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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    • "Methods derived from nonlinear dynamics explore data dynamics by investigating the strange attractor in phasespace [34] based on attractor invariants and some useful diagrams. Popular techniques include the Lyapunov exponent [35], fractal dimension [36], Kolmogorov entropy [37], phase portrait [38], Poincare section diagram [39] and RP [40]. Entropy, a thermodynamic quantity, describes the disorder state of the data dynamics [41], which mainly includes SE [42], WE [43], PE [44], ApEn [45], SampEn [46], and FuzzyEn [47]. "
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    • "To that aim, the Russian mathematician Alexandr Lyapunov introduced definitions and criteria to establishes unambiguously chaotic, periodic, quasi-periodic or stable behavior by studying the linearization of the equations of motion to determine the behavior of any system, through the now so-called Lyapunov exponents. The signs and the values of the Lyapunov exponents allow us to determine the qualitative and quantitative patterns of behavior of any system [30]. The Lyapunov exponents have been proved useful for determining and distinguishing the various types of orbits and behavior of our system. "
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