一类非线性系统周期解存在唯一性.doc

师范学院 本科生毕业论文 论文题目: 一类非线性系统周期解的存在唯一性 学院、年级：数学与信息科学学院 2012级 学科、专业：数学 数学与应用数学 完 成 日 期：2016 曲靖师范学院教务处 一类非线性系统周期解的存在唯一性 摘 要 俄国的数学家Liapunov是微分方程运动稳定性理论的奠基人之一，他在《运动稳定性的一般问题》中给出了运动状态稳定性的严格数学定义，并且提出了两套分析方法：第一套方法常被称为直接法-函数法，它的特征是不需要求解微分方程的通解或者特解，但是却需要找到一些具有特殊性的辅助函数,并且用它来判断渐近稳定性.第二套分析方法是完全定性的，他在第二套分析方法中创造性地提出求解非线性系统中常微分方程的Liapunov函数法，也称之为直接法，它把解具有稳定性与具备特殊性质的函数(现在叫李雅普诺夫函数)的存在性紧密联系起来.Liapunov稳定性理论的应用范围正在日益扩大，所以对研究微分方程的解尤为重要.文章研究了一类四阶非线性系统，运用Liapunov函数方法，得到了该系统存在唯一渐近稳定的周期解的充分条件.这对今后类似的研究奠定了基础，具有十分重要的意义. 关键词：非线性；周期解；微分方程 Existence and Uniqueness of Periodic Solutions for a Class of Nonlinear Systems Abstract:The Russian mathematician Liapunov is one of the founders of the theory of differential equation of motion stability, he has given the strict mathematical definition that Motion stability on the movement stability of the general issues,and two sets of analysis method is proposed: the first method is often referred to as direct method or function method, its characteristic is that dont need to solve the differential equation of the general or particular solution, but there is a special need to find some auxiliary function , and use it to judge asymptotic stability. The second set of analysis method is completely qualitative, he in the second set of analysis method is put forward creatively to solve the differential equation of nonlinear system in the Liapunov function method,also called the direct method, the stability of the solution is to have special properties function (now called lyapunov function) the existence of closely linked. The application scope of Liapunov stability theory is expanding increasingly, so the solution of differential equation is particularly important . In this paper, a class of four order nonlinear systems is studied. By suing the method of Liapunov function, a sufficient condition of the existence, uniqueness and asymptotic stability of the periodic solutions of this systems is obtained. Keywords: nonlinear, periodic solution，differential equ