On a Form of Lyapunov Exponents (I: Establishment of the Form) (original) (raw)
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Physical Review Letters, 1998
We present a new method for the computation of Lyapunov exponents utilizing representations of orthogonal matrices applied to decompositions of M or MM where M is the tangent map. This method uses a minimal set of variables, does not require renormalization or reorthogonalization, can be used to efficiently compute partial Lyapunov spectra, and does not break down when the Lyapunov spectrum is degenerate.
Comparison of Different Methods for Computing Lyapunov Exponents
have to be integrated simultaneously with the differential equations (2) in order to obtain Dx¢t as the mX m matrix Y. Here J= J(y(x; t))=«av;jayJly(x;t)) denotes the Jacobi matrix of partial derivatives of the vector field v at the pointy(x; t). The by guest on February 15, 2014 http://ptp.oxfordjournals.org/ Downloaded from by guest on February 15, 2014 http://ptp.oxfordjournals.org/ Downloaded from
Meccanica, 1980
Da diversi anni gli esponenti caratteristici di Lyapunov sono divenuti di notevole interesse nello studio dei sistemi dinamici al fine di caratterizzare quantitativamente le proprietà di stocasticità, legate essenzialmente alla divergenza esponenziale di orbite vicine. Si presenta dunque il problema del calcolo esplicito di tali esponenti, già risolto solo per il massimo di essi. Nel presente lavoro si dà un metodo per il calcolo di tutti tali esponenti, basato sul calcolo degli esponenti di ordine maggiore di uno, legati alla crescita di volumi. A tal fine si dà un teorema che mette in relazione gli esponenti di ordine uno con quelli di ordine superiore. Il metodo numerico e alcune applicazioni saranno date in un sucessivo articolo. Since several years Lyapunov Characteristic Exponents are of interest in the study of dynamical systems in order to characterize quantitatively their stochasticity properties, related essentially to the exponential divergence of nearby orbits. One has thus the problem of the explicit computation of such exponents, which has been solved only for the maximal of them. Here we give a method for computing all of them, based on the computation of the exponents of order greater than one, which are related to the increase of volumes. To this end a theorem is given relating the exponents of order one to those of greater order. The numerical method and some applications will be given in a forthcoming paper.
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Nonlinear Analysis: Modelling and Control, 2018
The problem of the inverse Lyapunov exponent was formulated and solved, involving to find such chaotic transformation for which the value of the Lyapunov exponent is given in advance. The solution procedure was presented by a numerical example. Furthermore, applications of the discussed model in chaos based cryptography were discussed.
A Comparison Between Methods to Compute Lyapunov Exponents
The Astronomical Journal, 2001
Lyapunov characteristic exponents measure the rate of exponential divergence between neighboring trajectories in the phase space. For a given autonomous dynamical system, the maximum Lyapunov characteristic exponent (hereafter LCE) is computed from the solution of the variational equations of the system. There are many dynamical systems in which the formulation and solution of the variational equations is a cumbersome task. In those cases an alternative procedure, Ðrst introduced by Benettin et al., is to replace the variational solution by computing two neighbor trajectories (the test particle and its shadow) and calculating the mutual distance. In this paper, we deal with a comparison between these two di †erent techniques for the calculation of LCE : the variational method and the two-particle method. We point out a problem that can appear when the two-particle method is used, which can lead to a false estimation of a positive LCE. The explanation of this phenomenon can be analyzed in two di †erent situations : (1) for relatively large initial separations the two-particle method is not a good approximation to the solution of the variational equations, and (2) for small initial separations the two-particle method have problems related to the machine precision, even when the separation can be many order of magnitudes larger than the machine precision. We show some examples of false estimates of the LCE that have already appeared in the literature using the two-particle method, and Ðnally we present some suggestions to be taken into account when this method has to be used.
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Physics Letters A, 1993
A new method for computing the largest Lyapunov exponent from chaotic time series is presented, whereby more robust convergence to the exact value is obtained. It aims to serve as a bridge between the two widely known existing methods: the Wolfand the Jacobian methods.
On the concept of stationary Lyapunov basis
Physica D: Nonlinear Phenomena, 1998
We propose the concept of stationary Lyapunov basis-the basis of tangent vectors eU)(x) defined at every point x of the attractor of the dynamical system, and show that one can reformulate some algorithms for calculation of Lyapunov exponents ~-i so that each)~i can be treated as the average of a function Si (x). This enables one to use measure averaging in theoretical arguments thus proposing the rigorous basis for a number of ideas for calculation of Lyapunov exponents from time series. We also study how the Lyapunov vectors in Benettin's algorithm converge to the stationary basis and show that this convergence rate determines continuity of the field of stationary Lyapunov vectors.