Bifurcation Problems Of Homoclinic Loop Or Heteroclinic Loop With Inclination Flip Or Orbit Flip In Higher Dimensional Systems

In this thesis we consider the bifurcation problems of homoclinic loop or heteroclinic loops with inclination flip or orbit flips in higher dimensional systems. By using a local coordinate system established in a neighborhood of the homoclinic loop or heteroclinic loop(This was first introduced in paper[74]), we construct the Poincare map and induce the bifurcation equation. Our aim is to analyse the bifurcation behaviour of a four dimensional system by solving this bifurcation equation. The existence, nonexistence, uniqueness and coexistence of the 1-periodic orbit, the 1-homoclinic loop and heteroclinic loop are studied. The existence of the two-fold periodic orbit and three-fold periodic orbit are also obtained. For the homoclinic loop or heteroclinic loop with one inclination flip or two inclination flips, we show that the number of periodic orbits bifurcated from this kind of homoclinic loop or heteroclinic loop depends heavily on their strength of the inclination flip. In section 2.1 we study codimension 3 non-resonant bifurcations of homoclinic orbits with two inclination flips in 4 dimensional systems. For this case,we obtain that if two invariant manifolds along the original homoclinic orbit (?)hom are not strong inclination flip then the perturbed system can exists at most one 1-periodic orbit near (?)hom, if only one of the invariant manifolds is strong inclination flip then the perturbed system can exists at most two 1-periodic orbits near (?)hom, if two invariant manifolds are strong inclination flip then the perturbed system can exists at most three 1-periodic orbits near (?)hom. And, there is similar character when the perturbed system have a homoclinic orbit and some 1-periodic orbits at the same time. That is, premising the perturbed system has a homoclinic orbit (?)μ, if two invariant manifolds are not strong inclination flip then the perturbed system have no any 1-periodic orbit near (?)hom , if only one of the invariant manifolds is strong inclination flip then the perturbed system can have at most one 1-periodic orbit near (?)hom , if all of two invariant manifolds are strong inclination flip then the perturbed system can have at most two 1-periodic orbits near (?)hom. In section 2.2 we study codimension 3 non-resonant bifurcations of homoclinic loop with orbit flips and inclination flips in 4 dimensional systems. We show that the perturbed system can have three 1-periodic orbits if only if a homoclinic orbit is strong inclination flip. In section 3.1 we study codimension 3 non-resonant bifurcations of heteroclinic loop with orbit flips in 4 dimensional systems.