Two Types Of Limit Cycles Of Polynomial Systems

The main content of this paper is the number of limit cycles for the planarpolynomial system and we study the bifurcation of a planar polynomial system withnon-zero critical points and a system has Lienard type. The study about the numberof the planar system’s limit cycles has a long history, this problem is so difficult thateven the situation of quadratic system is still open. However, a lot of mathematiciansdevote to study these questions which derive from the problem and get a largenumber of meaningful and excellent results.This paper consists of four chapters. In chapter1, this paper introduce thedevelopment survey and main research directions of dynamical systems, discuss thenecessity and important meaning of study the limit cycles of the planar polynomialsystems, and then, the author briefly introduce the research work of this paper. Inchapter2, we discuss some basic theory that relate to this paper. In chapter3, weconsider the planar system x_=-yF(x, y)+εP(x, y), y_=xF(x, y)+εQ(x, y), where theset {F(x, y)=0} consists of m non-zero points (ai, bi)(i=1,···,m) with multiplemultiplicities, P(x, y) and Q(x, y) are arbitrary real polynomials. We study the numberof limit cycles bifurcating from the periodic annulus surrounding the origin by usingAbelian integrals and residues integration. In chapter4, we study the planar systemof the form x_=pk(y), y_=-gm(x)-εfn(x)y, where pk(y) is polynomial of degree k in y,and fn(x), gm(x) are polynomials in x with degree of n and m, respectively. Let H(m, n,k) denote the maximal number of limit cycles of this system, we give the lowerbounds of H(m, n, k) in some cases.