Determining Lyapunov Exponents From a Time Series

Department of Physics, University of Texas, Austin, Texas 78712, USA
Physica D Nonlinear Phenomena (Impact Factor: 1.83). 07/1985; 16(3):285-317. DOI: 10.1016/0167-2789(85)90011-9

ABSTRACT 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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Available from: Alan Wolf, Aug 30, 2015
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    • "Notably, although most EEG data are now digital, and numerous protocols for automated seizure detection are available [2-16], the EEG is still largely analyzed by visual inspection. Here, we present a novel method for automated detection of seizure and epileptic interictal discharge based on the maximal short-term Lyapunov exponent (STLmax), a measure of dynamic system instability which has been extensively used in EEG analysis [3] [17] Previous studies show nonstationarity and chaotic nature of EEG data, and thus justify a measure of entropy such the STLmax [3] [8] [9] [10]. "
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    ABSTRACT: Over the past few decades, application of neural networks and chaos theory to electroencephalogram (EEG) analysis has grown rapidly due to the complex and nonlinear nature of EEG data. We report a novel method for epileptic seizure detection that is depending on the maximal short-term Lyapunov exponent (STLmax). The proposed approach is based on the automatic segmentation of the EEG into time segments that correspond to epileptic and non-epileptic activity. The STL-max is then computed from both categories of EEG signal and used for classification of epileptic and non-epileptic EEG segments throughout the recording. Neural network techniques are proposed both for segmentation of EEG signals and computation of STLmax. The data set from hospital have been used for experiments performing. It consists of 21 records during 8 seconds of eight adult patients. Furthermore the publicly available data were used for experiments. The main advantages of presented neural technique is its ability to detect rapidly the small EEG time segments as the epileptic or non-epileptic activity, training without desired data set about epileptic and non-epileptic activity in EEG signals. The proposed approach permits to detect exactly the epileptic and non-epileptic EEG segments of different duration and shape in order to identify a pathological activity in a remission state as well as detect a paroxysmal activity in a preictal period.
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    • "Another is to tackle far-from-equilibrium challenges by exploring the full dynamic characteristics of both societal systems and individuals. Key will be identifying behavioral symmetry breaks and formally testing for catastrophic or chaotic responses (Wolf et al. 1985) to economic, cultural, social and political crisis (Abdollahian and Kang 2008), coupling or decoupling. One approach is to endogenize perturbations and shocks across Y, RS, SE and D values and agent space, to map equilibrium stability and system phase transitions. "
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    • "Such features have been used for timeseries classification [42]. Other more complex time-domain features such as the Lyapunov exponent [43] have also been used for machine learning [35]. Recently, shapelets which represent local features in the data, have been used for "
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