The Bifurcation Of Homoclinic Cycles Accompanied With Transcritical Bifurcation

The bifurcation theory of ordinary differential equations is one of the most important components of the theory of ordinary differential equations. It is important in a lot of fields of science, especially in the chaos and the reaction diffusion equations and so on. Since the local coordinate method appears, the bifurcation problem is extended.In this paper, two type problems of homoclinic bifurcation accompanied with transcriti-cal bifurcation are studied for the high dimensional system under the nongeneric hypothesis.In chapter 2, the problem of homoclinic bifurcation accompanied with transcritical bifur-cation and orbit-inclination flips is investigated. By constructing the local coordinate systems, we established a Poincare map to investigate the existence of sufficiently small nonegative solutions of the induced bifurcation equations. Firstly, we discuss bifurcations of homoclinic orbit when transcritical bifurcation dose not happen, the persistence of homoclinic orbit, the existence of homoclinic orbit and periodic orbit are established; Secondly, we analyze bi-furcations of homoclinic orbit accompanied with transcritical bifurcation, the persistence of homoclinic orbit and bifurcation of periodic orbit from the homoclinic orbit are obtained. Meanwhile, we also consider the influence of the inclination strength in the two conditions.In chapter 3, the problem of homoclinic bifurcation accompanied with transcritical bi-furcation is studied under the hypothesis of double orbit-flip. Using the moving frame and Pioncare map, we obtain the bifurcation equations. Based on the equations, sufficient con-ditions are given to guarantee the persistence of homoclinic orbits, the existence of periodic orbits and heteroclinic orbits. Meanwhile, some known results are extended.