Chapter 7: Mechanisms in Programs and Nature

Section 4: Chaos Theory and Randomness from Initial Conditions

History of chaos theory

The idea that small causes can sometimes have large effects has been noted by historians and others since antiquity, and captured for example in "for want of a nail ... a kingdom was lost". In 1860 James Clerk Maxwell discussed how collisions between hard sphere molecules could lead to progressive amplification of small changes and yield microscopic randomness in gases. In the 1870s Maxwell also suggested that mechanical instability and amplification of infinitely small changes at occasional critical points might explain apparent free will (see page 1135). (It was already fairly well understood that for example small changes could determine which way a beam would buckle.) In 1890 Henri Poincaré found sensitive dependence on initial conditions in a particular case of the three-body problem (see below), and later proposed that such phenomena could be common, say in meteorology. In 1898 Jacques Hadamard noted general divergence of trajectories in spaces of negative curvature, and Pierre Duhem discussed the possible general significance of this in 1908. In the 1800s there had been work on nonlinear oscillators—particularly in connection with models of musical instruments—and in 1927 Balthazar van der Pol noted occasional "noisy" behavior in a vacuum tube oscillator circuit presumably governed by a simple nonlinear differential equation. By the 1930s the field of dynamical systems theory had begun to provide characterizations of possible forms of behavior in differential equations. And in the early 1940s Mary Cartwright and John Littlewood noted that van der Pol's equation could exhibit solutions somehow sensitive to all digits in its initial conditions. The iterated map x->4x(1-x) was also known to have a similar property (see page 918). But most investigations centered on simple and usually repetitive behavior—with any strange behavior implicitly assumed to be infinitely unlikely. In 1962, however, Edward Lorenz did a computer simulation of a set of simplified differential equations for fluid convection (see page 998) in which he saw complicated behavior that seemed to depend sensitively on initial conditions—in a way that he suggested was like the map x->FractionalPart[2x]. In the mid-1960s, notably through the work of Steve Smale, proofs were given that there could be differential equations in which such sensitivity is generic. In the late 1960s there began to be all sorts of simulations of differential equations with complicated behavior, first mainly on analog computers, and later on digital computers. Then in the mid-1970s, particularly following discussion by Robert May, studies of iterated maps with sensitive dependence on initial conditions became common. Work by Robert Shaw in the late 1970s clarified connections between information content of initial conditions and apparent randomness of behavior. The term "chaos" had been used since antiquity to describe various forms of randomness, but in the late 1970s it became specifically tied to the phenomenon of sensitive dependence on initial conditions. By the early 1980s at least indirect signs of chaos in this sense (see note below) had been seen in all sorts of mechanical, electrical, fluid and other systems, and there emerged a widespread conviction that such chaos must be the source of all important randomness in nature. So in 1985 when I raised the possibility that intrinsic randomness might instead be a key phenomenon this was greeted with much hostility by some younger proponents of chaos theory. Insofar as what they had to say was of a scientific nature, their main point was that somehow what I had seen in cellular automata must be specific to discrete systems, and would not occur in the continuous systems assumed to be relevant in nature. But from many results in this book it is now clear that this is not correct. (Note that James Gleick's 1987 popular book Chaos covers somewhat more than is usually considered chaos theory—including some of my results on cellular automata from the early 1980s.)

From Stephen Wolfram: A New Kind of Science [citation]