Exponential Dichotomies and Homoclinic Orbits from Heteroclinic Cycles

Tiejun Chen,Yu Xiang,Yuxiao Chen

doi:10.4236/ajcm.2012.22014

Abstract

In this paper, we investigate the homoclinic bifurcations from a heteroclinic cycle by using exponential dichotomies. We give a Melnikov—type condition assuring the existence of homoclinic orbits form heteroclinic cycle. We improve some important results.

Highlights

We consider the n-dimensional differential equations x f x, v, (1.1)where x Rn, is a small parameter, v R2 is a parameter
We investigate the homoclinic bifurcations from a heteroclinic cycle by using exponential dichotomies
We give a Melnikov—type condition assuring the existence of homoclinic orbits form heteroclinic cycle

Summary

Introduction

Where x Rn , is a small parameter, v R2 is a parameter. In studying the global bifurcation, we usuaally assume unperturbed differential equations x f x, 0, 0. In studying the pulses solutions of some recation-diffusion equations, we often meet the problem of homoclinic bifurcations from the heteroclinic cycles, refer to Kokubu [6], Chow, Deng and Terman [7], Gambaudo [8] and reference therein. In Kokubu [6], he needs to divide the problem into critical and non-critical two cases He needs that the heteroclinic orbits approach the hyperbolic equilibriums along the eignspaces associated with the principal eigenvalues. Deng and Terman [9] studied the same problem as this paper, Kokubu [7] did not need to divide the problem into critical and non-critical two cases and unified the two cases and didn’t ndde that the heteroclinic orbits approach the hyperbolic equilibriums along the eigenspaces associated with the principal eigenvalues. We say Equation (1.3) admits an exponential dichotomy on interval J if ther exist con stants K, α, a projection P and the fundamental matrix X(t) of Equation (1.3) satisfying;

Main Result

Findings

The Proof of the Main Result