The Qualitative Analysis Of Some Polynomial Systems

This paper includes three chapters. The first chapter is overview, in which we introduce the developing situation about bifurcation theory and study situation about the weakened Hilbert' s 16th problem posed by Anold.In the second chapter, we deal with a particular form of the Hamiltonianfunction H(x,y) = (x) + (x) (y)given in [3]of Chengzhi Li and Zhifen Zhang.We obtain a criterion of the monotonicity of a ratio of two Abelian integrals by straightforward computation.The third section is concerned with the limit cycles for a class of higher degree polynomial systemWe transform this polynomial system into Lienard equation. Then we consider the number and distribution of limit cycles of it using the abundant results of Lienard equation. Finally, we consider the existence and uniqueness of limit cycles for the polynomial system.