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40
A practical method for calculating largest Lyapunov exponents from small data sets
 PHYSICA D
, 1993
"... Detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest Lyapunov exponent. Lyapunov exponents quantify the exponential divergence of initially close statespace trajectories and estimate the amount of chaos in a system. We present a new m ..."
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Cited by 81 (0 self)
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Detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest Lyapunov exponent. Lyapunov exponents quantify the exponential divergence of initially close statespace trajectories and estimate the amount of chaos in a system. We present a new method for calculating the largest Lyapunov exponent from an experimental time series. The method follows directly from the definition of the largest Lyapunov exponent and is accurate because it takes advantage of all the available data. We show that the algorithm is fast, easy to implement, and robust to changes in the following quantities: embedding dimension, size of data set, reconstruction delay, and noise level. Furthermore, one may use the algorithm to calculate simultaneously the correlation dimension. Thus, one sequence of computations will yield an estimate of both the level of chaos and the system complexity.
Estimating fractal dimension
 Journal of the Optical Society of America A
, 1990
"... Fractals arise from a variety of sources and have been observed in nature and on computer screens. One of the exceptional characteristics of fractals is that they can be described by a noninteger dimension. The geometry of fractals and the mathematics of fractal dimension have provided useful tools ..."
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Cited by 75 (3 self)
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Fractals arise from a variety of sources and have been observed in nature and on computer screens. One of the exceptional characteristics of fractals is that they can be described by a noninteger dimension. The geometry of fractals and the mathematics of fractal dimension have provided useful tools for a variety of scientific disciplines, among which is chaos. Chaotic dynamical systems exhibit trajectories in their phase space that converge to a strange attractor. The fractal dimension of this attractor counts the effective number of degrees of freedom in the dynamical system and thus quantifies its complexity. In recent years, numerical methods have been developed for estimating the dimension directly from the observed behavior of the physical system. The purpose of this paper is to survey briefly the kinds of fractals that appear in scientific research, to discuss the application of fractals to nonlinear dynamical systems, and finally to review more comprehensively the state of the art in numerical methods for estimating the fractal dimension of a strange attractor. Confusion is a word we have invented for an order which is not understood.Henry Miller, "Interlude," Tropic of Capricorn Numerical coincidence is a common path to intellectual perdition in our quest for meaning. We delight in catalogs of disparate items united by the same number, and often feel in our gut that some unity must underlie it all.
On Selecting Models for Nonlinear Time Series
 Physica D
, 1995
"... Constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models, or to choose between competing classes. This paper discusses a method of building nonlinear models of possibly chaotic systems from data, while maintainin ..."
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Cited by 49 (14 self)
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Constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models, or to choose between competing classes. This paper discusses a method of building nonlinear models of possibly chaotic systems from data, while maintaining good robustness against noise. The models that are built are close to the simplest possible according to a description length criterion. The method will deliver a linear model if that has shorter description length than a nonlinear model. We show how our models can be used for prediction, smoothing and interpolation in the usual way. We also show how to apply the results to identification of chaos by detecting the presence of homoclinic orbits directly from time series. 1 The Model Selection Problem As our understanding of chaotic and other nonlinear phenomena has grown, it has become apparent that linear models are inadequate to model most dynamical processes. Nevertheless, linear models...
Reconstruction Expansion as a GeometryBased Framework for Choosing Proper Delay Times
 PHYSICA D
, 1994
"... The quality of attractor reconstruction using the method of delays is known to be sensitive to the delay parameter, t . Here we develop a new, computationally efficient approach to choosing t that quantifies reconstruction expansion from the identity line of the embedding space. We show that reconst ..."
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Cited by 27 (4 self)
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The quality of attractor reconstruction using the method of delays is known to be sensitive to the delay parameter, t . Here we develop a new, computationally efficient approach to choosing t that quantifies reconstruction expansion from the identity line of the embedding space. We show that reconstruction expansion is related to the concept of reconstruction signal strength and that increased expansion corresponds to diminished effects of measurement error. Thus, reconstruction expansion represents a simple, geometrical framework for choosing t . Furthermore, we describe the role of dynamical error in attractor expansion and argue that algorithms for determining t should be considered as attempts at estimating an upper bound to the optimal delay.
Is breathing in infants chaotic?, Dimension estimates for respiratory patterns during quiet sleep
 J. Appl. Physiol
"... in infants chaotic? Dimension estimates for respiratory patterns during quiet sleep. J. Appl. Physiol. 86(1): 359–376, 1999.—We describe an analysis of dynamic behavior apparent in timesseries recordings of infant breathing during sleep. Three principal techniques were used: estimation of correla ..."
