Non-Resonant Bifurcations Of Heteroclinic Loops With Two Saddle Points And One Inclination Flip

In the dissertation we consider the bifurcation problems of heteroclinic loops with one inclination flip.We obtain Poincare map and bifurcation equations by establishing the local coordinate and using the transformation of Silnikov coordinate.Moverover, the existence of periodic orbits or homoclinic orbits is equivalent to the existence of positive solutions or zero solutions of the bifurcation equations.In the first chapter,some notations and definitions which are mentioned in the paper,at the same time,research background of the paper are introduced.In section 2.1 and 2.2,system(2.1)is firstly revised to normal form under the hypotheses.Moreover,we induce the bifurcation equation by using a local coordinate system established in a neighborhood of the heteroclinic 100ps.Then we consider bi-furcation problems under two different situations.In section 2.3,if(?),(?),the following results can be derived:System(2.1)has not any periodic orbit,homoclinic orbit or heteroclinic loops under some conditionsï¼›We prove the uniqueness of the 1-periodic orbit,1-homoclinic and heteroclinic loops and obtain their existence regionï¼›We obtain that periodic orbit and homoclinic orbit don't coexistï¼›System(2.1)has not any double 1-periodic orbit and 1-homoclinic orbit.In section 2.4,if(?),(?), we have the following results:The propositions which are similar to the first two results in section 2.3 are given and we proof the perturbed system can exist at most two 1- periodic orbits; 1-periodic orbit can coexist with 1-heteroclinic loops and 1-homoclinic orbit respectively; Lastly, we gain the existence of 2-periodic orbit and the bifurcation surface.