Determining Lyapunov Exponents From a Time Series

Department of Physics, University of Texas, Austin, Texas 78712, USA
Physica D Nonlinear Phenomena (Impact Factor: 1.64). 07/1985; 16(3):285-317. DOI: 10.1016/0167-2789(85)90011-9


We present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov exponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems: we monitor the long-term growth rate of small volume elements in an attractor. The method is tested on model systems with known Lyapunov spectra, and applied to data for the Belousov-Zhabotinskii reaction and Couette-Taylor flow.

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    • "Additionally, multivariate techniques and nonlinear methods have been developed in recent years. The independent component analysis (ICA)[22,23], the principal component analysis (PCA)[24,25]and the nonegative matrix factorization (NMF)[26]have been widely used for the EMG channel minimization272829and classification2930313233; the correlation dimension[34], the maximum Lyapunov Exponent[35], the complexity[36]and the information entropy[37]have been applied into the nonlinear analysis of the EMG signals. Especially, the exrtensive entropy-based measures, such as the Kolmogorov–Sinai (K-S) entropy[38], the approximate entropy (ApEn)[39,40]and the sample entropy (SampEn)[41,42], have achieved many applications in bioelectricity signals processing. "

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    • "In this study, the Wolf method is used to predict the traffic flow[18]. In the method, Y j in phase space will be assumed as the predicted center point, and Y k is the closest point with a distance d j (0) from Y j . "

    Preview · Article · Jan 2016 · Entropy
    • "Lyapunov exponents are used to measure the rate at which nearby trajectories of a dynamical system diverge (see for example[60]),[16]). As a dynamic dissipative system is chaotic if its biggest Lyapunov's exponent is a positive number ([43]), we have adopted the Wolf algorithm (see[67],[66]) to calculate the biggest Lyapunov exponent. In our simulations, the calculated value has always been positive (see Table 1).The entropy K of Kolmogorov-Sinai (see[18],[58]) has been added to supplement the above mentioned analysis. "
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    ABSTRACT: This paper, following Kaldor’s approach, is written with the intention of interpreting fluctuations of economic systems (i.e trade cycles). In particular, a new discretized Kaldor model is proposed, which is also useful to explain what appears to be random and unpredictable, such as economic shocks. Moreover, by using numerical analysis, the chaoticity of the model is demonstrated.
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