Study Of Limit Cycles Of The Class Of Differential Systems

This thesis is devoted to studying the number and existence and uniqueness of limit cycles for some differential systems. It consists of six chapters.In Chapter 1, the background and present conditions are introduced and summarized for the study of the number and existence and uniqueness of limit cycles for differential systems.In Chapter 2, a class of nonlinear differential system is investigated.For this system withα= 0,b≠0andα=b²/4,GSansone theorem and rotatedvector fields are applied to discuss the existence , uniqueness and nonexi-stence of limit cycles for the differential systems.In Chapter 3, two classes of high order differential systems are investigated.andwhere F(x)=-[δx+ax^2m+lx²ⁿ⁺¹G(x)]=(?)f(x)dx,2n+1>2m(m,n∈N⁺).In this chapter, we have obtained some results of the existence, uniqueness and nonexistence of limit cycles for two classes of high order diffeential system by the generalized rotated vector field theory and the popularized Sansone theorem.In Chapter 4, the following two classes of plane differential systems are studied. The complete results of existence, uniqueness and nonexistence of limit cycles for two classes of plane differential systems are obtained by means of the method of the theory of N. Levison,O.K.Smith, Poincare tangent curve, the theory of rotation vector field, the ring region theroem and Zhang Zhifen theorem.In Chapter 5, a class of polynomial differential system is studied.The complete results of existence, uniqueness and nonexistence of limit cycles for a class of polynomial differential system is obtained, by means of the method of Poincare tangent curve, the theory ofА．�'．Драгилёвand N. Levison,O.K.Smith.In Chapter 6, the number of limit cycles of a planar near-Hamiltoni-an Systems under higher-order perturbations with multiple parameters is studied.where a > 0,εis a small parameter,δ∈D(?)Rⁿ, with D bounded, n≥1, and p(x,y,δ)=(?)δ_2i+1,2jx²ⁱ⁺¹y^2j,q(x,y,δ)=(?)δ_2i,2j+1x²ⁱy^2j+1(i,j=0,1,2…) Westudy the Melnikov functions of a planar near-Hamiltonian Systems under high order perturbations with multiple parameters, and discuss the maximal number of limit cycles in Hopf bifurcations by using the first-order Melnikov functions, and give some examples for application.