8 - Sensitive dependence

Summary

When a difference equation is used to model a real world problem, the particular solution of interest is specified by an initial value, which is fed into the computer to start the iteration. The initial value, however, is usually known only approximately. Errors can arise from the limited precision of the measuring instruments and also from the limited precision to which a computer accepts numbers. There are thus two initial values to consider:the “true” initial value x₀ and the approximate value y₀, with which the computer begins its calculation of the iterates.

The scientist investigating the real world problem tries to ensure that the approximate initial value y0 is as close to the true initial value x0 as possible. To make valid long term predictions, the scientist would need to know, furthermore, that the two solutions stay close together over many iterations. It is now known, however, that even for very simple difference equations, the two solutions can diverge so rapidly that long term predictions are impossible.

This rapid divergence of solutions, which are close together initially, is called sensitive dependence on initial conditions or sensitive dependence or just simply sensitivity. When present in a dynamical system it makes long term predictions impossible and hence is regarded as one of the key features of ‘chaotic’ behaviour.

DIVERGING ITERATES

In this section we show how rapidly two sequences of iterates of the logistic map Q₄ can diverge from each other.