Analysis Of Equilibrium Points Of Two Kinds Of Planar Polynomial Systems

The theory of planar differential systems is widely applied in the natural and social sciences.The position and behavior of the equilibrium point determine the trajectory direction of the differential system.Thus the equilibrium point plays an important role in in describing the evolution of the object.This paper mainly studies the equilibrium points of two kinds of planar polynomial differential systems,which are divided into four parts.In Chapter 1,the recent domestic and overseas research on equilibrium points of planar polynomial systems,especially the central focus and the order of focus is introduced.In Chapter 2,we introduce the basic concepts of planar polynomial system and some methods to determine whether a singularity is a center or focus.These methods include the method of finding a periodic solution directly and method for formal series.We also introduce some important lemmas which are useful in this paper.Chapter 3 is devoted to studying the location and properties of equilibrium points of planar cubic polynomial systems with two parameters of the form???.Firstly,it is proved that the origin is a fourth-order fine focus of the system by using the method of finding a periodic solution directly,and the stability of the focus is determined according to the range of parameters.Then it is proved that when ???,the system has four infinite equilibrium points which are saddles,and three finite equilibrium points which are focuses.Moreover,the positions,orders and stability of these three focuses are given.In Chapter 4,by using the method of formal series,we discuss the order of weak focus of polynomial system expressed by the complex equation ???.An algorithm for calculating the order of focus of such system is provided.Furthermore,by employing this algorithm,we prove that the origin of the system is a center or a fine focus whose order is not less than 4n-7(where n>100).