Homoclinic Orbits and Chaos in Discretized Perturbed NLS Systems

Part I. Homoclinic Orbits

Research output: Contribution to journalArticle

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
StatePublished - May 1997

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Homoclinic Orbit
Chaos theory
Geometric Singular Perturbation Theory
chaos
Chaos
Orbits
Invariant Manifolds
orbits
Submanifolds
Codimension
perturbation theory
Smale Horseshoe
Schroedinger equation
Symbolic Dynamics
Fibration
Periodic Boundary Conditions
Nonlinear equations
Damped
Persistence
Finite Difference

Keywords

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

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mathematics(all)
  • Applied Mathematics
  • Mathematical Physics
  • Statistical and Nonlinear Physics

Cite this

Homoclinic Orbits and Chaos in Discretized Perturbed NLS Systems : Part I. Homoclinic Orbits. / Li, Y.; McLaughlin, D. W.

In: Journal of Nonlinear Science, Vol. 7, No. 3, 05.1997, p. 211-269.

Research output: Contribution to journalArticle

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