Invariant Spectra of Orbits in Multidimensional Symplectic Maps

Abstract

We study the spectra of ‘stretching numbers’ of orbits of 2n-dimensional symplectic maps with n ≥ 2. Let (0) be the distance at time T = 0 between a given orbit and a neighbouring orbit. This distance grows to (j) at time T = j, and to (j + 1) at time T = j + 1. The ‘stretching number’ [1], or ‘local Lyapunov indicator’ [2] at time T = j is defined as: a j = In | (j + 1)/ (j)|, i-e. the stretching number is a one-period Lyapunov number. The limit = lim gives the maximal Lyapunov Characteristic Number (LCN). The spectrum S(a) of stretching numbers is defined as the probability density for a, i.e. S(a) = limn→∞ dN(a)/Nda where dN(a) is the number of times that the values of a appear in the interval (a, a + da) after N periods of integration. The main property of the spectra of chaotic orbits of 2D symplectic maps is their invariance with respect (a) to the initial point along an orbit, (b) to the direction of the deviation (0) and (c) to the initial conditions of orbits, if they are on the same chaotic domain [1]. Spectra of regular orbits are invariant with respect to the initial conditions on the same invariant curve. Invariant spectra were found in conservative Hamiltonian systems [3], dissipative maps [1], and Hamiltonian systems depending periodically on time [4]