IDENTIFICATION OF TRUE AND SPURIOUS LYAPUNOV EXPONENTS FROM TIME SERIES

Abstract

A new method for the identification of true and spurious Lyapunov exponents computed from time series is presented. It is based on the observation that the true Lyapunov exponents change their signs upon time reversal whereas the spurious exponents do not. Furthermore by comparison of the spectra of the original data and the reversed time series suitable values for the free parameters of the algorithm used for computing the Lyapunov exponents (e.g., the number of nearest neighbors) are determined. As an example for this general approach an algorithm using local nonlinear approximations of the flow map in embedding space by radial basis functions is presented. For noisy data a regularization method is applied in order to get smooth approximating functions. Numerical examples based on data from the Hénon map, a four-dimensional analog of the Hénon map, a quasiperiodic time series, the Lorenz model, and Duffing’s equation are given.