Ergodic theory of chaos and strange attractors

Abstract

Summary form only given, as follows. Deterministic chaos arises in a variety of nonlinear dynamical systems in physics, and in particular in optics. One has now gained a reasonable understanding of the onset of chaos in terms of the geometry of bifurcations and strange attractors. This geometric approach does not work for attractors of more than two or three dimensions. For these, however, ergodic theory provides new concepts: characteristic exponents, entropy, information dimension, which are reproducibly estimated from physical experiments. The characteristic exponents measure the rate of divergence by nearby trajectories of a dynamical system, the entropy measures the rate of information creation by the system, and the information dimension is a fractal dimension of particular interest. There are inequalities (or even identities) relating the entropy and information dimension to the characteristic exponents. The experimental measure of these ergodic quantities provides a numerical estimate of the instability of chaotic systems, and of the number of 'degrees of freedom' which they possess