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

TL;DR

This paper investigates how specific periodic orbits in a Hamiltonian lattice influence the largest Lyapunov exponent and macroscopic variables, revealing orbit-dependent and orbit-independent behaviors in a nonlinear Schrödinger model.

Contribution

It introduces a modulational estimate method to predict Lyapunov exponents and macroscopic variables, highlighting their dependence on the chosen periodic orbit.

Findings

Lyapunov exponent depends on the periodic orbit used

Macroscopic variable expectation value is orbit-independent at high energy

Method provides a link between microscopic orbits and macroscopic behavior

Abstract

We study which and how a periodic orbit in phase space links to both the largest Lyapunov exponent and the expectation values of macroscopic variables in a Hamiltonian system with many degrees of freedom. The model which we use in this paper is the discrete nonlinear Schr\"odinger 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.