This thesis is devoted to the problems of integral conditions, center-focus determination and bifurcation of limit cycles at degeneratesingular point and the infinity of planar polynomial differential system. Itis composed of seven chapters.In chapter 1, the historical background and the present progress ofproblems about center-focus determination and bifurcation of limit cyclesof planar polynomial differential system were introduced and summarized.At the same time, the main work of this paper was concluded.In chapter 2, center conditions and bifurcation of limit cycles fromthe equator for a class of cubic polynomial system with no singular pointat the infinity were studied. By converting real planar system intocomplex system, the recursion formula for the computation of singularpoint quantities were given, and, with computer algebra systemMathematica, the first 7 singular point quantities were deduced. At thesame time, the conditions for the infinity to be a center and 7 degree finefocus were derived respectively. A cubic system that bifurcates 7 limitcycles from the infinity was obtained. This result was published on《Applied Mathematics and Computation》2006, V. 177（1）.In chapter 3, center conditions and bifurcation of limit cycles fromthe equator for a class of quintic polynomial system with no singularpoint at the infinity were studied. The recursion formula to compute thesingular point quantities of quintic polynomial system at the infinity wasgiven. With this formula, the first eleven singular point quantities of aclass of quintic polynomial differential system at the infinity werecomputed with computer algebra system Mathematica. The conditions forthe infinity to be a center were derived as well. At last, a system that allows the appearance of eleven limit cycles in a small enoughneighborhood of the infinity was constructed at the first time. This resulthas been accepted on《Computers and Mathematics with Applications》.In chapter 4, center conditions and bifurcation of limit cycles fromthe equator in a class of polynomial system of degree seven were studied.The method was based on converting real planar system into complexsystem, the reeursion formula for the computation of singular pointquantities of the infinity were given, which allows us to compute thegeneralized Lyapunov constants （the singular point quantities） for theinfinity. The first 14 singular point quantities of the infinity were deduced.At the same time, the conditions for the infinity to be a center and 14degree fine focus were derived respectively. A system of degree 7 thatbifurcates 13 limit cycles from infinity was constructed at the first time.In chapter 5, Center conditions and bifurcation of limit cycles at thedegenerate singular point and infinity （the equator） in a class of quinticpolynomial differential system with two small parameters and eightnormal parameters was studied. The method was based on twohomeomorphic transformations of the infinity and degenerate singularpoint into linear singular point, which allows us to compute thegeneralized Lyapunov constants （the singular point quantities） for theorigin and infinity. The center conditions for the degenerate singular pointand infinity were derived respectively. The limit cycle configurations of{（7）, 2} and {（2）, 6} were obtained under simultaneous perturbation atthe origin and infinity. This result was published on《AppliedMathematics and Computation》2006, V. 181 （1）.In chapter 6, the origin of a class of general complex autonomouspolynomial differential system (with z, w, T independent complex variables,α_{αβ}，b_{αβ}complex constants, （p, q）=1, n natural)was studied. It is a degenerate singular point. Theextended singular point quantity of the singular point was defined, at thesame time, the construction of the extended singular point quantity werestudied. The linear recursion formula to compute the extended singularpoint quantity was given and necessary and sufficient condition of thesingular point to be a extended complex center was obtained. Theconclusion of this chapter is the expansion of that of [41, 46].In chapter 7, the infinity of a class of general complex autonomouspolynomial differential system(with z, w, T independent complex variables, a_{αβ}, b_{αβ} complex constants, （p, q）=1, n natural)was studied. The extended singular point quantity ofthe infinity was defined, at the same time, the construction of theextended singular point quantity of the infinity were studied. The linearrecursion formula to compute the extended singular point quantity of theinfinity was given and necessary and sufficient condition of the infinity tobe a extended complex center was obtained. The conclusion of thischapter is the expansion of that of [46]. |