Bifurcation Of Homoclinic Orbit Accompanied With Nilpotent Singular Point

In this paper,we mainly study a class of the bifurcation problems of heteroclinic orbit with nilpotent singular point for higher dimensional systems.Firstly,we give the normal form of the system in a sufficiently small neighborhood of singular point Pi and reduce dimensionality of the center manifold by the polar coordinates trans-formation.Thus,we study the m-1 dimensional system instead of the original m dimensional system.Secondly,we construct maps Fi0 of in a sufficiently small neigh-borhood Ui0 of singular poiit pi and we construct maps Fi1 in a sufficiently small tube neighborhood Ui1 of homoclinic orbit by setting up local moving coordinate system,then combinate the two map and obtain Poincare maps,then,the succes-sor functions and bifurcation equations obtained.Finally,the study of bifurcation problem is converted into the study of the existence of nonnegative solutions of the bifurcation equations.Then we get some results about the existence,coexistence,periodicity of bifurcated homoclinic orbits,periodic orbits and heterolinic orbits.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,it is the bifurcation of het-erolinic orbit accompanied with nilpotent singular point.This chapter is introduced in four parts:the basic assumptions,the establishment of normative and local coordinate systems,the construction of the Poincare map and branch equations,bifurcations of non-twisted heterolinic orbit under nonresonant condition.the main results of bifurcation has two small parts:the result of bifurcation when the dis-turbance parameter of the singularity of ? = 0,the result of bifurcation when the disturbance parameter of the singularity of ??0.In chapter three,it is the future research of bifurcation of heterolinic orbit accompanied with nilpotent singular point for higher dimensional systems.