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Determining Lyapunov Exponents from a Time Series
 Physica
, 1985
"... We present the first algorithms that allow the estimation of nonnegative 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 n ..."
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We present the first algorithms that allow the estimation of nonnegative 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 longterm 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 BelousovZhabotinskii reaction and CouetteTaylor flow. Contents 1.
Lyapunov Exponents From Random Fibonacci Sequences To The Lorenz Equations
 Department of Computer Science, Cornell University
, 1998
"... this paper (Mathematical Reviews:29 #648) with the words "This is a profound memoir." 9 will show in Chapter 3, there are simple algorithms for bounding the Lyapunov exponents in this setting. The advanced state of the theory for random matrix products is a peculiar situation because dete ..."
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Cited by 11 (1 self)
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this paper (Mathematical Reviews:29 #648) with the words "This is a profound memoir." 9 will show in Chapter 3, there are simple algorithms for bounding the Lyapunov exponents in this setting. The advanced state of the theory for random matrix products is a peculiar situation because deterministic matrix products that govern sensitive dependence on initial conditions are barely understood; it is as if the strong law of large numbers were well understood without a satisfactory theory of convergence of infinite series. The elements of the theory of random matrix products are carefully explained in the beautiful monograph by Bougerol [16]. The basic result about Lyapunov exponents, lim
FINDING OPTIMAL ORBITS OF CHAOTIC SYSTEMS
, 2005
"... Chaotic dynamical systems can exhibit a wide variety of motions, including periodic orbits of arbitrarily large period. We consider the question of which motion is optimal, in the sense that it maximizes the average over time of some given scalar “performance function. ” Past work indicates that opt ..."
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Chaotic dynamical systems can exhibit a wide variety of motions, including periodic orbits of arbitrarily large period. We consider the question of which motion is optimal, in the sense that it maximizes the average over time of some given scalar “performance function. ” Past work indicates that optimal motions tend to be periodic orbits with low period, but does not describe, beyond a brute force approach, how to determine which orbit is optimal in a particular scenario. For onedimensional expanding maps and higher dimensional hyperbolic systems, we have found constructive methods for calculating the optimal average and corresponding periodic orbit, and by carrying them out on a computer have found them to work quite well in practice.
Physica 7D (1983) 153180 NorthHolland Publishing Company THE DIMENSION OF CHAOTIC ATTRACTORS
"... Dimension is perhaps the most basic property of an attractor. In this paper we discuss a variety of different definitions of dimension, compute their values for a typical example, and review previous work on the dimension of chaotic attractors. The relevant definitions of dimension are of two gener ..."
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Dimension is perhaps the most basic property of an attractor. In this paper we discuss a variety of different definitions of dimension, compute their values for a typical example, and review previous work on the dimension of chaotic attractors. The relevant definitions of dimension are of two general types, those that depend only on metric properties, and those that depend on the frequency with which a typical trajectory visits different regions of the attractor. Both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value, which we call the "dimension of the natural measure", and all of the metric dimensions take on a common value, which we call the "fractal dimension". Furthermore, the dimension of the natural measure is typically equal to the Lyapunov dimension, which is defined in terms of Lyapunov numbers, and thus is usually far easier to calculate than any other definition. Because it is computable and more physically relevant, we feel that the dimension of the natural measure is more important than the fractal dimension.
Numéro d'ordre: 489 Numéro attribué par la bibliothèque: 07ENSL0489 THÈSE
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