Estimation of Lyapunov exponents from a time series for n-dimensional state space using nonlinear mapping

Abstract

It has been demonstrated that when estimating Lyapunov exponents using a time series, nonlinear mapping used for characterizing the evolution of the neighbors leads to more accurate negative exponents and is more robust to noise in the times series. However, the number of unknown elements of the matrices associated with nonlinear mapping increases significantly with the embedding dimensions of the state space where the dynamics is reconstructed. Such unknown coefficients are solved from a set of linear algebraic equations based on the least square-root fit method. Derivation of such linear equations and computer programming are tedious and error prone especially for the systems with high embedding dimensions. In this work, we develop a general form of the linear algebraic equations and the corresponding computer program in terms of arbitrary embedding dimensions. A stable robotic system with all negative Lyapunov exponents and the Lorenz system with positive, zero, and negative exponents are used to demonstrate the efficacy of the proposed method. The work can contribute significantly to estimating Lyapunov exponents for systems with large embedding dimensions.