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

Abstract

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

Complex statistics and diffusion in nonlinear disordered particle chains

“…Having, thus, access to the deviation vectors, we compute the Lyapunov exponents in descending order (i.e., k 1 > k 2 … > k 2N ) following the so-called standard method. [35][36][37] B. Complex statistics shows persisting chaos in the Klein-Gordon chain…”

Lyapunov stability of charge transport in miniband semiconductor superlattices

“…Although a continuous spatially extended system could be discretized in space and thus reduced to the set of ODEs, the result of the direct application of the standard procedures 11,14 to the calculation of the spectrum of Lyapunov exponents depends on the step of discretization. 17 Moreover, the physical meaning of such Lyapunov exponents is vague since their number and the corresponding Lyapunov vectors depend on the method of discretization.…”

“…[11][12][13][14] However, a straightforward application of these methods to the spatially extended systems is only possible for the systems which are naturally discrete in space, [15][16][17] e.g., to the arrays of coupled oscillators or maps. Importantly, the direct application of numerical techniques for the calculation of Lyapunov exponents, developed for systems with finite dimension of the phase space, to continuous spatially extended systems is rather ambiguous 17 and unreliable.…”

The chaotic rotation of Hyperion

“…Thus in the spin-orbit problem only three LCEs are independent; it is sufficient to only study those which are positive or zero. (For a review of the mathematical results regarding LCEs see Benettin et al (1980a)). …”