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September  2003, 9(5): 1293-1322. doi: 10.3934/dcds.2003.9.1293

Bifurcation from a homoclinic orbit in partial functional differential equations

Shigui Ruan 1, , Junjie Wei 2, and  Jianhong Wu 3,

1. 

Department of Mathematics, The University of Miami, P.O. Box 249085, Coral Gables, Florida 33124, United States
2. 

Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China
3. 

Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada

Received  May 2002 Revised  November 2002 Published  June 2003

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: stable and unstable manifolds, bifurcation, Homoclinic orbit, variation of constant formula, periodic solutions., partial functional differential equations.
Mathematics Subject Classification: 35R10, 34K15, 58F1.
Citation: Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293
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