The Central Focus Of The Class Of Polynomial Differential Systems To Determine And Limit Cycles

This thesis is devoted to the problems of center-focus and limit cycle bifurcation for the differential system. It is composed of five chapters.In chapter 1, the historical background and the present progress of problem about center-focus and limit cycle bifurcation in planar polynomial differential system which have nilpotent singular point are introduced and summarized.In chapter 2, the successor functions and focal value of polynomial differential systems in which origin is elementary singular point are described. The definition of center and focus is also introduced.Several classical conclusions about the problem of center-focus are introduced. The definition of the system about symmetric and Hamilton and the center about symmetric and Hamilton is given.In chapter 3, quasi-Lyapunov constant and limit cycle bifurcation about cubic nilpotent singular point are introduced. the successor functions and focal value of cubic nilpotent singular point in generalized polar coordination are given. When the origin is a center, the property of that formal series become inverse integral factor is given, which is the property only beyonds to cubic singular point. A recursive formula to compute quasi-Lyapunov constant about cubic singular point is given too.In chapter 4, Using the recursive formula which is given by chapter 3 and computer system-Mathematica, the first seven quasi-Lyapunov constants of a class of five-order polynomial system are given. Seven limit cycles which origin is enclosed in a small neighborhood of origin are obtained when the system is in a small perturbation.In chapter 5, the first ten quasi-Lyapunov constants of a class of septic polynomial system are given by using the same method in chapter 4. Ten limit cycles which origin is enclosed in a small neighborhood of origin are obtained when the system is in a small perturbation.