Bifurcation from a homoclinic orbit in partial functional differential equations

Shigui Ruan; Junjie Wei; Jianhong Wu

doi:10.3934/dcds.2003.9.1293

Article Contents

2003, Volume 9, Issue 5: 1293-1322. Doi: 10.3934/dcds.2003.9.1293

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Bifurcation from a homoclinic orbit in partial functional differential equations

1.
Department of Mathematics, The University of Miami, P.O. Box 249085, Coral Gables, Florida 33124
2.
Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001
3.
Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3

Received: April 30, 2002

Revised: October 31, 2002

Published: August 2003

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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 Šil'nikov for ordinary differential equations and of Chow and Deng for semilinear parabolic equations and retarded functional differential equations.

Keywords:

Homoclinic orbit,
partial functional diﬀerential equations,
stable and unstable manifolds,
variation of constant formula,
bifurcation,
periodic solutions.

Mathematics Subject Classification: 35R10, 34K15, 58F14.

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