Limit Cycles For Several Kinds Of Multi - Top Systems

The main content of this paper is the number of limit cycles for the planar polynomial system and we study the bifurcation of a planar polynomial system with non-zero critical points and a system has Lienard type. The study about the number of the planar system’s limit cycles has a long history, this problem is so difficult that even the situation of quadratic system is still open. However, a lot of mathematicians devote to study these questions which derive from the problem and get a large number of meaningful and excellent results.This paper consists of four chapters. In chapter1, this paper introduce the development survey and main research directions of dynamical systems, discuss the necessity and important meaning of study the limit cycles of the planar polynomial systems, and then, the author briefly introduce the research work of this paper. In chapter2, we discuss some basic theory that re-late to this paper. In chapter3, we consider the planar system x=-yK(x,y)+εP(x,y), y xK(x,y)+εQ(x,y), where the set{K(x,y)=0} consists of m, non-zero points (ai, bi)(i=1,…,m) with multiple multiplicities, P(x,y) and Q(x,y) are arbitrary real poly-nomials.In this paper, we use Abel integral and Maple and Matlab to study the m=1, k1=3, m=2, k1=2,k2=2and m=3,k1=1,k2=1,k3=2quadratic and cubic polynomial system, study the number of limit cycles bifurcating from the periodic annulus surrounding the origin by using Abelian integrals and residues integration.