【作者】 李晓月；

【导师】 黄启昌； 王克；

【作者基本信息】 东北师范大学 ， 应用数学， 2005， 博士

【摘要】 本文主要研究了Lienard系统轨线的拓扑分类的问题, 按照无闭轨线和有闭轨线两种情形来分别考虑此问题. 首先, 在文献^[1-3] 对无闭轨Lienard系统进行拓扑分类, 得到64种可能的拓扑结构的基础之上, 给出了这每一种拓扑结构具体存在的例子, 从而得到了这种分类的可行性和实现性. 接下来, 对有闭轨Lienard系统进行了拓扑分类, 给出了具体的分类原则和方法. 全文共分成五章: 第一章作为准备知识给出了本文要用到的相关知识内容, 其中包括微分方程的基本定性理论, Lienard系统极限环存在定理, 以及泛函分析中关于直线上开集的构造的基本理论等. 第二章给出了王克在文献^[1-3]中的主要结论以及相关证明, 介绍了对无闭轨Lienard系统进行拓扑分类的方法和所得到64种可能的拓扑结构. 第三章是关于如何去实现无闭轨Lienard系统存在的64种可能的拓扑结构.众所周知, 对于无闭轨Lienard系统进行拓扑分类是重要的, 但是对于拓扑分类的可行性和实现性的证明更加重要, 因为只有这样, 才更能说明该种分类的合理性和意义. 为此, 在王克对无闭轨Lienard系统进行完整拓扑分类的基础上, 利用Filippov变换的方法以及相平面的拼接方法, 举出具体的满足一定条件的Lienard方程的例子并证明其具有64种可能拓扑结构中的一种. 通过64个具体的满足一定条件的Lienard方程的例子的举出就证明了无闭轨Lienard系统可能存在的64种拓扑结构全部都是可以实现的. 第四章在前面的工作基础上, 继续对有闭轨Lienard系统进行拓扑分类. 我们知道Lienard系统在平面动力系统极限环问题研究中占有重要的理论地位, 因此对有闭轨的Lienard系统进行拓扑分类在理论研究中意义更为深刻, 问题要复杂得多. 本文接下去考虑了有闭轨Lienard系统的拓扑分类问题, 按照Y⁺上点集和Y^-上点集的对应关系给出了对于有闭轨Lienard系统的拓扑分类原则和方法,并证明了有闭轨Lienard系统有∞多种可能的拓扑结构, 按照我们给出的分类原则和方法可以将这∞多种可能的拓扑结构分为40类. 在论文的最后, 总结了论文的创新点提出了论文的改进方向以及研究中所参考的主要文献.更多还原

【Abstract】 This paper discuss mainly topological classification of Lienard systems which in-clude two cases: （1） the systems without closed orbit; （2） the systems with at least oneclosed orbit. The paper ^[1-3] gave a complete classification of Lienard systems withoutclosed orbit and obtained 64 possible topological structures. With this understandingwe provide some concrete equations which ensure the existence of every topologicalstructure and thereby yield the feasibility and realization of the classification. Then, wediscuss the topological classification of Lienard systems with at least one closed orbitand give the concrete principle and method of classification. The whole contents is divided into five chapters. Chapter 1, as the beginning of this paper, offers some relative knowledge, such aspreliminary qualitative theory of differential equation, the existence theorem of limitcycles of Lienard equation, and basis of functional analysis about open set constitutionon a line. Chapter 2, as the second part of this thesis, gives the main results obtained by ^[1-3]and corresponding proof, introduces the method of classification and the 64 possibletopological structures. Chapter 3, as one of the main parts of this thesis, focuses on how to realize theclassification in Chapter 2. As we all know, the topological classification of Lienardsystems without closed orbit is important, however, the proof of feasibility and real-ization of the classification is more important. Only the feasibility and realization canilluminate the rationality and signification of the classification. with this understandingutilizing Filippov transformation and method of patching phase plane we provide someconcrete examples about Lienard equations satisfying some conditions and prove theseexamples possess possible topological structures. According 64 different examples,that all of 64 possible topological structures can be realized is prove. Chapter 4, as the other main part of this thesis, resumes to discuss the topologicalclassification of the Lienard systems with at least one closed orbit. As we know thatLienard systems hold an important place about the study of limit cycles in the theory ofplane dynamical systems. Therefore, the study of topological classification of Lienardsystems with at least one closed orbit is more profound and more complicated. Thenwe consider this problem, give the concrete principle and method of classification bythe corresponding relation between the point set of Y+ and the point set of Y-, and atlast prove that the systems have ∞ possible topological structures which can be dividedinto 40 groups. At the end of the paper, it is proposed the remark of this paper and the furtherstudy direction, many related references are listed.更多还原