistic map,” a perfect parabola, defined by the function y = rx(1–x), where the value of r, from 0 to 4, determines the parabola’s steepness. But Feigenbaum discovered that it did not matter precisely what sort of arch he used; the details of the equation were beside the point. What mattered was that the function should have a “hump.”
The behavior depended sensitively, though, on the steepness—the degree of nonlinearity, or what Robert May called “boom-and–bustiness.” Too shallow a function would produce extinction: Any starting population would lead eventually to zero. Increasing the steepness produced the steady equilibrium that a traditional ecologist would expect; that point, drawing in all trajectories, was a one-dimensional “attractor.”
Beyond a certain point, a bifurcation produced an oscillating population with period two. Then more period-doublings would occur, and finally (bottom right) the trajectory would refuse to settle down at all.
Such images were a starting point for Feigenbaum when he tried to construct a theory. He began thinking in terms of recursion: functions of functions, and functions of functions of functions, and so on; maps with two humps, and then four….
Numbers and functions were his object of study, instead of mesons and quarks. They had trajectories and orbits. He needed to inquire into their behavior. He needed—in a phrase that later became a cliché of the new science—to create intuition. His accelerator and his cloud chamber were the computer. Along with his theory, he was building a methodology. Ordinarily a computer user would construct a problem, feed it in, and wait for the machine to calculate its solution—one problem, one solution. Feigenbaum and the chaos researchers who followed needed more. They needed to do what Lorenz had done, to create miniature universes and observe their evolution. Then they could change this feature or that and observe the changed paths that would result. They were armed with the new conviction, after all, that tiny changes in certain features could lead to remarkable changes in overall behavior.
Feigenbaum quickly discovered how ill-suited the computer facilities of Los Alamos were for the style of computing he wanted to develop. Despite enormous resources, far greater than at most universities, Los Alamos had few terminals capable of displaying graphs and pictures, and those few were in the Weapons Division. Feigenbaum wanted to take numbers and plot them as points on a map. He had to resort to the most primitive method conceivable: long rolls of printout paper with lines made by printing rows of spaces followed by an asterisk or a plus sign. The official policy at Los Alamos held that one big computer was worth far more than many little computers—a policy that went with the one problem, one solution tradition. Little computers were discouraged. Furthermore, any division’s purchase of a computer would have to meet stringent government guidelines and a formal review. Only later, with the budgetary complicity of the Theoretical Division, did Feigenbaum become the recipient of a $20,000 “desktop calculator.” Then he could change his equations and pictures on the run, tweaking them and tuning them, playing the computer like a musical instrument. For now, the only terminals capable of serious graphics were in high-security areas—behind the fence, in local parlance. Feigenbaum had to use a terminal hooked up by telephone lines to a central computer. The reality of working in such an arrangement made it hard to appreciate the raw power of the computer at the other end of the line. Even the simplest tasks took minutes. To edit a line of a program meant pressing