Class Of Differential Autonomous System Limit Cycle Branch

This thesis is devoted to the problems of integral conditions, center-focus determination and bifurcation of limit cycles at the origin and the infinity of planar polynomial differential system. It is composed of four chapters.In chapter 1, it is introduced and summarized about the historical background and the present progress of problems about center-focus determination and bifurcation of limit cycles of planar polynomial differential system. At the same time, the main work of this paper is concluded.In chapter 2, center conditions and integral conditions for a class of cubic polynomial differential system are studied. Recursion formula for the computation of singular point quantities are given, and with computer algebra system Mathematica the first 3 singular point quantities are deduced. At the same time, the integral conditions are studied carefully, and the conditions to be a center are derived.In chapter 3, center conditions and bifurcation of limit cycles from the equator for a class of quintic polynomial system are studied. The recursion formula for the quintic polynomial system in the infinity are given. With the formula, the first ten singular point quantities of a class of quintic polynomial system for the infinity are computed with computer algebra system Mathematica. The conditions for the infinity to be a center are derived as well. At last, it is the conducted that the system allows the appearance of seven limit cycles in a small neighbourhood of the infinity.In chapter 4, center conditions and bifurcation of limit cycles at the origin and infinity (the equator) in a class of quintic polynomial differ-rential system with one small parameter and eight normal parameters are studied. From computing the generalized Lyapunov constants (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 {(5),3}and {(3),3} are obtained under simultaneous perturbation at the origin and infinity.