几类光滑与非光滑系统的周期解问题

发布时间：2018-01-08 17:00

本文关键词：几类光滑与非光滑系统的周期解问题　出处：《上海师范大学》2017年博士论文　论文类型：学位论文

【摘要】：本文首先对推广的Melnikov函数方法进行了补充和完善,建立了分段光滑系统的Melnikov函数零根与极限环个数之间的关系.其次,利用推广的Melnikov函数方法,研究了两类分段光滑系统.通过复杂的计算,求得Melnikov函数零根的个数,由此分别得到了这两类系统由闭轨扰动产生的极限环的个数.作为对动力系统周期解的另一项研究,本文还考虑了一维周期方程,并给出了几种方法来研究周期解的个数问题和稳定性问题.其中,我们通过建立Poincare映射、将方程化成规范型或利用平均方程,来确定周期解的个数.另外,通过研究零解的重数,可以了解零解的稳定性.本文分为五章,具体安排如下:本文的第一章是绪论,主要介绍了本文的研究背景,包括研究对象以及所用到的主要方法.本文的第二章介绍了分段光滑系统的Melnikov函数方法.针对分段光滑的近哈密顿系统,我们对与推广的Melnikov函数方法相关的结论进行总结归纳,并且在这些已知结果的基础上,给出了周期带分支定理,以及在Hopf分支问题中的Melnikov函数的性质.本文的第三章研究了一类非光滑的Lienard系统的极限环分支问题.针对该切换系统的两种情况:切换直线在x轴上和切换直线在y 轴上,我们先分别讨论了带双参数扰动的两类系统,并利用带双参数的Melnikov函数的第一项和第二项表达式,计算出这两类系统的极限环在闭轨附近的个数.然后,再根据这两类系统与原系统之间的关系,得出原系统在这两种情况下的极限环的个数,并举例说明结论.本文的第四章主要考虑了一类具有Lienard形式的切换系统,该系统的左右子系统均为多项式系统.通过分段光滑系统的Melnikov函数方法,我们得到了系统的Melnikov函数M(h)的表达式.之后,为了能得到该式的零根个数,从而确定系统在闭轨附近的极限环个数,我们将M(h)中与h有关的项分为三部分来讨论,最终的结果与这三部分有关.最后,我们按照系统中多项式的次数大小分类讨论,得到一系列关于这类系统的极限环在闭轨附近的个数的结果.本文的第五章讨论了一维周期系统的周期解的存在性、稳定性及其个数问题,并给出了研究这些问题的若干理论与方法.本章的主要结果包含四个部分:第一部分阐述了周期解个数和Poincare映射之间的关系,并且总结和改进了已有结论;在第二部分中,我们得到了一维周期方程零解的重数与稳定性之间的关系,并且分别给出了 x = 0是奇数重、偶数重和中心型的条件;第三部分给出了一个适用于一般方程的规范型定理;第四部分是平均法理论,阐述了如何利用平均方程来研究周期方程周期解的个数问题.
[Abstract]:In this paper, the generalized Melnikov function method is supplemented and perfected, and the relation between the zero root of Melnikov function and the number of limit cycles of piecewise smooth system is established. Two classes of piecewise smooth systems are studied by using the generalized Melnikov function method. The number of zero roots of Melnikov function is obtained by complex calculation. The number of limit cycles generated by the closed-orbit perturbation is obtained respectively. As another study of the periodic solution of the dynamical system, the one-dimensional periodic equation is also considered in this paper. Several methods are given to study the number and stability of periodic solutions. By establishing Poincare maps, the equations are transformed into normal form or mean equations. In addition, by studying the multiplicity of zero solutions, we can understand the stability of zero solutions. This paper is divided into five chapters, the specific arrangements are as follows: the first chapter of this paper is an introduction. This paper mainly introduces the research background. The second chapter introduces the Melnikov function method of piecewise smooth system, aiming at the piecewise smooth near Hamiltonian system. We sum up the conclusions related to the generalized Melnikov function method and give the theorem of periodic band bifurcation on the basis of these known results. In chapter 3, we study the limit cycle bifurcation of a class of nonsmooth Lienard systems. Situation:. Switch the straight line on the x axis and switch the line on the y axis. We first discuss two kinds of systems with two parameters perturbation, and use the first and second expressions of Melnikov function with two parameters. Then, according to the relationship between the two kinds of systems and the original system, the number of limit cycles of the original system in these two cases is obtained. In chapter 4th, we mainly consider a class of switched systems with Lienard form. The left and right subsystems of the system are polynomial systems. By using the Melnikov function method of piecewise smooth systems, we obtain the expression of the Melnikov function of the system. In order to obtain the number of zero roots of the formula and determine the number of limit cycles of the system near the closed orbit, we divide the term of h into three parts to discuss, and the final result is related to these three parts. Finally. We discuss it according to the degree of polynomial in the system. We obtain a series of results about the number of limit cycles near closed orbits of this kind of systems. In chapter 5th, we discuss the existence, stability and number of periodic solutions of one-dimensional periodic systems. The main results of this chapter include four parts: the first part describes the relationship between the number of periodic solutions and Poincare mapping. And summarized and improved the existing conclusions; In the second part, we obtain the relation between the multiplicity and stability of zero solution of one-dimensional periodic equation, and give the condition that x = 0 is odd, even and central. In the third part, a normal form theorem for general equations is given. The 4th part is the theory of the averaging method. How to use the average equation to study the number of periodic solutions of the periodic equation is discussed.
【学位授予单位】：上海师范大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175

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