The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations

Changrong  Zhu; Bin  Long

doi:10.3934/dcdsb.2016121

Article Contents

2016, Volume 21, Issue 10: 3793-3808. Doi: 10.3934/dcdsb.2016121

This issue Previous Article Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case Next Article

The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations

Changrong Zhu^1, and
Bin Long^1,

1.
College of Mathematics and Statistics, Chongqing University, Chongqing 401331

Received: August 2015

Revised: September 2016

Published: December 2016

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Abstract

Assume that the unperturbed parabolic equation has a degenerate homoclinic orbit. Under $T$-periodic perturbations, the periodic solutions bifurcated from the homoclinic solution are studied. By Fredholm alternative and Lyapunov-Schmidt reduction, the bifurcation functions defined between two finite-dimensional spaces are obtained. Some solvable conditions for the bifurcation functions are given. It is shown that, for any large $n>0$, the perturbed parabolic differential equation has a periodic solution with period $nT$.

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Mathematics Subject Classification: Primary: 34C23, 34C25; Secondary: 34C45, 34C40.

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