Bifurcations Of Heteroclinic2-point-loop With Resonant Eigenvalues

This paper is devoted to study the bifurcation problems of heteroclinic loop-s with resonant eigenvalues for higher dimensional systems. It consists of threechapters:In chapter one, the background of problems and current research status ofbifurcation theory are briefly given and the main results obtained in the paper areintroduced.In chapter two, the bifurcation problems of heteroclinic2-point-loop with reso-nant eigenvalues for higher dimensional systems are discussed in detail. We mainlyconsider the following Crsystem（z|˙）=f（z）+g（z, μ）,and its unperturbed system（z|˙）=f（z）,where r≥3, z∈R^m+n, μ∈R^l, l≥2,0≤|μ|≤1, g（z,0）=0. We assumef（p_i）=0, g（p_i, μ）=0for i=1,2. The chapter is composed of five sections. Insection one and in section two, we obtain some preparation. In section one, the fun-damental hypotheses are given. In section two, we set up the P oincare′map whichis composed of two maps F_i⁰and F_i¹which is defined in the small neighborhood U_iof the saddle point piand defined in the small tube neighborhood of the heteroclinicorbit Γ_i, outside of Ui, respectively, where F_i⁰will be induced by the flow of the linear approximate system, F_i¹will be constructed from the flow of the perturbedsystem by using the Silnikov coordinate. The P oincare′map will be given by thecomposition of the above two maps, and then the successor function and bifurcationequation are obtained. In section three, we discuss the persistence of the heteroclinicloop Γ and the bifurcation problems of1homoclinic loop. In section four and insection five, we study the bifurcation problems of heteroclinic loops under nontwist-ed and twisted conditions, respectively. That is to say, the existence, uniqueness,incoexistence and existence fields of1homoclinic loop,1heteroclinic loop and1periodic orbit, moreover, we study the bifurcation problems of2heteroclinicloop and2homoclinic loop and give the corresponding discussion.In chapter three, the main methods of thinking and conclusions summarize tobe introduced. We point out the direction for studying the bifurcation problems ofheteroclinic2-point-loop with resonant eigenvalues in higher dimensional systems,and some work given needs to be solved.