Two Types Of Limit Cycle Problem For Planar Systems

In this thesis, we investigate the limit cycle problem into two topics of planar autonomous quadratic system and polynomial Lienard system of degree 4.This thesis is divided into 3 chapters.Chapter 1 is considered as an introduction, involving the review and some concepts of the theory of limit cycles.Chapter 2 is devoted to the study of limit cycle problem of type â…¢ quadratic systems with n = 0 in Ye' s classification, i. e. , consider system;x= -y + dx + lx² +mxy≡P(x,y), y=x(l +ax + by) ≡Q(x,y).It includes 4 sections. Section 1 contains a general analysis of the previous system, which shows the fundamental qualitative properties related to the limit cycles possibly around O.Section 2 considers the limit cycle problem when d = 0 and yields the following conclusions:â‘  When a, b, l satisfy a(b +2l) ≤0, then the system has no limit cycle;â‘¡ For fixed l, we consider the regions in (a,b) plane and by using the generalized Hopf bifurcation we can get the regions see Fig 2.4, for which the system may have one or two limit cycles.Other than the above regions, we obtain four parts numbered by â…  , â…¡ , â…¢ , â…£, in which we prove that there is no limit cycle around O.At the end of this section , we discuss a related conjecture in [ 5 ], which says that there is no limit cycle when d, V₃, V₅, V 7 are all positive or all negative for type â…¢ system. By using the conclusion above we conclude it is al-most true for system (IK)n=0.In section 3, we study the uniqueness of limit cycles of ( W) n=0-For the regions with no limit cycles obtained in section 2 for d = 0, then it is easy to prove there is also no limit cycle as d[l-a(b+21)] > 0. By using the theory of rotated vector fields, if d[l-a(b +21)] < 0, we conjecture that the limit cycle is at most one. We prove that it is true in the case a^0,b +21 ^0 and a^Q,b +2/=sO with certain additional conditions, see Theorems 2.1, 2.8.In seetion 4, we study the conjecture in [5] : if the quadratic system has a weak focus of order a and (3 limit cycles around it, than a +/3=S3. For systems ( HI ) n=0 with d = 0, Vj t^O , then O is a weak focus of order one, then we prove that the number of limit cycles around O is at most two, thus the conjecture is true. We obtain appropraite conditions under which the conclusion is true, see Theorems 2.9, 2. 10.Chapter 3 concerns with the polynomial Lienard system:dx ?, NJt=y-nx),dtwhere F(x) =alx + a2x2 + â– ?â–  +anxn.[ 7 ] gave a conjecture that when n = 2k + 1 or 2k + 2, then the system has at most k limit cycles. In this chapter we study the case n =4, under certain conditions of the coefficients ax, a2, a3 and a4, we proves that the system has at most one limit cycle, see Theorems 3.2, 3.3, 3.4, 3.5, 3.6.