解析系统的正规形及其应用
发布时间：2018-10-15 16:03
【摘要】：正规形理论的基本思想是：对一个给定的非线性微分系统，如何寻找形式简单的微分系统，，同时保持其“本质性质”不变，也就是所求得的简单微分系统与原微分系统是“等价”的。这里面临的一个问题是如何界定两个微分系统是等价的？现有的文献都把这种等价描述为所求得的简单微分系统与原微分系统具有相同的拓扑结构。由于拓扑结构与定性结构应该是两个不同的概念，并且定性结构比拓扑结构更能体现一个非线性微分系统的动力学行为。比如平面非退化线性系统的结点与焦点具有相同的拓扑结构，但显然它们的动力学行为是完全不同的！然而，到目前为止，在国内外的文献中还没有给出平面解析系统定性结构的严格定义。本文将给出平面解析系统定性结构的严格定义，同时按照已有的拓扑结构的定义和我们所给出的定性结构的定义分别对平面非退化解析系统的奇点进行分类。结果表明：我们的定义是合理的，并且对于平面非退化解析系统，按定性结构进行分类比按拓扑结构进行分类能更好地刻画系统的动力学行为。 同一个非线性微分系统的正规形一般是不唯一的，因此研究两个正规形之间的关系是有意义的。本文的另一个工作是利用向量场的内积，给出了幂零系统两种不同正规形的单项式系数之间的关系。 幂零系统是一类具有广泛应用价值的非线性微分系统，例如在研究偏微分方程行波解的存在性时，通过一个行波变换，常常把原来的偏微分方程化为一个常微分的幂零系统进行研究。本文的最后一个工作是利用正规形理论及拟齐次极坐标Blow up变换研究幂零系统奇点的单值性问题。 最后，我们对全文进行了总结与展望。
[Abstract]:The basic idea of normal form theory is how to find a simple form differential system for a given nonlinear differential system, while keeping its "essential property" unchanged. In other words, the obtained simple differential system is equivalent to the original differential system. One of the problems here is how to define that two differential systems are equivalent. The existing literatures describe this kind of equivalent as that the obtained simple differential system has the same topological structure as the original differential system. Because the topological structure and the qualitative structure should be two different concepts, the qualitative structure can reflect the dynamic behavior of a nonlinear differential system better than the topological structure. For example, the nodes and focal points of planar nondegenerate linear systems have the same topological structure, but obviously their dynamic behaviors are completely different! However, up to now, there is no strict definition of qualitative structure of plane analytic system in domestic and foreign literature. In this paper, the strict definition of qualitative structure of planar analytic systems is given, and the singularities of planar nondegenerate analytic systems are classified according to the existing definitions of topological structures and our definitions of qualitative structures. The results show that our definition is reasonable and that qualitative structure classification is better than topological structure classification for planar nondegenerate analytic systems. The normal form of the same nonlinear differential system is generally not unique, so it is meaningful to study the relationship between two normal forms. Another work of this paper is to give the relationship between the coefficients of the monomial expressions of two different normal forms of nilpotent systems by using the inner product of vector fields. Nilpotent systems are a class of nonlinear differential systems with wide application value. For example, in the study of the existence of traveling wave solutions of partial differential equations, a traveling wave transformation is used. The original partial differential equation is often studied as an ordinary differential nilpotent system. The last work of this paper is to study singularities of nilpotent systems by using normal form theory and quasi-homogeneous polar coordinate Blow up transformation. Finally, we summarize and look forward to the full text.
【学位授予单位】：浙江理工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175
本文编号：2273050
[Abstract]:The basic idea of normal form theory is how to find a simple form differential system for a given nonlinear differential system, while keeping its "essential property" unchanged. In other words, the obtained simple differential system is equivalent to the original differential system. One of the problems here is how to define that two differential systems are equivalent. The existing literatures describe this kind of equivalent as that the obtained simple differential system has the same topological structure as the original differential system. Because the topological structure and the qualitative structure should be two different concepts, the qualitative structure can reflect the dynamic behavior of a nonlinear differential system better than the topological structure. For example, the nodes and focal points of planar nondegenerate linear systems have the same topological structure, but obviously their dynamic behaviors are completely different! However, up to now, there is no strict definition of qualitative structure of plane analytic system in domestic and foreign literature. In this paper, the strict definition of qualitative structure of planar analytic systems is given, and the singularities of planar nondegenerate analytic systems are classified according to the existing definitions of topological structures and our definitions of qualitative structures. The results show that our definition is reasonable and that qualitative structure classification is better than topological structure classification for planar nondegenerate analytic systems. The normal form of the same nonlinear differential system is generally not unique, so it is meaningful to study the relationship between two normal forms. Another work of this paper is to give the relationship between the coefficients of the monomial expressions of two different normal forms of nilpotent systems by using the inner product of vector fields. Nilpotent systems are a class of nonlinear differential systems with wide application value. For example, in the study of the existence of traveling wave solutions of partial differential equations, a traveling wave transformation is used. The original partial differential equation is often studied as an ordinary differential nilpotent system. The last work of this paper is to study singularities of nilpotent systems by using normal form theory and quasi-homogeneous polar coordinate Blow up transformation. Finally, we summarize and look forward to the full text.
【学位授予单位】：浙江理工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175
【参考文献】
相关期刊论文 前2条
1 王铎;An Introduction to the Normal Form Theory of Ordinary Differential Equations[J];数学进展;1990年01期
2 姜永,李静,黄民海;Bogdanov-Takens唯一正规形的一种情形[J];厦门大学学报(自然科学版);1999年04期
本文编号：2273050
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