A robust method on estimation of Lyapunov exponents from a noisy time series

Lyapunov exponents can indicate the asymptotic behaviors of nonlinear systems, and thus can be used for stability analysis. However, it is notoriously difficult to estimate these exponents reliably from experimental data due to the measurement error (noise). In this paper, a novel method for estimating Lyapunov exponents from a time series in the presence of additive noise corruption is presented. The method combines the ideas of averaging the noisy data to form new neighbors and of nonlinear mapping to determine neighborhood mapping matrices. Two case studies of balancing control of a bipedal robot and the Lorenz systems are presented to demonstrate the efficacy of the proposed method. The bipedal robot system has two negative Lyapunov exponents while the Lorenz system has one positive, zero, and negative exponents, respectively. It is shown that, as compared with the existing methods, our proposed one is more robust to the ratio of signal to noise, and is particularly effective in estimating negative Lyapunov exponents. We believe that the work can contribute significantly to the stability analysis of nonlinear systems using a noisy time series.