A Category Five System And The Proposed Center And The Bifurcation Of Limit Cycles Of The Five Systems,

This thesis is devoted to the problems of the center conditions, bifurcation of limit cycles for a class of fifth analytic system and quasi fifth analytic system. It is composed of three chapters.In chapter 1, 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.In chapter 2, the center conditions of a class of fifth analytic system and the bifurcation of limit cycles of the corresponding real system are investigated. The recursion formula for the computation of singular point quantities are given and the first 12 singular point quantities are deduced with computer algebra system Mathematica. Then, the conditions for the origin to be a center and 10-th and 12-th fine singular point with the critical form are derived respectively. An instance is given for the corresponding real fifth system that bifurcates 5 limit cycles.In chapter 3, the center conditions and the bifurcation of limit cycles of quasi fifth analytic system are investigated. The system is showed by some special kinds of real planar polynomial differential systems, when x is an arbitrary rational number. Thus, the situations of determining center-focus and bifurcation of limit cycles for elementary critical point, higher critical point and infinite point are solved. Then by converting the origin (or the singular point at infinity) into the origin of an equivalent complex differential system under the right transformation, the recursion formula for the computation of singular point quantities is given and the first 7 singular point quantities are deduced with computer algebra system Mathematica. So, the conditions for 6-th and 7-th fine singular point with the critical form are derived and the 4 and 5 limit cycles are given respectively.