Local Lyapunov exponents computed from observed data

Abarbanel, H. D. I.; Brown, Robert G.; Kennel, Matthew B.

doi:10.1007/bf01208929

Cited by 161 publications

(118 citation statements)

References 16 publications

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“…In continuation, it would be possible to apply methods of nonlinear time series analysis that yield invariant quantities of the system, such as for example Lyapunov exponents [40][41][42] or dimension estimates [43][44][45], in order to obtain deeper insights into the system dynamics. However, all calculated quantities would be meaningless if the studied time series did not originate from a deterministic stationary system.…”

Section: Resultsmentioning

confidence: 99%

Nonlinear time series analysis of the human electrocardiogram

Perc

2005

Eur. J. Phys.

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We analyse the human electrocardiogram with simple nonlinear time series analysis methods that are appropriate for graduate as well as undergraduate courses. In particular, attention is devoted to the notions of determinism and stationarity in physiological data. We emphasize that methods of nonlinear time series analysis can be successfully applied only if the studied data set originates from a deterministic stationary system. After positively establishing the presence of determinism and stationarity in the studied electrocardiogram, we calculate the maximal Lyapunov exponent, thus providing interesting insights into the dynamics of the human heart. Moreover, to facilitate interest and enable the integration of nonlinear time series analysis methods into the curriculum at an early stage of the educational process, we also provide user-friendly programs for each implemented method.

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Section: Resultsmentioning

confidence: 99%

Nonlinear time series analysis of the human electrocardiogram

Perc

2005

Eur. J. Phys.

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“…(5) and the observation function is the same as Eq. (6). From this we obtain the same embedding equations as (7) and (8).…”

Section: Spurious Lyapunov Exponentsmentioning

confidence: 67%

“…The search for an algorithm to calculate Lyapunov exponents with desirable finite sample properties has gained momentum in the last few years. Abarbanel et al [4][5][6], Ellner et al [7], McCaffrey et al [8], Gençay and Dechert [9] and Dechert and Gençay [10] came up with improved algorithms for the calculation of the Lyapunov exponents from observed data. Gençay [11] worked on the calculation of the Lyapunov exponents with noisy data when feedforward networks were used as the estimation technique.…”

Section: Introductionmentioning

confidence: 99%

Is the largest Lyapunov exponent preserved in embedded dynamics?

Dechert

Gençay

2000

Physics Letters A

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The method of reconstruction for an n-dimensional system from observations is to form vectors of m consecutive observations, which for m > 2n, is generically an embedding. This is Takens' result. Our analytical examples show that it is possible to obtain spurious Lyapunov exponents that are even larger than the largest Lyapunov exponent of the original system. Therefore, we present examples where the largest Lyapunov exponent may not be preserved under Takens' embedding theorem.

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“…Hurst exponents were determined as explained in Hurst (1951). The embedding dimension (and separation values) were determined according to the average auto-mutual information method described by Kennel et al (I 992) (see also Kennel 1992;Abarbanel et al, 1992Abarbanel et al, , 1993. The tit to the non-linear predictor was obtained as follows, using a methodology derived from the work of Sughara and May (1990).…”

Section: Resultsmentioning

confidence: 99%

Oscillatory, stochastic and chaotic growth rate fluctuations in permittistatically controlled yeast cultures

Davey

Woodward

et al. 1996

Biosystems

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We describe a continuous culture syste,m related to the turbidostat, but using a feedback system based on biomass estimation from the dielectric permittivity of the cell suspension rather than its optical density. It is shown that this system provides an excellent method of maintaining a constant biomass level within a fermentor. The computercontrolled system was able to effect the essentially continuous registration of growth rate by monitoring the rate of medium addition via the time-dependent activity of the pump. At some biomass setpoints for aerobically grown cultures of baker's yeast substantial time-dependent fluctuations in the growth rate of the culture were thereby observed. At some biomass setpoints, however, or under anaerobic conditions, or when using a nonCrabtree yeast, the growth rate was constant, indicating that the fluctuations were inherent to the biological system and not simply a property of the fermentor and control system. A variety of time series analyses (Fourier transformations, Hurst and Lyapunov exponents, the determination of embedding dimension, and non-linear time series predictions based on the methodology of Sugihara and May) were used to demonstrate, for the first time, that as well as stochastic and periodic components these fluctuations exhibited deterministic chaos. 'Trivial predictors' were unable to give accurate predictions of the growth rate in these cultures. The growth rate fluctuations were studied further by means of offline measummenta of changes in percentage viability, bud count, and in the external ethanol and glucose concentrations; these data and other evidence suggested that the growth rate fluctuations were closely linked to the primary respirofermentative metabolism of this organism. The identification of chaotic growth rates in cell cultures suggests that there may be novel methods for controlling the growth of such cultures.

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Local Lyapunov exponents computed from observed data

Cited by 161 publications

References 16 publications

Nonlinear time series analysis of the human electrocardiogram

Nonlinear time series analysis of the human electrocardiogram

Is the largest Lyapunov exponent preserved in embedded dynamics?

Oscillatory, stochastic and chaotic growth rate fluctuations in permittistatically controlled yeast cultures

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