三维分段光滑系统的周期轨分支

发布时间：2018-05-15 06:36

本文选题：三维系统 + 分段光滑　；参考：《上海师范大学》2017年硕士论文

【摘要】：本文考虑一类有一个不变平面的三维分段光滑系统且在这个不变平面上有一个k重闭轨.通过使用分支技巧和分析系统分支方程的解,我们研究了k重闭轨附近的特殊的分支现象并得到了从这个k重闭轨分支出周期轨的条件.此外,我们还研究了平面高次分段哈密顿系统的闭轨的重数.本文分为六章,具体内容介绍如下:第一章主要介绍了所研究课题的实际背景,给出平面分段光滑系统的k重闭轨的定义.第二章主要介绍了与研究课题相关的一些基本概念和引用一些已有结果来作为本文的引理.第三章用曲线坐标变换对原系统进行处理,进而得到处理后系统的分支方程.第四章通过分析分支方程的解来研究系统的极限环.进一步地,我们得到极限环存在的条件.第五章研究了平面高次分段哈密顿系统的闭轨的重数.对于哈密顿函数是一个高次多项式的分段系统,我们得到了其闭轨重数的上界.第六章给出了一个具体的三维分段光滑系统,通过前面的定理,我们分析了其闭轨附近的极限环.此外,我们还给出了一个具体的二维分段哈密顿系统并得到了其闭轨的最大重数.
[Abstract]:In this paper, we consider a three dimensional piecewise smooth system with an invariant plane and have a K closed orbit on this invariant plane. By using the branch technique and analyzing the solution of the bifurcation equation of the system, we have studied the special branch phenomena near the K closed rail and obtained the conditions for the bifurcation of the periodic rails from the K closed rail. The weight of the closed orbit of a planar higher-level Hamiltonian system is also studied. This paper is divided into six chapters. The specific content is introduced as follows: the first chapter introduces the actual background of the subject, and gives the definition of the K closed orbit of a planar piecewise smooth system. The second chapter introduces some basic concepts and references related to the research topic. The results have been used as the lemma in this paper. The third chapter deals with the original system by curvilinear coordinate transformation, and then obtains the branch equation of the system after processing. The fourth chapter studies the limit cycle of the system by analyzing the solution of the branch equation. Further, we get the condition of the existence of the limit ring. The fifth chapter studies the high order piecewise Hamilton. The weight number of the closed orbit of the system. The Hamilton function is a piecewise system of a high order polynomial. We get the upper bound of the number of closed orbit. In the sixth chapter, a concrete three-dimensional piecewise smooth system is given. Through the previous theorem, we analyze the limit ring near the closed orbit. In addition, we give a specific two-dimensional partition. The Hamiltonian system is obtained and its maximum number of closed orbits is obtained.

【学位授予单位】：上海师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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