The aim of this chapter is to provide an easy-to-use test for showing that a mapping f : [0, 1] → [0, 1] has chaotic behaviour. The test is applicable to symmetric one-hump mappings which are differentiable three times and involves the concept of the Schwarzian derivative of f.

In earlier chapters we approached the idea of chaotic behaviour with the aid of the concept of wiggly iterates. Hence we begin this chapter by investigating the behaviour of some mappings which do not have wiggly iterates. This leads to the concept of a fat wiggle or a woggle.

Schwarzian derivatives are then introduced as a device to test for the absence of woggles.

WIGGLES AND WOGGLES

If f : [0, 1] → [0, 1] is a one-hump mapping, then its nth iterate will consist of 2n-1 humps. Each hump has the interval between two consecutive zeroes of fn as its base. As n → ∞ the number of wiggles in the nth iterate also approaches infinity.

How can such a mapping fail to have wiggly iterates? The answer to this question is suggested by looking at the computer plots of some examples.

Failure to wiggle

Graphs of two examples of one-hump mappings which do not have wiggly iterates are shown in Figure 10.1.1. Each mapping was obtained by modifying the formula for the logistic mapping Q4.