Limit Cycles Of Z₄-equivariant System And A Class Of Four-point Boundary-value Problem

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.