Bifurcations Of Degenerated Double Homoclinic Loops For Higher Dimensions

This paper is devoted to study the bifurcation problems of degenerated double homoclinic loops for higher dimensional systems. We will study the bifurcation problems of double homoclinic loops through analyzing Poincare map on the cross section. Firstly, we simplify the system in some sufficiently small neighborhood of the saddle point, and use the foundational solutions of the linear variational equations of the unperturbed system along the double homoclinic orbits as the demanded local coordinate system. Then, in some sufficiently small neighborhood of the saddle point, we select the Poincare sections to construct the Poincare map which is the compositions of two maps. One of the maps defined in the small neighborhood of the saddle point, will be induced by the linear approximate system. And, the other map defined in the small tube neighborhood of the homoclinic orbits, will be induced by a transformation. Then, the Poincare map will be constructed by the two maps. After that, the successor functions and bifurcation equations are obtained. The Poincare map and the bifurcation equations we obtained by the above method are more simple and easy to analyze.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 problem about degenerated double homoclinic loops for higher dimensions is discussed. After the assumptions and preparation, in section four, we study the bifurcation problems of the degenerated double homoclinic loops under non-twisted conditions. Particularly, We study the existence, uniqueness and non-coexistence problem of double homoclinic loops, big1-homoclinic loops, big1-period orbit. In section five, we study the bifurcation problems of the degenerated double homoclinic loops under twisted conditions. Particularly, We study the exis-tence, uniqueness and non-coexistence problem of double homoclinic loops,1-lbig homoclinic loops,2-lbig homoclinic loops,2-lbig period orbit,2-lright period orbit.In chapter three, the main methods of thinking and conclusions summarize to be introduced. We point out the direction for studying the bifurcation problems of double homoclinic loops in higher dimensional systems, and some work given needs to be solved.