One-Parameter Families of Maps

Abstract

The iterations of maps of an interval into itself certainly present one of the easiest models or examples of nonlinear (dissipative) dynamical systems. In fact, iterations of the form ${\rm x}_{{\rm n}} \rightarrow {\rm x}_{{\rm n + 1}} = {\rm f}({\rm x}_{{\rm n}})$, where f maps [-1,1] into itself, can be viewed as a discrete time version of a continuous dynamical system as we shall see below. Here, n plays the role of the time variable. Such iterations have been advocate with more or less success as models for biological, chemical or physical systems. We shall take here a more general point of view by asking questions which are not related to any specific map, but whose answer is, to some extent, independentof the particular map.