Homoclinic Manifold Branch In The High-dimensional System

In this paper, the homoclinic bifurcations taken in the homoclinic manifold for the 4-dimensional system are considered. The results of [5] are extended to the homoclinic manifold consisting of a series of homoclinic orbits F(a),a G / C R. The existence of 1-periodic orbit and 1-homoclinic orbit is proved respectively near the homoclinic orbit T(a) corresponding to a = a. Especially the conditions for the existence of 2-fold 1-periodic orbit are given under the resonant conditions. Moreover, the approximate expressions of the corresponding bifurcation surfaces are given.The paper is consist of two chapters. In chapter one, we look back the history and current development of the study of bifurcation problems of homoclinic orbits. Then, a general summarization of this thesis is given. In chapter two, we study the problems of the homoclinic manifold bifurcations for the 4-dimensional system. Considering Cr systemwhere (H1) If e system (1.2) has a 2-dimensional homoclinic manifold consisting of homoclinic orbits , I is an open interval, and are the eigenvalues satisfying (H2) Let Ws, Wu be the stable and unstable manifold of saddle O respectively.then e+ T0Wu ande- G T0Ws are unit eigenvalues corresponding to 1 and respectively.At first, we give a fine normal form for the system in some sufficiently small neighborhood of the saddle point under a suitable transformation, and use the foundational solutions of the linear variational equation of the unperturbed system along the homoclinic or heteroclinic orbits as the demanded local coordinate system to establish another normal form. Then, in the small neighborhood of the saddle point, the Poincaresections are selected to construct the Poincare map which is the composition of two maps. One of the maps, which is defined in the small neighborhood of the saddle point, is induced by the flow of the linear approximate system. The other map, which is defined in a small tube neighborhood of the homoclinic or heteroclinic orbits and outside of a neighborhood of the saddle points, is constructed from the flow of perturbed system under the new coordinates. The composition of the above two maps produces the Poincare map, which in turn yields the successor function and bifurcation equation. The Poincare' map and the bifurcation equation obtained by the above method are much simpler and easily to be used. This method has not only the theoretical significance, but also is more applicable to real system.The main results are included in the section three and section four of the chapter two.In section three, we study the problems of the homoclinic bifurcation under the nonresonant condition. The existence, uniqueness and incoexistence of 1-homoclinic orbit and 1-periodic orbit are obtained under the nonresonant condition. Moreover, the approximate expressions of the corresponding bifurcation surfaces are given.In section four, we study the problems of the homoclinic bifurcation which has resonant eigenvalues. First, the homoclinic bifurcation is studied which satisfies the conditions . The existence and uniqueness of 1-homoclinic orbit, the existence of 1-periodic orbit, 2-fold 1-periodic orbit, two 1-periodic orbits, the coexistence of 1-homoclinic orbit and 1-periodic orbit are obtained. Moreover, the corresponding bifurcation surfaces are given. Second, we' study the homoclinic bifurcation which satisfies the conditions Now, in the process of the change of the parameter , the saddle O keeps being a weak saddle. We confine to consider the case , then the existence of 1-homoclinic orbit, 1-periodic orbit, the corresponding existing region and the incoexistence of 2-fold 1-periodic orbit are obtained., the existence of 1-homoclinic orbit, l-periodic orbit, 2-fold 1-periodic orbit, two 1-periodic orbits, the coexistenc of 1-homocinic orbit and 1-periodic orbit and the corresponding existingdomain are also given.