Bifurcations Of Degenerated Heteroclinic Loops For Higher Dimensional Systems

In this paper,we chiefly study the bifurcation problems of degenerated hetero-clinic loops for higher dimensional systems under the nonresonant condition.This paper contains three chaptersIn chapter one,we give the related concepts of rough heteroclinic loop,1-periodic orbit and 1-homoclinic loop,then illustrate the background of bifurcation problems and latest research trends of bifurcation theory,in the end of paper,we introduce the main resultsIn chapter two,we discuss the bifurcations problems of degenerated heteroclinic loops which have a multiple-2 eigenvalues in the higher dimensional systems for the hyperbolic ratios ?i,i=1,2 satisfying ?1>1,?2<1 and ?1?2<1.We chiefly consider the following Cr system z=f(z)+g(z,?)and its unperturbed system z=f(z),where r>4,z?Rm+n,??Rl,l?2,0?|?|?1,g(0,?)=g(z.0)=0.For i=1,2.we assume f(pi)=0,g(Pi,?)=0.This chapter is made up of three parts.In part one,we give the some basic hypotheses.In part two,we build up the local active coordinate system under the degenerated situation.In part three,we construct the Poincare map which is made up of two maps Fi0 and Fi1 which is defined in the small neighborhood Ui of the saddle point pi and defined in the small tube neighborhood of the heteroclinic orbit ?i,where Fi0 will be induced by the flow of the linear approximate system,Fil will be constructed from the flow of the perturbed system by using the Silnikov coordinate,then obtain the successor function and bifurcation equation.In part four,we study the persistence of the heteroclinic loop and the bifurcation problems of 1-homoclinic loop under the degenerated situation,at the same time discuss the existence of the corresponding areas.In part five and part six,we study the 1-periodic orbit bifurcations problems of heteroclinic loop under twisted and nontwisted conditions,even discuss the more complicated bifurcations problems which including 2-heteroclinic loop,2-homoclinic loop and 2-periodic orbit problems.In chapter three,we summarize the content and the methods of this paper briefly.In addition,we give the further ideas for readers.