Homoclinic Orbits and Chaos in Discretized Perturbed NLS Systems: Part I. Homoclinic Orbits
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چکیده
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 parameters on a codimension 1 submanifold, the existence of homoclinic orbits is established through an argument which combines Melnikov analysis with a geometric singular perturbation 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].
منابع مشابه
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. ...
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تاریخ انتشار 1997