Dynamical Complexity of Strange Sets

Abstract

Systems governed by deterministic dynamics may exhibit unpredictable time evolution, thus appearing chaotic. Such phenomena are ubiquitous in nature, ranging from turbulent fluid flows to heart arrythmia. The unpredictability of chaotic systems arises from the abundancy of trajectories that exist for slight changes of the initial conditions. The entropy provides a measure of the multitude of possible time evolutions a system may exhibit, but does not provide a quantification of the ease or difficulty with which the set of all possible motions can be organized and encoded. Many of the proposed definitions [1] for dynamical complexity reduce to entropy related quantities such as the Kolmogorov entropy, and as such are measures of randomness. In this communication a measure of complexity unrelated to the entropy is introduced in order to quantify the difficulty in organizing the possible motions of a chaotic system. Other proposed topological definitions of complexity [2], such as the algorithmic complexity usually diverge for a generic chaotic system.