Center Problems And Bifurcation Of Limit Cycles For Several Classes Of Planar Polynomial Systems

This thesis is devoted to the problems of integral conditions,centerfocus determination and bifurcation of limit cycles at the origin and the infinity of planar polynomial differential system.It is composed of five chapters.In chapter 1 and chapter 2,it is introduced and summarized about the historical background and the present progress of problems about centerfocus determination and bifurcation of limit cycles of planar polynomial differential system.At the same time,the main work of this paper is concluded and some preliminary knowledge is introduced.In chapter 3,the problem of center conditions and bifurcation of limit cycles at infinity for a class of cubic system are investigated,the method is based on a proper transformation of the infinity into the origin,the first 21 singular point quantities are computed by computer algebra system Mathematica, the conditions of the origin to be a center and the 21st degree fine focus are derived correspondingly.Finally,we construct a cubic system which can bifurcate 7 limit cycles from infinity by a small perturbation of parameters.In chapter 4,we study the center conditions and bifurcation of limit cycles at the higher-degree singular point and infinity in a class of septic polynomial differential system in which the higher-degree singular point and infinity can be transformed into the origin by two proper transforma- tions.From computing the singular point quantities for the origin and infinity, we can get conditions for the origin and infinity to be centers.The limit cycle configurations of {(9),2} and {(2),9} are obtained under simultaneous perturbation at the origin and infinity.In chapter 5,using an indirect method to study center conditions and bifurcation of limit cycle for a class of quasi quintic system,We obtain 7 limit cycles.It is the first time that 7 limit cycles can bifurcated from the infinity for a class of quasi quintic system.