Homoclinic Orbits in a Four Dimensional Model of a Perturbed NLS Equation: A Geometric Singular Perturbation Study

Abstract

This paper contains a detailed and explicit study of an important mechanism leading to chaotic dynamics in a perturbation of the integrable nonlinear Schrödinger equation, a perturbation which contains damping and driving terms. Specifically, we study, both analytically and numerically, homoclinic and chaotic behavior in a two mode ode truncation. First, we summarize recent results of numerical experiments which establish the presence of irregular (chaotic) temporal behavior in this two mode system. We then establish the existence of an orbit which is homoclinic to a fixed point q _∈ - a saddle in a “resonance band”. This analytical argument begins from a representation of certain invariant manifolds by fibers, a representation which we explicitly illustrate with concrete examples. The existence of the homoclinic orbit then follows from a Melnikov argument combined with methods from geometric singular perturbation theory. Next these homoclinic orbits are constructed, and studied, numerically with a bifurcation algorithm. These numerical studies find some members of the family of homoclinic orbits which were predicted by the theory. Finally, the existence of a chaotic symbol dynamics is established through a “Smale horseshoe” which is shown to exist near the homoclinic orbit.