【作者】 徐伟骄；

【导师】 韩茂安；

【作者基本信息】 上海师范大学 ， 应用数学， 2009， 硕士

【摘要】 本文第一章为引言,主要内容是介绍所研究课题的来源,现状,以及本文的研究方法和主要结论.第二章主要研究近哈密顿系统在Z₄等变七次扰动下产生极限环的个数.利用Hopf分支和异宿环分支的方法,扰动可得16个极限环,并且给出了它们的分布.本章中我们应用了最新的理论和方法,涉及相当复杂的计算.第三章主要研究近哈密顿系统在Z₄等变五次扰动下产生极限环的个数及近哈密顿系统在Z₄等变三次扰动下产生极限环的个数,利用第二章的有关结果和方法,并应用了一些技巧,分别得到13个和5个极限环,并且给出了它们的分布.第四章主要研究了方程（u丨¨）+q（t）f（t）=0,t∈（0,1）,边值条件为（?）（0）=0,u（1）=a₁u（ξ）+a₂u（η）,其中0＜ξ,η＜1,a₁+a₂＜1.应用锥上的不动点定理可得正解存在.本章主要创新是将已有的一类三点边值问题复杂化至四点边值问题,三点边值问题的所得结果在四点边值问题下同样成立.更多还原

【Abstract】 As an introduction,in the first chapter we introduce the background of our research and main topics that we will study in the following chapters.We also give a description of our methods and results detained in this thesis in the first chapter.In the second chapter,our main purpose is to concern with the number of limit cycles of a near-Hamiltonian system under Z₄-equivariant septic perturbation.Using the methord of Hopf and heteroclinic bifurcation,we found that the perturbed system can have 16 limit cycles.In this chapter,we use the latest theorems and methods, concerning with quite complicated computation,and the distributions are given.In the third chapter,we study the number of limit cycles of a near-Hamiltonian system under Z₄-equivariant quintic perturbation and the number of limit cycles of a near-Hamiltonian system under Z₄-equivariant cubic perturbation.Using the methods and results of the second chapter,then by some skills,we found that the perturbed systems respectively have 13 and 5 limit cycles,the distributions are given.In the fourth chapter,we study equation（u|¨）+q（t）f（t） = 0,t∈（0,1） with boundary conditions u（0） = 0,u（1） = a_u（ξ） + a₂u（η）,where 0 <ξ,η< 1,a₁ + a₂ < 1.The existence result of positive solution is obtained by applying the fixed point theorem in cones.The approaches developed here extend the ideas and techniques derived in recent literaures.The main innovative point of this chapter,we complicate a class of threepoint boundary problem into four-point boundary problem,the results of three-point boundary problem still holds in four-point boundary problem.更多还原