Exponential dichotomies, the Fredholm Alternative, and transverse homoclinic orbits in partial functional differential equations

Shigui Ruan; Weinian Zhang

doi:10.1007/s10884-005-8274-9

Exponential dichotomies, the Fredholm Alternative, and transverse homoclinic orbits in partial functional differential equations

Shigui Ruan, Weinian Zhang

Mathematics

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

In this paper we investigate the transversality of homoclinic orbits in partial functional differential equations. We first discuss the exponential dichotomies for linear operator equations. Then we show that the Fredholm Alternative holds if the homogeneous equation has exponential dichotomies on R. Transversality of homoclinic orbits for periodically perturbed partial functional differential equations is studied using the Liapunov-Schmidt method and the Melnikov integral.

Original language	English (US)
Pages (from-to)	759-777
Number of pages	19
Journal	Journal of Dynamics and Differential Equations
Volume	17
Issue number	4
DOIs	https://doi.org/10.1007/s10884-005-8274-9
State	Published - Oct 2005

Keywords

Exponential dichotomy
Liapunov-Schmidt method
Melnikov integral
Partial functional differential equations
The Fredholm Alternative
Transverse homoclinic orbit

ASJC Scopus subject areas

Analysis

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abstract = "In this paper we investigate the transversality of homoclinic orbits in partial functional differential equations. We first discuss the exponential dichotomies for linear operator equations. Then we show that the Fredholm Alternative holds if the homogeneous equation has exponential dichotomies on R. Transversality of homoclinic orbits for periodically perturbed partial functional differential equations is studied using the Liapunov-Schmidt method and the Melnikov integral.",

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AU - Ruan, Shigui

AU - Zhang, Weinian

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N2 - In this paper we investigate the transversality of homoclinic orbits in partial functional differential equations. We first discuss the exponential dichotomies for linear operator equations. Then we show that the Fredholm Alternative holds if the homogeneous equation has exponential dichotomies on R. Transversality of homoclinic orbits for periodically perturbed partial functional differential equations is studied using the Liapunov-Schmidt method and the Melnikov integral.

AB - In this paper we investigate the transversality of homoclinic orbits in partial functional differential equations. We first discuss the exponential dichotomies for linear operator equations. Then we show that the Fredholm Alternative holds if the homogeneous equation has exponential dichotomies on R. Transversality of homoclinic orbits for periodically perturbed partial functional differential equations is studied using the Liapunov-Schmidt method and the Melnikov integral.

KW - Exponential dichotomy

KW - Liapunov-Schmidt method

KW - Melnikov integral

KW - Partial functional differential equations

KW - The Fredholm Alternative

KW - Transverse homoclinic orbit

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