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Cited by 15 (12 self)
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in infants chaotic? Dimension estimates for respiratory patterns during quiet sleep. J. Appl. Physiol. 86(1): 359–376, 1999.—We describe an analysis of dynamic behavior apparent in timesseries recordings of infant breathing during sleep. Three principal techniques were used: estimation of correlation dimension, surrogate data analysis, and reduced linear (autoregressive) modeling (RARM). Correlation dimension can be used to quantify the complexity of time series and has been applied to a variety of physiological and biological measurements. However, the methods most commonly used to estimate correlation dimension suffer from some technical problems that can produce misleading results if not correctly applied. We used a new technique of estimating correlation dimension that has fewer problems. We tested the significance of dimension estimates by comparing estimates with artificial data sets (surrogate data). On the basis of the analysis, we conclude that the dynamics of infant breathing during quiet sleep can best be described as a nonlinear dynamic system with largescale, lowdimensional and smallscale, highdimensional behavior; more specifically, a noisedriven nonlinear system with a twodimensional periodic orbit. Using our RARM technique, we identified the second period as cyclic amplitude modulation of the same period as periodic breathing. We conclude that our data are consistent with respiration being chaotic. control of breathing; periodic breathing; dynamical systems; chaos; surrogate data techniques THIS PAPER DESCRIBES and summarizes a study of infant breathing by using dataanalysis techniques derived from dynamic systems theory (DST). Such techniques have been useful for examining other complex physi
Monitoring changing dynamics with correlation integrals: Case study of an epileptic seizure
, 1996
"... We describe a procedure (and the motivation behind it) which rapidly and accurately tracks the onset and progress of an epileptic seizure. Roughly speaking, one monitors changes in the relative dispersion of a reembedded time series. The results are robust with respect to variation of adjustable pa ..."
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Cited by 11 (0 self)
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We describe a procedure (and the motivation behind it) which rapidly and accurately tracks the onset and progress of an epileptic seizure. Roughly speaking, one monitors changes in the relative dispersion of a reembedded time series. The results are robust with respect to variation of adjustable parameters such as embedding dimension, lag time, and critical distances. Moreover, the general method is virtually unaffected when the data is significantly corrupted by external noise. When the information computed for the individual channels is displayed in an appropriate spacetime plot, the progress and geometric location of the seizure are easily seen. An interpretation of these results in terms of a cloud of particles moving in an abstract phase space is examined. 1 Introduction Epilepsy is a disease characterized by recurrent, unprovoked seizures accompanied by pathological electrical activity in the brain[1]. This activity can be monitored and recorded using electrodes attached to the...
Fractal and Chaotic Dynamics in Nervous Systems
 Prog. Neurobiol
, 1991
"... : This paper presents a review of fractal and chaotic dynamics in nervous systems and the brain, exploring mathematical chaos and its relation to processes, from the neurosystems level down to the molecular level of the ion channel. It includes a discussion of parallel distributed processing models ..."
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Cited by 8 (0 self)
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: This paper presents a review of fractal and chaotic dynamics in nervous systems and the brain, exploring mathematical chaos and its relation to processes, from the neurosystems level down to the molecular level of the ion channel. It includes a discussion of parallel distributed processing models and their relation to chaos and overviews reasons why chaotic and fractal dynamics may be of functional utility in central nervous cognitive processes. Recent models of chaotic pattern discrimination and the chaotic electroencephalogram are considered. A novel hypothesis is proposed concerning chaotic dynamics and the interface with the quantum domain. Contents : 0 : Introduction 2 1 : Concepts and Techniques in Chaos 2 (a) Chaotic Systems 2 (b) Indicators of Chaos 4 (i) Liapunov Exponent and Entropy 4 (ii) Power Spectrum 6 (iii) Hausdorff dimension and Fractals 7 (iv) Correlation Integral 8 (c) Iterations as Examples of Chaos 10 (i) The Logistic Map 10 (ii) The Transition from Q...
Detecting Nonlinearity in Experimental Data
 International Journal of Bifurcation and Chaos Submitted
, 1997
"... The technique of surrogate data has been used as a method to test for membership of particular classes of linear systems. We suggest an obvious extension of this to classes of nonlinear parametric models and demonstrate our methods with respiratory data from sleeping human infants. Although our data ..."
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Cited by 7 (7 self)
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The technique of surrogate data has been used as a method to test for membership of particular classes of linear systems. We suggest an obvious extension of this to classes of nonlinear parametric models and demonstrate our methods with respiratory data from sleeping human infants. Although our data are clearly distinct from the different classes of linear systems we are unable to distinguish between our data and surrogates generated by nonlinear models. Hence we conclude that human respiration is likely to be a nonlinear system with more than 2 degrees of freedom with a limit cycle that is driven by high dimensional dynamics or noise.
Spatial extension of brain activity fools the singlechannel reconstruction of EEG dynamics. Hum Brain Mapp 5:26–47
, 1997
"... r r Abstract:We report here on a first attempt to settle the methodological controversy between advocates of two alternative reconstruction approaches for temporal dynamics in brain signals: the singlechannel method (using data from one recording site and reconstructing by timelags), and the multi ..."
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Cited by 6 (2 self)
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r r Abstract:We report here on a first attempt to settle the methodological controversy between advocates of two alternative reconstruction approaches for temporal dynamics in brain signals: the singlechannel method (using data from one recording site and reconstructing by timelags), and the multiplechannel method (using data from a spatially distributed set of recordings sites and reconstructing by means of spatial position). For the purpose of a proper comparison of these two techniques, we computed a series of EEGlike measures on the basis of wellknown dynamical systems placed inside a spherical model of the head. For each of the simulations, the correlation dimension estimates obtained by both methods were calculated and compared, when possible, with the known (or estimated) dimension of the underlying dynamical system. We show that the singlechannel method fails to reliably quantify spatially extended dynamics, while the multichannel method performs better. It follows that the latter is preferable, given the known spatially distributed nature of brain processes.Hum. Brain Mapping 5:26–47, 1997. r 1997WileyLiss, Inc. Key words: nonlinear dynamics; spatially extended systems; EEG; singlechannel reconstruction; multichannel reconstruction; correlation dimension; spatiotemporal chaos; coupled map lattices r r