几类微分系统的极限环研究

【作者】 杨晓霞；

【导师】 陈海波；

【作者基本信息】 中南大学 ， 应用数学， 2007， 硕士

【摘要】 本文研究了几类微分系统的极限环的存在唯一性及个数问题，由六章组成。第一章主要对微分系统的极限环存在唯一性及个数问题的历史背景与现状进行了综述。第二章研究了一类非线性微分系统。运用G.Sansone定理和旋转向量场理论在给定已知条件下对此类微分系统分析，得到了该系统在参数a=0,b≠0与a=b~2/4等两种情况下的极限环的存在性与唯一性及不存在性的较为完整的分析。第三章研究了两类高次微分系统。及其中(?)，2n+1>2m(m，n为正整数)的极限环的存在唯一性。本章运用推广的G．Sansone定理和广义旋转向量场理论分别对两类高次微分系统进行了定性分析，得到了较为完整的结果。第四章研究了两类平面微分系统，借助N．Levison,O．K．Smith定理、Poincare切性曲线法、旋转向量场理论、环域定理和张芷芬定理对这两类平面微分系统进行全面分析，得到其极限环的存在性、唯一性与不存在性的较完整结果。第五章研究了一类多项式微分系统。通过运用庞卡来(H·Poincare)切性曲线法先证明极限环的不存在性，再运用А．В．Драгилёв定理及N．Levison,O．K．Smith定理来证明平面微分系统极限环的存在性与唯一性。第六章研究了一类带参数高次扰动的平面近Hamilton系统的极限环个数。其中a>0，ε是一个小参数，δ∈D(?)R~n，D有界，n≥1，并且本章研究带参数高次扰动的平面近Hamilton系统的Melnikov函数，利用一阶Melnikov函数来确定其在Hopf分支中极限环的个数，并给出了实例。更多还原

【Abstract】 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．В．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.更多还原