A System To Be Three Times The Center Of The Bifurcation Of Limit Cycles And When The Center

This thesis is devoted to the problems of the center conditions, bifurcation of limit cycles and the isochronous conditions for quasi analytic systems. It is composed of three chapters.In chapter 1, the historical background and the present progress of problem about center conditions, bifurcation of limit cycles and isochronous conditions of planar polynomial differential system are introduced and summarized.In chapter 2, center condition and bifurcation of limit cycles of quasi cubic system are investigated. By converting the origin or the singular point at infinity of quasi analytic system into the origin of a equivalent complex analytic system, the recursion formula for computation of singular point quantities are given, and, with computer algebra system Mathematica, the first 18 singular point quantities are deduced. At the same time, the conditions for the origin to be a center and 15-order and 18-order fine focus are derived respectively. A quasi cubic system that bifurcated 5 limit cycles, three of which are stable, at the origin is obtained. This result was published on《Journal of Heilongjiang Institute of Science&Technology》2007,17(3).In chapter 3, the problem of the isochronous conditions of quasi cubic system in chapter 2 is investigated. Firstly, by using a new method, the recursion formula for calculating periodic constants are given, then the first 9 periodic constants are deduced. At the same time, necessary conditions for center to be an isochronous center are given, and proof of isochronous these systems by using some effective methods are given, too.