Bifurcations Of Heteroclinic Loop Accompanied By Saddle Node Bifurcation

This paper is devoted to study the bifurcation problems of heteroclinic loops accompanied by saddle node bifurcation for high dimensional systems. It consists of three chapters:In chapter one, the background of problems and current research status of bifurcation theory are briefly given and the main results obtained in the paper are introduced.In chapter two, the bifurcation problems of heteroclinic loop accompanied by saddle node bifurcation are discussed in detail. We mainly consider the following Cr system z=F(z)+G(z,?,?), and its unperturbed system z=F(z), where r?3, z ? Rm+n, ? ? R, ? ? J. J is a open neighbohood of Rl, we assume F(pi)=0 for i=1,2, G(z,0,0)= 0, G(0,?,?)=0. The chapter is composed of seven sections. In section one, the fundamental hypotheses are given. In section two, we set up the Poincare map which is composed of two maps Fi0 and Fi1 which is defined in the small neighborhood Ui of the equilibrium pi and defined in the small tube neighborhood of the heteroclinic orbit ?i, outside of Ui, respectively, where Fi0 will be induced by the flow of the linear approximate system, Fl will be constructed from the flow of the perturbed system by a transformation. The Poincare map will be given by the composition of the above two maps, and then the successor function and bifurcation equation are obtained. In section three under ?=0 condition, the transcritical bifurcation does not happen, we discuss the persistence of the heteroclinic loop ?, and the existence of 1-homoclinic loop. In section four and five, under the nonresonant and resonant conditions, we study the existence of 1-periodic orbit for the case of A= 0. In section six, when ??0, the transcritical bifurcation happens, we discuss the persistence of the heteroclinic loop ? and, the existence of 1-homoclinic loop. In section seven and eight, under the nonresonant and resonant conditions, we study the existence of 1-periodic orbit for the case of ??0.In chapter three, the main methods of thinking and conclusions summarize to be introduced, and some suggestions are given to the further investigation.