| The second part of the famous16th Hilbert problem asks for the maximum number and relative positions of limit cycles of planar polynomial vector fields of a given degree n. This problem, opened for more than a century, has made a lot of results. However, it is still not solved even for n=2. In this paper, based on the averaging method of bifurcation theory of dynamical systems, the maximum number and expression of limit cycles of two classes of polynomial differential systems are studied. The full text of the content is divided into three chapters.The first chapter is the preface, in which we mainly introduce the background and known results of the limit cycles of polynomial differential systems and bifurcation, and briefly illustrate the main works of the thesis.In the second chapter, we investigate the number of limit cycles of the generalized Lienard polynomial differential systems of the formx=y-ε(g11(x)+f11(x)y)-kε2(g12(x)+f12(x)y),y=-x-ε(g21(x)+f21(x)y+h21(x)y2)-kε2(g22(x)+f22(x)y+h22(x)y2), for k=0and1, respectively. We provide the maximum number of limit cycles that the above differential systems can have bifurcating from the periodic orbits of the linear center x=y, y=-x using averaging method of first and second order. The contents of this chapter improves related results in [36].In the third chapter, the global shape of limit cycles for the quintic polynomial differential systems x=P1(x,y)+P5(x,y), y=Q1(x,y)+Q5(x,y), are studied. More precisely, by using the averaging method, we obtain the approximate expressions of limit cycles bifurcating from a Hopf bifurcation and also from periodic orbits of the linear center x=y, y=-x. |
