Bifurcation Analysis Of Several Types Of Planar Polynomial Systems

In this paper, by using the qualitative theories and bifurcation method of ordinary differential equations, several planar polynomial systems are studied, qualitative behaviors of trajectories are obtained. The whole paper consists of five chapters.The first chapter is the introduction, in which we introduce the developing history and the present progress of bifurcation theories, some fundamental definitions and lemmas of dynamics systems such as bifurcation and stability theory that can be used in this paper, and briefly represent the main works of the thesis.In the second chapter, the behaviors of all the singular points of a class ofE₃³ system are studied whenβ< 0 .And then the existence and uniqueness of limit cycle is proved by using the Hopf bifurcation theory; when|α| <2, -1/4γ²≤β<0 andα=γ= 0 ,at the same time, all the possible global structures of the system are obtained.In the third chapter, we give the sufficient condition for a class of E₃¹system to generate the Hopf bifurcation. When O(0,0) is a center, we obtain allthe possible global structures of the system through the analysis of the infinite singular points of the system.In the fourth chapter, the limit cycles bifurcated from a family of planar closed orbits of a class of Hamilton system under higher-degree polynomial perturbation are studied by using the method of bifurcation theory. According to that, three judgment theorems on existence and uniqueness of the limit cycles and their stability are obtained, the number and distributions of compound limit cycles are given, and then some results of predecessors are extended.In the fifth chapter, by using the bifurcation method to analyze the relative distance between the stable manifold and the unstable manifold after the Homoclinic loop of the unperturbed system is perturbed to break, we study the existing problem of limit cycles of a class of planar cubic systems and give the existence conditions of limit cycles, and then some results of predecessors are improved and extended.