Chapter 4: Systems Based on Numbers

Section 7: Iterated Maps and the Chaos Phenomenon

Smooth iterated maps

In the main text, all the functions used as mappings consist of linear pieces, usually joined together discontinuously. But the same basic phenomena seen with such mappings also occur when smooth functions are used. A particularly well-studied example (see page 918) is the so-called logistic map x -> a x (1 - x). The base 2 digit sequences obtained with this map starting from x = 1/8 are shown below for various values of a. The quadratic nature of the map typically causes the total number of digits to double at each step. But at least for small a, progressively more digits on the left show purely repetitive behavior. As a increases, the repetition period goes through a series of doublings. The detailed behavior is different for every value of a, but whenever the repetition period is 2^j, it turns out that with any initial condition the leftmost digit always eventually follows a sequence that consists of repetitions of step j in the evolution of the substitution system {1->{1, 0}, 0->{1, 1}} starting either from {0} or {1}. As a approaches 3.569946, the period doublings get closer and closer together, and eventually a point is reached at which the sequence of leftmost digits is no longer repetitive but instead corresponds to the nested pattern formed after an infinite number of steps in the evolution of the substitution system. (An important result discovered by Mitchell Feigenbaum in 1975 is that this basic setup is universal to all smooth maps whose functions have a single hump.) When a is increased further, there is usually no longer repetitive or nested behavior. And although there are typically some constraints, the behavior obtained tends to depend on the details of the digit sequence of the initial conditions. In the special case a = 4, it turns out that replacing x by Sin[π u]^2 makes the mapping become just u-> FractionalPart[2 u], revealing simple shift map dependence on the initial digit sequence. (See pages 1090 and 1098.)

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