Sensitive Dependence on Parameters in Nonlinear Dynamics

Abstract:

Two qualitatively different types of dynamical behaviour can be so tightly interwoven that it becomes impossible to predict when a small change in parameters will cause a change in qualitative properties. For Quadratic mappings of the interval, for example, the chaotic parameter vales form a Cantor set of positive measure, broken up by periodic intervals. This set can be described by a global scaling law, which makes it possible to form a good estimate of the fraction of chaotic parameter values. Sensitive dependence on parameters occurs when the scaling exponent is conjectures to display universal behaviour.