### 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 language | English (US) |
---|---|

Pages (from-to) | 211-269 |

Number of pages | 59 |

Journal | Journal of Nonlinear Science |

Volume | 7 |

Issue number | 3 |

State | Published - May 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

- 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.

Research output: Contribution to journal › Article

*Journal of Nonlinear Science*, vol. 7, no. 3, pp. 211-269.

}

TY - JOUR

T1 - Homoclinic Orbits and Chaos in Discretized Perturbed NLS Systems

T2 - Part I. Homoclinic Orbits

AU - Li, Y.

AU - McLaughlin, D. W.

PY - 1997/5

Y1 - 1997/5

N2 - 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].

AB - 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].

KW - Discrete nonlinear Schroedinger equation

KW - Fenichel fibers

KW - Homoclinic orbits

KW - Melnikov analysis

KW - Persistent invariant manifolds

KW - Spectral theory

UR - http://www.scopus.com/inward/record.url?scp=0000038726&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000038726&partnerID=8YFLogxK

M3 - Article

VL - 7

SP - 211

EP - 269

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

SN - 0938-8974

IS - 3

ER -