3D Homoclinic Orbit And High Dimensional Heterodimensional Cycle Bifurcations With Flip

Many researchers are devoted to the study of homoclinic bifurcation and heterodimensional cycle bifurcation. It’s of great importance to study bifurcations of singular cycles both theoretically and practically. The main work of this paper is to research the 3D homoclinic bifurcation with orbit-flip and the heterodimensional cycles bifurcation with inclination-flip in high dimensional space. The main methods are the local moving frame method and Shil’nikov time transformation method.The so called 3D homoclinic bifurcation with orbit-flip means the homoclinic orbit occurs orbit-flip in its positive direction, that is to say, the homoclinic orbit goes into the equilibrium along the strong stable direction. Firstly, in the small neighborhood of the equilibrium, the local linearization approximation is employed to get the singular flow map. Secondly, on the large scale, the local moving frame and Melnikov integral is used to establish the regular flow map.By composition of the two maps introduced above, the Poincaré return map is obtained.Consequently, we get the bifurcation equations. Through the discussion of the bifurcation equations, we obtain the bifurcation results of the bifurcation surface of 1-homoclinic orbit, the existence and non-existence of of 1-periodic orbit, the coexistense of 1-homoclinic orbit and1-periodic orbit, the existence of 2-fold periodic orbit and the existense of two periodic orbits.The so called heterodimensional cycle bifurcation with inclination-flip in higher dimensional space refers to the heterodimensional cycle whose second orbit occurs inclination-flip. More precisely, it refers to the inclination-flip of its unstable manifold. Firstly,we use the improved local linearization approximation in the small neighborhoods of the two equilibria respectively, and establish the corresponding singular flow maps. Then, we use the local moving frame and Melnikov integral to establish regular maps on the large scale.Correspondingly, we get the Poincaré return maps, and finally establish two bifurcation equtions.Through discussion of bifurcation equations, we have the bifurcation results as follows: the bifurcation surface of the persisted heterodimensional cycle, the existense and non-existence of the bifurcated homoclinic orbit, the existense and non-existence of the bifurcated periodic orbit.