Nonlinear Maps

Abstract

In the study of forced oscillator phenomena, we have avoided plunging into heavy analysis because generally the details can be gory and are probably soon forgotten by the student. The virtue of maps, and the logistic map in particular, is that they are amenable to relatively simple, easily understandable, analysis because they are governed by finite difference equations rather than nonlinear differential equations. Despite their relative simplicity, nonlinear maps can guide us along the road to understanding many of the features that are seen in forced nonlinear oscillator systems such as the period doubling route to chaos, the stretching and folding of strange attractors, and so on. The emphasis will be on understanding rather than trying to establish the direct connection of a given map with a particular nonlinear ODE which is a nontrivial task beyond the scope of this text. Some new concepts like bifurcation diagrams and Lyapunov exponents, which will be encountered in this chapter, could have been introduced in the last chapter but are more easily dealt with in the framework of nonlinear maps.

In all chaos there is a cosmos, in all disorder a secret order. Carl Jung (1875–1961), Swiss psychiatrist