Center Conditions And Bifurcation Of Limit Cycles For Two Classes Of Nilpotent Systems

This thesis is devoted to the problems of center-focus determination and bifurca-tion of limit cycles at the origin for two classes of nilpotent systems. It is composed of four chapters.In chapter1, it is introduced and summarized about the historical background and the present progress of problems about center-focus determination and bifurcation of planar polynomial differential system and the nilpotent singular point. At the same time, the main work of this paper is concluded.In chapter2, it is introduced some preliminary knowledge.In chapter3, the Problem of center conditions and bifurcation of limit cycles at the origin for a class of quartic nilpotent system are investigated. The first9Quasi-Lyapunov constants are computed by the recursive formula which was given in chapter2and the computer algebra system Mathematica, the conditions of the origin to be a center and the9st degree fine focus are derived correspondingly.9limit cycles which origin was surrounded in the neighbothood of origin are obtained when the system is Perturbed finely.In chapter4, the Problem of center conditions and bifurcation of limit cycles at the origin for a class of quinitic nilpotent system are investigated. The first11Quasi-Lyapunov constants are given, from which the conditions of the origin to be a center and the11st degree fine focus are derived correspondingly.11limit cycles which origin is surrounded in the neighbothood of origin are obtained when the system is Perturbed finely.