Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application

Abstract

The present paper, together with the previous one (Part 1: Theory, published in this journal) is intended to give an explicit method for computing all Lyapunov Characteristic Exponents of a dynamical system. After the general theory on such exponents developed in the first part, in the present paper the computational method is described (Chapter A) and some numerical examples for mappings on manifolds and for Hamiltonian systems are given (Chapter B)

Routes to chaos in high-dimensional dynamical systems: A qualitative numerical study

“…A primary tool of analysis will be the Lyapunov exponent spectrum ( [6] [32]), since it is a good measure of the tangent space of the mapping along its orbit. Negative Lyapunov exponents correspond to global stable manifolds or contracting directions; positive Lyapunov exponents correspond to global unstable manifolds or expanding directions and are ( [32]), in a computational framework, the hallmark of chaos.…”

“…an analytical normal form calculation -again starting with the early bifurcations along the route; iii. a study of Lyapunov spectrum calculation technique a la [16], [15], [37], [6], or [35] -these networks form a nice set of high-dimensional mappings to study Lyapunov spectrum calculation schemes since our mappings are high dimensional, not pathological, and are not, as high-dimensional maps go, computationally intensive to use; very often, upon the presentation of a new Lyapunov spectrum computation algorithm the test cases are very low dimensional dynamical systems for which numerical stability is rarely a problem; iv. a brute force -by hand -statistical study of the bifurcation sequences in these networks; v. a more numerically accurate Lyapunov spectrum calculation routine that can be used for a full statistical analysis of the routes to chaos in this set of dynamical systems; vi.…”

Dynamics of Oscillator Chains

“…Besides the largest Lyapunov exponent λ, in a dynamical system made of N degrees of freedom, each one described by a pair of canonical coordinates (position and momentum) one can define a spectrum of Lyapunov exponents, λ i , where the index i = 1, · · · , 2N labels the exponents from the largest to the smallest one. An effective algorithmic procedure for evaluating the spectrum of Lyapunov exponents is discussed in [152]. Beyond rigorous mathematical definitions, an interpretation of the Lyapunov spectrum can be obtained by considering that the partial sum h n = n i=1 λ i (n ≤ 2N ) measures the average exponential rates of expansion, or contraction, of a generic volume of geometric dimension n in phase space.…”

“…2.3, we have discussed the numerical determination of the statistical quantities λ and n eff from which T eq (or some fraction of it) is found. In the higher energy regime, these and other statistical measures of the diffusion have been explored in the late 1970s and 1980s, particularly from the group in Firenze (see [50,51,52,123,128,151]) and in Milano (see [25,48,49,152,153]). Here, and in Sect.…”

“…The only terms to transfer energy to high-frequency modes are the ones where j = i, since then the product of the two low-frequency angles does not have a fast phase associated with it. Additionally, the selection rule requires that B = 0 unless 152) which reduces the index to a single sum. This result for a single low-frequency mode i is an estimate for the average energy decay rate, which, from (2.151) is 153) where ω i = πi/N .…”