The Qualitative Analysis For Some Kinds Of Planar Differential Systems

In this thesis, by applying the qualitative theory and stability theory of ordinary differential equations, we mainly study the properties of equilibrium, existence and uniqueness of limit cycle for some kinds of planar differential systems.This paper is composed of six chapters.In the first chapter, we introduce the background of problems which will be investigated and the main results of this paper. At the same time, some basic theory of the qualitative theory and stability theory of ordinary differential equations are given.Chapter 2 deals with the study of two types of planar differential systems with 2n+1 degree. By the singular theory, the properties of the finite singular points and infinite singular points are obtained. By the Dulac function, the nonexistence of the closed orbits in whole plane is proved. Under respective conditions, the phase portraits about global structure on the Poincaré-disc are given.Chapter 3 mainly considers a class of a biochemistry reaction system. The sufficient conditions for the boundedness of solutions and the existence for limit cycle are obtained. Some known results are generalized.In chapter 4, we study a class of higher planar differential system. By the formal series method, the center and focus are judged, by the Dulac function, the non-existence of closed orbit is discussed, by the Hopf bifurcation theory, some sufficient conditions for the existence and stability of limit cycles bifurcated from the equilibrium point is analyzed, then, under the change of time, the system is changed into Liénard equation, and then, by constructing comparison system and using the comparison principle of differential equation, all conditions in the theorem ofЛ.А.ЧеркасandЛ.И.Жилевычit can be proved, finally, by applying of the theorem about uniqueness, some sufficient conditions for the uniqueness and stability of limit cycles are established.In chapter 5, by being added higher degree disturbance terms and deleted the assumption on parameters, a class of planar differential system is studied. By Hopf bifurcation theory, some sufficient conditions for the existence of limit cycles of such systems are obtained. By applying of theЛ.А.ЧеркасandЛ.И.Жилевыч's theorem about uniqueness, some sufficient conditions for the uniqueness of limit cycles of such system are established. Some known results about the properties of limit cycles around origin are generalized.In the last chapter, by the same method as chapter 5, we consider a class of planar differential system. Some sufficient conditions for the existence, uniqueness and stability of limit cycles of such system are obtained.