Sensitive dependence on parameters in nonlinear dynamics

Abstract

Two qualitatively different types of dynamical behavior can be so tightly interwoven that it becomes impossible predict when a small change in parameters will cause a change in qualitative properties. For quadratic mappings of the interval, for example, the chaotic to parameter values 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 ββ is conjectured to display universal behavior.