Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory

SommarioDa 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.SummarySince 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.

[1]  A. Liapounoff,et al.  Problème général de la stabilité du mouvement , 1907 .

[2]  O. Perron,et al.  Die Stabilitätsfrage bei Differentialgleichungen , 1930 .

[3]  J. Gillis,et al.  Probability and Related Topics in Physical Sciences , 1960 .

[4]  L. Cesari,et al.  Asymptotic Behaviour and Stability Problems in Ordinary Differential Equations , 1960 .

[5]  Richard A. Silverman,et al.  An introduction to the theory of linear spaces , 1963 .

[6]  Lamberto Cesari,et al.  Asymptotic Behavior and Stability Problems in Ordinary Differential Equations , 1963 .

[7]  S. Sternberg.  Lectures on Differential Geometry , 1964 .

[8]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[9]  P. Walters.  Ergodic theory: Introductory lectures , 1975 .

[10]  L. Galgani,et al.  Numerical computations on a stochastic parameter related to the Kolmogorov entropy , 1976 .

[11]  G. Benettin,et al.  Kolmogorov Entropy and Numerical Experiments , 1976 .

[12]  Y. Pesin.  CHARACTERISTIC LYAPUNOV EXPONENTS AND SMOOTH ERGODIC THEORY , 1977 .

[13]  G. Benettin,et al.  Numerical experiments on the free motion of a point mass moving in a plane convex region: Stochastic transition and entropy , 1978 .

[14]  George Contopoulos,et al.  On the number of isolating integrals in Hamiltonian systems , 1978 .

[15]  G. Benettin,et al.  Kolmogorov entropy of a dynamical system with an increasing number of degrees of freedom , 1979 .

[16]  G. Benettin,et al.  Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application , 1980 .