From an unstable periodic orbit to the Lyapunov exponent and a macroscopic variable in a Hamiltonian lattice - Periodic orbit dependencies

摘要

We study the problem of determining which periodic orbits in phase space can predict the largest Lyapunov exponent and the expectation values of macroscopic variables in a Hamiltonian system with many degrees of freedom. We also attempt to elucidate the manner in which these orbits yield such predictions. The model which we use in this paper is a discrete nonlinear Schrodinger equation. Using a method based on the modulational estimate of a periodic orbit, we predict the largest Lyapunov exponent and the expectation value of a macroscopic variable. We show that (i) the predicted largest Lyapunov exponent generally depends on the periodic orbit which we employ, and (ii) the predicted expectation value of the macroscopic variable does not depend on the periodic orbit, at least in a high energy regime. In addition, the physical meanings of these dependencies are considered.