HOMOCLINIC ORBITS AND CHAOS IN DISCRETIZED PERTURBED NLS SYSTEMS .1. HOMOCLINIC ORBITS

The existence of homoclinic orbits, for a finite-difference discretize d form of a damped and driven perturbation of the focusing nonlinear S chroedinger equation under even periodic boundary conditions, is estab lished. More specifically, for external parameters on a codimension I submanifold, the existence of homoclinic orbits is established through an argument which combines Melnikov analysis with a geometric singula r perturbation theory and a purely geometric argument (called the ''se cond measurement'' in the paper). The geometric singular perturbation theory deals with persistence of invariant manifolds and fibration of the persistent invariant manifolds. The approximate location of the co dimension 1 submanifold of parameters is calculated. (This is the mate rial in Part I.) Then, in a neighborhood of these homoclinic orbits, t he existence of ''Smale horseshoes'' and the corresponding symbolic dy namics are established in Part II [21].