Estimating Lyapunov Exponents from Time Series

Abstract

Lyapunov exponents are important statistics for quantifying stability and deterministic chaos in dynamical systems. In this review article, we first revisit the computation of the Lyapunov spectrum using model equations. Then, employing state space reconstruction (delay coordinates), two approaches for estimating Lyapunov exponents from time series are presented: methods based on approximations of Jacobian matrices of the reconstructed flow and so-called direct methods evaluating the evolution of the distances of neighbouring orbits. Most direct methods estimate the largest Lyapunov exponent, only, but as an advantage they give graphical feedback to the user to confirm exponential divergence. This feedback provides valuable information concerning the validity and accuracy of the estimation results. Therefore, we focus on this type of algorithms for estimating Lyapunov exponents from time series and illustrate its features by the (iterated) Henon map, the hyper chaotic folded-towel map, the well known chaotic Lorenz-63 system, and a time continuous 6-dimensional Lorenz-96 model. These examples show that the largest Lyapunov exponent from a time series of a low-dimensional chaotic system can be successfully estimated using direct methods. With increasing attractor dimension, however, much longer time series are required and it turns out to be crucial to take into account only those neighbouring trajectory segments in delay coordinates space which are located sufficiently close together.