Chapter 4: Systems Based on Numbers

Section 7: Iterated Maps and the Chaos Phenomenon

History of iterated maps

Newton's method from the late 1600s for finding roots of polynomials (already used in specific cases in antiquity) can be thought of as a smooth iterated map (see page 920) in which a rational function is repeatedly applied (see page 1101). Questions of convergence led in the late 1800s and early 1900s to interest in iteration theory, particularly for rational functions in the complex plane (see page 933). There were occasional comments about complicated behavior (notably by Arthur Cayley in 1879) but no real investigation seems to have been made. In the 1890s Henri Poincaré studied so-called return maps giving for example positions of objects on successive orbits. Starting in the 1930s iterated maps were sometimes considered as possible models in fields like population biology and business cycle theory—usually arising as discrete annualized versions of continuous equations like the Verhulst logistic differential equation from the mid-1800s. In most cases the most that was noted was simple oscillatory behavior, although for example in 1954 William Ricker iterated empirical reproduction curves for fish, and saw more complex behavior—though made little comment on it. In the 1950s Paul Stein and Stanislaw Ulam did an extensive computer study of various iterated maps of nonlinear functions. They concentrated on questions of convergence, but nevertheless noted complicated behavior. (Already in the late 1940s John von Neumann had suggested using x 4x (1 - x) as a random number generator, commenting on its extraction of initial condition digits, as mentioned on page 921.) Some detailed analytical studies of logistic maps of the form x a x (1 - x) were done in the late 1950s and early 1960s—and in the mid-1970s iterated maps became popular, with much analysis and computer experimentation on them being done. But typically studies have concentrated on repetition, nesting and sensitive dependence on initial conditions—not on more general issues of complexity.

In connection with his study of continued fractions Carl Friedrich Gauss noted in 1799 complexity in the behavior of the iterated map x FractionalPart[1/x]. Beginning in the late 1800s there was number theoretical investigation of the sequence FractionalPart[an x] associated with the map x FractionalPart[a x] (see page 903), notably by G. H. Hardy and John Littlewood in 1914. Various features of randomness such as uniform distribution were established, and connections to smooth iterated maps emerged after the development of symbolic dynamics in the late 1930s.

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