Homoclinic Orbits and Chaos in Discretized Perturbed NLS Systems: Part II. Symbolic Dynamics

In Part I ([9], this journal), Li and McLaughlin proved the existence of homoclinic orbits in certain discrete NLS systems. In this paper, we will construct Smale horseshoes based on the existence of homoclinic orbits in these systems. First, we will construct Smale horseshoes for a general high dimensional dynamical system. As a result, a certain compact, invariant Cantor set3 is constructed. The Poincaré map on 3 induced by the flow is shown to be topologically conjugate to the shift automorphism on two symbols, 0 and 1. This gives rise to deterministic chaos. We apply the general theory to the discrete NLS systems as concrete examples. Of particular interest is the fact that the discrete NLS systems possess a symmetric pair of homoclinic orbits. The Smale horseshoes and chaos created by the pair of homoclinic orbits are also studied using the general theory. As a consequence we can interpret certain numerical experiments on the discrete NLS systems as “chaotic center-wing jumping.”