Computing the Lyapunov spectrum of a dynamical system from an observed time series

Brown, Robert G.; Bryant, Paul H.; Abarbanel, Henry D. I.

doi:10.1103/physreva.43.2787

Cited by 330 publications

(175 citation statements)

References 25 publications

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“…This conclusion was solely based on the resulting positive largest Lyapunov exponent using Lyapunov spectrum. 61 However, studies show that Lyapunov spectrum is not an appropriate method to estimate LLE and it cannot prevent spurious Lyapunov exponents. 58,59 There are numerous instances in the literature that lists the shortcomings of Lyapunov exponents.…”

Section: Introductionmentioning

confidence: 99%

On the dynamics of ocean ambient noise: Two decades later

Siddagangaiah

Guo

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Two decades ago, it was shown that ambient noise exhibits low dimensional chaotic behavior. Recent new techniques in nonlinear science can effectively detect the underlying dynamics in noisy time series. In this paper, the presence of low dimensional deterministic dynamics in ambient noise is investigated using diverse nonlinear techniques, including correlation dimension, Lyapunov exponent, nonlinear prediction, and entropy based methods. The consistent interpretation of different methods demonstrates that ambient noise can be best modeled as nonlinear stochastic dynamics, thus rejecting the hypothesis of low dimensional chaotic behavior. The ambient noise data utilized in this study are of duration 60 s measured at South China Sea.

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Section: Introductionmentioning

confidence: 99%

On the dynamics of ocean ambient noise: Two decades later

Siddagangaiah

Guo

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…However, noise is inevitably present in experimental and natural systems. The effect of noise on the estimation of the linear Lyapunov exponent spectrum has been noted in previous studies (e.g., Wolf et al, 1985;Brown et al, 1991). It is of interest and importance to explore the effects of noise on the calculation of the NLLE spectrum.…”

Section: Resultsmentioning

confidence: 99%

Determining the spectrum of the nonlinear local Lyapunov exponents in a multidimensional chaotic system

Ding

2017

Adv. Atmos. Sci.

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For an n-dimensional chaotic system, we extend the definition of the nonlinear local Lyapunov exponent (NLLE) from one-to n-dimensional spectra, and present a method for computing the NLLE spectrum. The method is tested on three chaotic systems with different complexity. The results indicate that the NLLE spectrum realistically characterizes the growth rates of initial error vectors along different directions from the linear to nonlinear phases of error growth. This represents an improvement over the traditional Lyapunov exponent spectrum, which only characterizes the error growth rates during the linear phase of error growth. In addition, because the NLLE spectrum can effectively separate the slowly and rapidly growing perturbations, it is shown to be more suitable for estimating the predictability of chaotic systems, as compared to the traditional Lyapunov exponent spectrum.Key words: Lyapunov exponent, nonlinear local Lyapunov exponent, predictability Citation: Ding, R. Q., J. P. Li, and B. S. Li, 2017: Determining the spectrum of the nonlinear local Lyapunov exponents in a multidimensional chaotic system.

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“…Therefore, we concentrate on quantifying the state-dependent short-term predictability through finite-time growth rates. Note that these differ from the finite-time Lyapunov exponents defined in [Lorenz, 1965;Abarbanel et al, 1991;Brown et al, 1991;Yoden & Nomura, 1993;Ziehmann et al, 1999] as well as the finitetime growth rates of [Nese, 1989], both of which require the knowledge of the tangent maps thus the equations governing the dynamics. In [Ziehmann et al, 1999] we have discussed the significant differences between these quantities.…”

Section: Finite-time Growth Ratesmentioning

confidence: 99%

Nonlinear Methods of Cardiovascular Physics and Their Clinical Applicability

Wessel

Malberg²,

Bauernschmitt

et al. 2007

Int. J. Bifurcation Chaos

102

107

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In this tutorial we present recently developed nonlinear methods of cardiovascular physics and show their potentials to clinically relevant problems in cardiology. The first part describes methods of cardiovascular physics, especially data analysis and modeling of noninvasively measured biosignals, with the aim to improve clinical diagnostics and to improve the understanding of cardiovascular regulation. Applications of nonlinear data analysis and modeling tools are various and outlined in the second part of this tutorial: monitoring-, diagnosis-, course and mortality prognoses as well as early detection of heart diseases. We show, that these data analyses and modeling methods lead to significant improvements in different medical fields.Keywords: Cardiovascular physics; nonlinear data analysis and modeling; heart rate and blood pressure variability; baroreflex sensitivity. List of Abbreviations

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Computing the Lyapunov spectrum of a dynamical system from an observed time series

Cited by 330 publications

References 25 publications

On the dynamics of ocean ambient noise: Two decades later

On the dynamics of ocean ambient noise: Two decades later

Determining the spectrum of the nonlinear local Lyapunov exponents in a multidimensional chaotic system

Nonlinear Methods of Cardiovascular Physics and Their Clinical Applicability

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