The existences of transverse homoclinic solutions and chaos for parabolic equations |
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Authors: | Changrong Zhu Guangping Luo |
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Institution: | a Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China b Department of Mathematics and Physics, Chongqing University, Chongqing 400044, PR China |
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Abstract: | By using Lyapunov-Schmidt reduction and exponential dichotomies, the persistence of homoclinic orbit is considered for parabolic equations with small perturbations. Bifurcation functions are obtained, where d is the dimension of the intersection of the stable and unstable manifolds. The zeros of H correspond to the existence of the homoclinic orbit for the perturbed systems. Some applicable conditions are given to ensure that the functions are solvable. Moreover the homoclinic solution for the perturbed system is transversal under the applicable conditions and hence the perturbed system exhibits chaos. The basic tools are shadowing lemma which was obtained by Blazquez (see C.M. Blazquez, Transverse homoclinic orbits in periodically perturbed parabolic equations, Nonlinear Anal. 10 (1986) 1277-1291]). |
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Keywords: | Homoclinic bifurcation Chaotic motion Lyapunov-Schmidt method Exponential dichotomy |
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