Bifurcation Of Homoclinic Orbit Accompanied With Hopf Bifurcation

In this paper, we mainly study a class of the bifurcation problems of homoclinic orbit with a saddle-focus point for higher dimensional systems. Firstly, we give the simplified expression of the system in some sufficiently small neighborhood of the saddle-focus point and reduce dimensionality of the center manifold by the polar coordinates transformation. Then, we study the n-1 dimensional system instead of the original n dimensional system. We construct a Poincare map of the system by setting up local moving coordinate system, and obtain the successor functions and bifurcation equations. Hence, the study of bifurcation problem is converted into discussing the existence of nonnegative solutions of the bifurcation equations.In section one, we mainly introduce the research background and current sit-uation of the bifurcation theory. In section two, we give the basic assumptions. In section three, we set up the local moving coordinate system. In section four, we construct the Poincare map and obtain the successor functions and bifurcation e-quations. In section five, based on the symbol of the perturbation parameter A and analyzing the bifurcation equations under untwisted condition, we give the persis-tent conditions of the homoclinic orbit and conditions of 1-periodic orbit. Hence, for the original system, we obtain persistent conditions of the homoclinic orbit and conditions of 1-heteroclinic manifold and 1-periodic orbit. In section six, we point out the future research of bifurcation of homoclinic (heteroclinic) orbit accompanied with Hopf bifurcation for higher dimensional systems.