Correlations between Chaos in a Perturbed Sine-Gordon Equation and a Truncated Model System

Abstract

The purpose of this paper is to present a first step toward providing coordinates and associated dynamics for low-dimensional attractors in nearly integrable partial differential equations (pdes), in particular, where the truncated system reflects salient geometric properties of the pde. This is achieved by correlating: (i) Numerical results on the bifurcations to temporal chaos with spatial coherence of the damped, periodically forced sine-Gordon equation with periodic boundary conditions; (ii) An interpretation of the spatial and temporal bifurcation structures of this perturbed integrable system with regard to the exact structure of the sine-Gordon phase space; (iii) A model dynamical systems problem, which is itself a perturbed integrable Hamiltonian system, derived from the perturbed sine-Gordon equation by a finite mode Fourier truncation in the nonlinear Schrödinger limit; and (iv) The bifurcations to chaos in the truncated phase space. In particular, a potential source of chaos in both the pde and the model ordinary differential equation systems is focused on: the existence of homoclinic orbits in the unperturbed integrable phase space and their continuation in the perturbed problem. The evidence presented here supports our thesis that the chaotic attractors of the weakly perturbed periodic sine-Gordon system consists of low-dimensional metastable attracting states together with intermediate states that are $O(1)$ unstable and correspond to homoclinic states in the integrable phase space. It is surmised that the chaotic dynamics on these attractors is due to the perturbation of these homoclinic integrable configurations.