The Central Focus Of The Two Types Of Differential Systems Nilpotent Singular Point Determined, And The Bifurcation Of Limit Cycles

This thesis is devoted to the problems of center-focus and limit cycle bifurcation for the differential system. It is composed of three chapters.In chapter 1, the historical background and the present progress of problems about center conditions, the highest degree fine focus conditions and the bifurcation of limit cycles of planar polynomial differential system were introduced and summarized, and also the historical background and the present progress about the differential system in which origin is nilpotent singular.In chapter 2, the works about the differential system in which origin is nilpotent singular, which is done by the people of the past, were introduced and summarized, and also the methods to get the center conditions, the highest degree fine focus conditions and the bifurcation of limit cycles for a class of differential system in which origin is nilpotent singular point. And all of those methods made a solid foundation for the chapter 3 which is the center of this thesis.In chapter 3, two classes of differential system in which origin is nilpotent singular point were studied. For one of classes which is cubic differential system, using the recursive formula which was given in chapter 2 and the computer system-Mathematic, the first six quasi-Lyapunov constants of the system were given, from which the conditions for origin to be a center and the highest degree fine focus were derived. Six limit cycles which origin was surrounded in the neighborhood of origin were obtained when the system was perturbed finely. And for the other class which is septic differential system, using the same method and the computer system-Mathematic, the first nine quasi-Lyapunov constants of the system were given, from which the conditions for origin to be a center and the highest degree fine focus were derived. Nine limit cycles which origin was surrounded in the neighborhood of origin were obtained when the system was perturbed finely.