LYAPUNOV EXPONENTS IN CHAOTIC SYSTEMS: THEIR IMPORTANCE AND THEIR EVALUATION USING OBSERVED DATA

Abstract

We review the idea of Lyapunov exponents for chaotic systems and discuss their evaluation from observed data alone. These exponents govern the growth or decrease of small perturbations to orbits of a dynamical system. They are critical to the predictability of models made from observations as well as known analytic models. The Lyapunov exponents are invariants of the dynamical system and are connected with the dimension of the system attractor and to the idea of information generation by the system dynamics. Lyapunov exponents are among the many ways we can classify observed nonlinear systems, and their appeal to physicists remains their clear interpretation in terms of system stability and predictability. We discuss the familiar global Lyapunov exponents which govern the evolution of perturbations for long times and local Lyapunov exponents which determine the predictability over a finite number of time steps.