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

Abstract

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.

Computation of the Lyapunov exponents in the compass-gait model under OGY control via a hybrid Poincaré map

“…Here, the controlled hybrid Poincaré map (21)-(22) is three-dimensional compared with that defined by (6)- (7). Let z − * be the one-periodic fixed point of the reduced controlled hybrid Poincaré map (21).…”

“…We stress that calculation of the two terms S 3 (τ k ) and S 4 (τ k ) must be done also by means of the 4-dimensional controlled hybrid Poincaré map (6)- (7). Indeed, as S(z −…”

“…Generally, the Lyapunov exponents are used for chaos identification in nonlinear dynamical systems and as a powerful tool for analyzing the stability of nonlinear dynamic systems, especially when the mathematical models of the systems are available. Many research studies focused on the design of numerical methods to estimate the spectrum of Lyapunov exponents or the largest one from time series or data sets [3][4][5][6][7][8][9][10][11][12][13][14][15]. In addition, several papers dealt with the design of numerical/analytical methods for the computation of the spectrum of Lyapunov exponents.…”

Lyapunov-Exponent Spectrum From Noisy Time Series

“…Yao et al [2012] proposed a novel method for estimating simultaneously the largest Lyapunov exponent and noise level based on the invariant of the largest Lyapunov exponent. Yang and Wu [2011] presented an averaging algorithm based on the BBA method to estimate Lyapunov exponents from a time series in the presence of measurement noise. But because the BBA method is based on the mappings from small displacements into small displacements and the displacements in the BBA method are all from the same central point, any noise on that central point is never addressed if the averaging method is directly based on the BBA method as in the work of Yang and Wu [2011].…”

“…Yang and Wu [2011] presented an averaging algorithm based on the BBA method to estimate Lyapunov exponents from a time series in the presence of measurement noise. But because the BBA method is based on the mappings from small displacements into small displacements and the displacements in the BBA method are all from the same central point, any noise on that central point is never addressed if the averaging method is directly based on the BBA method as in the work of Yang and Wu [2011]. It is well known that, trajectory tracing method usually can only estimate the largest Lyapunov exponent, and local linear mappings method is not reliable for estimating negative exponents due to the local data-set curvature [Brown et al, 1991].…”