Bifurcation from a homoclinic orbit in partial functional differential equations

Shigui Ruan; Junjie Wei; Jianhong Wu

doi:10.3934/dcds.2003.9.1293

Bifurcation from a homoclinic orbit in partial functional differential equations

Shigui Ruan, Junjie Wei, Jianhong Wu

Mathematics

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

4 Scopus citations

Abstract

We consider a family of partial functional differential equations which has a homoclinic orbit asymptotic to an isolated equilibrium point at a critical value of the parameter. Under some technical assumptions, we show that a unique stable periodic orbit bifurcates from the homoclinic orbit. Our approach follows the ideas of Šil'nikov for ordinary differential equations and of Chow and Deng for semilinear parabolic equations and retarded functional differential equations.

Original language	English (US)
Pages (from-to)	1293-1322
Number of pages	30
Journal	Discrete and Continuous Dynamical Systems
Volume	9
Issue number	5
DOIs	https://doi.org/10.3934/dcds.2003.9.1293
State	Published - Sep 2003

Keywords

Bifurcation
Homoclinic orbit
Partial functional differential equations
Periodic solutions
Stable and unstable manifolds
Variation of constant formula

ASJC Scopus subject areas

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

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10.3934/dcds.2003.9.1293

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abstract = "We consider a family of partial functional differential equations which has a homoclinic orbit asymptotic to an isolated equilibrium point at a critical value of the parameter. Under some technical assumptions, we show that a unique stable periodic orbit bifurcates from the homoclinic orbit. Our approach follows the ideas of {\v S}il'nikov for ordinary differential equations and of Chow and Deng for semilinear parabolic equations and retarded functional differential equations.",

keywords = "Bifurcation, Homoclinic orbit, Partial functional differential equations, Periodic solutions, Stable and unstable manifolds, Variation of constant formula",

author = "Shigui Ruan and Junjie Wei and Jianhong Wu",

year = "2003",

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doi = "10.3934/dcds.2003.9.1293",

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T1 - Bifurcation from a homoclinic orbit in partial functional differential equations

AU - Ruan, Shigui

AU - Wei, Junjie

AU - Wu, Jianhong

PY - 2003/9

Y1 - 2003/9

N2 - We consider a family of partial functional differential equations which has a homoclinic orbit asymptotic to an isolated equilibrium point at a critical value of the parameter. Under some technical assumptions, we show that a unique stable periodic orbit bifurcates from the homoclinic orbit. Our approach follows the ideas of Šil'nikov for ordinary differential equations and of Chow and Deng for semilinear parabolic equations and retarded functional differential equations.

AB - We consider a family of partial functional differential equations which has a homoclinic orbit asymptotic to an isolated equilibrium point at a critical value of the parameter. Under some technical assumptions, we show that a unique stable periodic orbit bifurcates from the homoclinic orbit. Our approach follows the ideas of Šil'nikov for ordinary differential equations and of Chow and Deng for semilinear parabolic equations and retarded functional differential equations.

KW - Bifurcation

KW - Homoclinic orbit

KW - Partial functional differential equations

KW - Periodic solutions

KW - Stable and unstable manifolds

KW - Variation of constant formula

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EP - 1322

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 5

ER -

Bifurcation from a homoclinic orbit in partial functional differential equations

Abstract

Keywords

ASJC Scopus subject areas

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