Bifurcation from a homoclinic orbit in partial functional differential equations

,Shigui Ruan,Jianhong Wu,,Department Of Mathematics, Harbin Institute Of Technology, Harbin, Heilongjiang 150001 ,Junjie Wei

doi:10.3934/dcds.2003.9.1293

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.

Highlights

Where x ∈ Rn, ∈ R is a parameter and g is a smooth function, it is known that if x = 0 is a hyperbolic equilibrium for = 0 and the Jacobian matrix Dxf (0, 0) = A has a unique eigenvalue λ > 0 which is simple and the real parts of all other eigenvalues are strictly less than −λ, under certain additional transversality conditions, a unique stable periodic orbit bifurcates from the homoclinic orbit as the parameter changes
We study the Ck−smoothness of the stable and unstable manifolds of equation (2.1)
The exponentially asymptotic stability of γ follows from Theorem 5.3(a)

Summary

BIFURCATION FROM A HOMOCLINIC ORBIT IN PARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS

Department of Mathematics, University of Miami, P. O. Box 249085, Coral Gables, FL 33124-4250, USA Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada (Communicated by Peter Bates)

Introduction

This shows that

It follows that

Dφ us τ