Chaos - James Gleick [79]
WHEN FEIGENBAUM BEGAN to think about nonlinearity at Los Alamos, he realized that his education had taught him nothing useful. To solve a system of nonlinear differential equations was impossible, notwithstanding the special examples constructed in textbooks. Perturbative technique, making successive corrections to a solvable problem that one hoped would lie somewhere nearby the real one, seemed foolish. He read through texts on nonlinear flows and oscillations and decided that little existed to help a reasonable physicist. His computational equipment consisting solely of pencil and paper, Feigenbaum decided to start with an analogue of the simple equation that Robert May studied in the context of population biology.
It happened to be the equation high school students use in geometry to graph a parabola. It can be written as y = r(x-x2). Every value of x produces a value of y, and the resulting curve expresses the relation of the two numbers for the range of values. If x (this year’s population) is small, then y (next year’s) is small, but larger than x; the curve is rising steeply. If x is in the middle of the range, then y is large. But the parabola levels off and falls, so that if x is large, then y will be small again. That is what produces the equivalent of population crashes in ecological modeling, preventing unrealistic unrestrained growth.
For May and then Feigenbaum, the point was to use this simple calculation not once, but repeated endlessly as a feedback loop. The output of one calculation was fed back in as input for the next. To see what happened graphically, the parabola helped enormously. Pick a starting value along the x axis. Draw a line up to where it meets the parabola. Read the resulting value off the y axis. And start all over with the new value. The sequence bounces from place to place on the parabola at first, and then, perhaps, homes in on a stable equilibrium, where x and y are equal and the value thus does not change.
In spirit, nothing could have been further removed from the complex calculations of standard physics. Instead of a labyrinthine scheme to be solved one time, this was a simple calculation performed over and over again. The numerical experimenter would watch, like a chemist peering at a reaction bubbling away inside a beaker. Here the output was just a string of numbers, and it did not always converge to a steady final state. It could end up oscillating back and forth between two values. Or as May had explained to population biologists, it could keep on changing chaotically as long as anyone cared to watch. The choice among these different possible behaviors depended on the value of the tuning parameter.
Feigenbaum carried out numerical work of this faintly experimental sort and, at the same time, tried more traditional theoretical ways of analyzing nonlinear functions. Even so, he could not see the whole picture of what this equation could do. But he could see that the possibilities were already so complicated that they would be viciously hard to analyze. He also knew that three Los Alamos mathematicians—Nicholas Metropolis, Paul Stein, and Myron Stein—had studied such “maps” in 1971, and now Paul Stein warned him that the complexity was frightening indeed. If this simplest of equations already proved intractable, what about the far more complicated equations that a scientist would write down for real systems? Feigenbaum put the whole problem on the shelf.
In the brief history of chaos, this one innocent-looking equation provides the most succinct example of how different sorts of scientists looked at one problem in many different ways. To the biologists, it was an equation with a message: Simple systems can do complicated things. To Metropolis, Stein, and Stein, the problem was to catalogue