# American Institute of Mathematical Sciences

December  2016, 21(10): 3793-3808. doi: 10.3934/dcdsb.2016121

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

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

Received  August 2015 Revised  September 2016 Published  November 2016

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 $nT$.
Citation: Changrong Zhu, Bin Long. The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3793-3808. doi: 10.3934/dcdsb.2016121
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