Bifurcations Of Degenerated Rough Heteroclinic Loops With Two Saddle Points

In recent years, the theory of dynamical system has been used in the field of disciplines, including biology, chemistry, physics and so on, there are a wide range of applications. The homoclinic (heteroclinic) bifurcation of orbit has from the plane degenerated degree not high turned to the higher dimensional system. But when the space dimensions or degenerated degree increasing, there are few of people to study the complex phenomena of bifurcation. In the paper, we mainly study the bifurcations problems of 2-point degenerated heteroclinic loops in the high-dimensional system under the nonresonant condition. This paper includes four chapters:In chapter one, we give research present situation and trend of the main branch theory, and introduce the main results obtained in the paper. The concepts of heteroclinic loop, periodic orbit and homoclinic loop are also briefly given.In chapter two, bifurcations problems of degenerated heteroclinic loops which have a heavy characteristic root in the high dimensional systems are discussed in detail, which hyperbolic ratio is greater than 1(ρ11>λ11,ρ12>λ12). This chapter is composed of six sections. In the first three sections, we do some preparatory work. In section one, the fundamental hypotheses are given. In section two and section three, we set up the Poincare map by using the Silnikov coordinate, then the successor function and bifurcation equation are obtained under the degenerated case. In section four, we discuss the persistence of the heteroclinic loop and the birfucation problems of 1-homoclinic loop under the degenerated case. Then the existence of the corresponding area is discussed. In section five, we study the 1-periodic orbit bifurcation problems of heteroclinic loops under nontwisted and twisted conditions, namely to discuss the existence and the existence fields of 1-periodic orbit under these conditions. In section six, we discuss the more complex bifurcation problems, which include 2-heteroclinic loop,2-homoclinic loop and 2-periodic orbit problems.In chapter three, we simply give a four-dimensional heteroclinic loop system example conformed to assumptions. We can study this problem by using the method of chapter two.In chapter four, we summarize the content and ideas of this paper briefly. Furthermore, we point out the direction for studying the bifurcation problems of degenerated heteroclinic loops. Then, some suggestions and prospects are given to the further investigation.