Homoclinic Orbits and Chaos in Discretized Perturbed

Summary. The existence of homoclinic orbits, for a finite-difference discretized form of a damped and driven perturbation of the focusing nonlinear Schroedinger equation under even periodic boundary conditions, is established. More specifically, for external param-eters on a codimension 1 submanifold, the existence of homoclinic orbits is established through an argument which combines Melnikov analysis with a geometric singular per-turbation theory and a purely geometric argument (called the “second 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 codimension 1 submanifold of parameters is calculated. (This is the material in Part I.) Then, in a neighborhood of these homoclinic orbits, the existence of “Smale horseshoes ” and the corresponding symbolic dynamics are established in Part II [21]