Ergodic theory of chaos and strange attractors

TLDR

Deterministic noise, or chaos, is ubiquitous, yet while the onset of chaos is well understood, moderately excited chaotic systems require new tools, which ergodic theory supplies. This review aims to present the main mathematical ideas and their concrete implementation for analyzing chaotic experiments. The authors discuss dimensions, entropy, and characteristic exponents, and their interrelations and experimental determination, as key tools for analyzing chaotic systems. The systematic investigation of these quantities has enabled a reasonable understanding of dynamical systems beyond quasiperiodic regimes and represents a step toward comprehending highly turbulent fluids.

Abstract

Physical and numerical experiments show that deterministic noise, or chaos, is ubiquitous. While a good understanding of the onset of chaos has been achieved, using as a mathematical tool the geometric theory of differentiable dynamical systems, moderately excited chaotic systems require new tools, which are provided by the ergodic theory of dynamical systems. This theory has reached a stage where fruitful contact and exchange with physical experiments has become widespread. The present review is an account of the main mathematical ideas and their concrete implementation in analyzing experiments. The main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions). The relations between these quantities, as well as their experimental determination, are discussed. The systematic investigation of these quantities provides us for the first time with a reasonable understanding of dynamical systems, excited well beyond the quasiperiodic regimes. This is another step towards understanding highly turbulent fluids.