The Center Conditions And The Bifurcation Of Differential System

This paper focuses on several types of odd system problems, divided into the following four chapters.The first chapter,the historical background and the Present Progress of the Problems about center conditions and bifurcation of limit cycles of Planar Polynomial differential systems are introduced and summarized.The second chapter describes the relationship between the focus and fine singular Point with the critical form are derived receptive, and the amount of the recursive formula singular point of knowledge.The third chapter divided into four sections studied. Section one studied the center conditions about the origin of the secondary system with complex is a11=2b20,b11=2a20,a02b202-b02a202=0。The system corresponds to the real system can be divided into two limit cycles. Section two studied the necessary and sufficient conditions for the origin of cubic system with complex is a21=b21,a12=3a30,b12,12=3a30。Section three studied the conditions of the origin of the complex system of five fifth-order for the singularity-b21=b30b12-a30bu=b503/2b230=0. Section four studied fifth systems with complex can bifurcation limit five limit cycles.The fourth chapter,we consider a discrete delayed population model obtained by Euler method. Firstly, the linear stability of the model is studied. It is shown that the system will undergo Neimark-Sacker bifurcation when the delay passes through some critical values. Some numerical simulations are carried out to illustrate our results with the theoretical analysis.