On a Form of Lyapunov Exponents (II: Verification and Illustration) (original) (raw)
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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.
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
Lyapunov Exponents without Rescaling and Reorthogonalization
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.
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