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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    • "Fig. 1 we present two biparametric plots showing the different behaviors based on the Lyapunov exponents computed using the algorithm of Wolf et al. [32]. The only difference in the simulations is that the lower picture is done considering a transient time 3 × 10 4 before computing the exponents. "
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    ABSTRACT: Chaotic behavior is a common feature of nonlinear dynamics, as well as hyperchaos in high-dimensional systems. In numerical simulations of these systems it is quite difficult to distinguish one from another behavior in some situations, as the results are frequently quite “noisy”. We show that in such systems a global hyperchaotic invariant set is present giving rise to long hyperchaotic transient behaviors. This fact provides a mechanism for these noisy results. The coexistence of chaos and hyperchaos is proved via Computer-Assisted Proofs techniques.
    Physics Letters A 10/2015; 379(38):2300-2305. DOI:10.1016/j.physleta.2015.07.035 · 1.68 Impact Factor
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    • "The main advantages of the proposed algorithm are its numerical stability and short computation time. In the considered example, the reduction of CPU time 2 is about 90% compared to standard time-domain techniques [2] [3]. "
    Dataset: 6087
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    • "where ¯ A i = ¯ A(i∆t) and p = T /∆t is the number of sampling points per period. Numerous algorithms [7] [13] "
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    ABSTRACT: The paper presents two approaches to the stability analysis of flexible dynamical systems in the time domain. The first is based on the partial Floquet theory and proceeds in three steps. A preprocessing step evaluates optimized signals based on the proper orthogonal decomposition method. Next, the system stability characteristics are obtained from partial Floquet theory through singular value decomposition. Finally, a postprocessing step assesses the accuracy of the identified stability characteristics. The Lyapunov characteristic exponent theory provides the theoretical background for the second approach. It is shown that the system stability characteristics are related to the Lyapunov characteristic exponent closely, for both constant and periodic coefficient systems. For the latter systems, an exponential approximation is proposed to evaluate the transition matrix. Numerical simulations show the proposed approaches are robust enough to deal with the stability analysis of flexible dynamical systems and the predictions of the two approaches are found to be in close agreement.
    Journal of Computational and Nonlinear Dynamics 09/2015; DOI:10.1115/1.4031675 · 1.11 Impact Factor
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