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22
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 62 (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.
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 39 (11 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 25 (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.
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 10 (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 6 (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 5 (5 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.
A new chaotic system for better secure communication
 Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms
, 2002
"... Abstract. A 4dimensional nonlinear dynamical system is proposed for secure communications using chaos synchronization. It is shown that the chaotic attractor of the system has complex local structure that is sensitive to the perturbation of the systemâ€™s parameters. With the aid of the nonlinear for ..."
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Cited by 4 (3 self)
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Abstract. A 4dimensional nonlinear dynamical system is proposed for secure communications using chaos synchronization. It is shown that the chaotic attractor of the system has complex local structure that is sensitive to the perturbation of the systemâ€™s parameters. With the aid of the nonlinear forecasting technique, these properties have proven to be very useful in improving the security of communications via chaos encryption.
Automated Embedding and the Creep Phenomenon in Chaotic Time Series
, 2000
"... Embedding techniques represent a powerful advance in the development of experimental chaos. However there seems no universal method to find the best set of parameters to use. In this paper we present a new approach, an automated embedding method, to estimate a near optimum embedding dimension and de ..."
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Cited by 3 (1 self)
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Embedding techniques represent a powerful advance in the development of experimental chaos. However there seems no universal method to find the best set of parameters to use. In this paper we present a new approach, an automated embedding method, to estimate a near optimum embedding dimension and delay time based on the test [Stefnsson 1997]. A strange attractor can be regarded as the union of an infinite number of unstable periodic orbits. In order to extract these unstable periodic orbits from a set of embedding space vectors (used for reconstruction) we introduce a technique to estimate a suitable jump time. We also describe the `creep phenomenon', discovered using these techniques, which may allow us to make more accurate longer term predictions. Keywords: Delay coordinates, Embedding dimension, Gamma test, Prediction. Masayuki Otani DEPARTMENT OF COMPUTING IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY AND MEDICINE, 180 Queen's Gate London SW7 2BZ Telephone: +44(0)1715948281 Telefax: +44(0)1715818024 Antonia J. Jones DEPARTMENTOFCOMPUTER SCIENCE UNIVERSITY OF WALES, CARDIFF PO Box 916, Cardiff CF2 3XF Telephone: +44(0)1222874812 Telefax: +44(0)1222874598 Date/version: 14 October 2000 Copyright 1997. Masayuki Otani and Antonia J. Jones Automated embedding and the creep phenomenon in chaotic time series CONTENTS 1
by
, 1997
"... This report studies chaotic systems with particular emphasis on the recently developed method of E. Ott, C. Grebogi and J. A. Yorke [Ott 1990] (the OGY method) for controlling such systems. Concepts useful in understanding chaos in general are introduced. We can conceptualise chaotic systems as aris ..."
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This report studies chaotic systems with particular emphasis on the recently developed method of E. Ott, C. Grebogi and J. A. Yorke [Ott 1990] (the OGY method) for controlling such systems. Concepts useful in understanding chaos in general are introduced. We can conceptualise chaotic systems as arising from classes of differential equations having particularly intractable solutions sets. However, in many applications the underlying equations are unknown, one works from observations of measurable parameters of the system. The use of successive samples of a single variable (or few variables) to generate an embedding with a view to reconstructing the details of an attractor for a higher dimensional dynamic system was suggested in [Packard 1980] and a frequently quoted embedding theorem [Takens 1980] establishes the existence of such models for homogeneous systems: if the underlying state space of a system has ddimensions then the embedding space needs to have at most (2d + 1) dimensions to capture the dynamics of the system completely. These results were later generalised and improved by [Levin 1993]. It is a remarkable fact that much of the dynamics of a high dimensional system can be recovered from a suitable embedding of a single variable, but in practice a critical factor in the accuracy of such reconstructions is the sampling delay. In this report a number of existing techniques for deriving delay time, sampling delays, suitable for