Homoclinic Orbits and Chaos in Discretized Perturbed NLS Systems: Part I. Homoclinic Orbits

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Abstract

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].

Original languageEnglish (US)
Pages (from-to)211-269
Number of pages59
JournalJournal of Nonlinear Science
Volume7
Issue number3
DOIs
StatePublished - Jan 1 1997

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Keywords

  • Discrete nonlinear Schroedinger equation
  • Fenichel fibers
  • Homoclinic orbits
  • Melnikov analysis
  • Persistent invariant manifolds
  • Spectral theory

ASJC Scopus subject areas

  • Modeling and Simulation
  • Engineering(all)
  • Applied Mathematics

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