多项式系统的拟齐次分解与单值性问题
发布时间：2018-08-05 20:28
【摘要】：在平面微分系统定性理论研究中，重要课题之一是对系统的孤立奇点进行分类，并建立各类奇点的判别准则，其中一个经典的问题是判定何时它是单值的（即判定一个奇点是不是中心-焦点类型）。根据平面解析系统如果有轨线进入系统的孤立奇点，则它只能螺旋形地进入或沿固定方向进入的事实可知：平面解析微分系统的孤立奇点是焦点-中心类型当且仅当没有轨线沿固定方向离开（进入）。当奇点是非强退化（即系统在奇点的线性化矩阵非零）时，单值性问题已基本上解决，而对强退化的情形，即使是解析系统，，截止到目前仍然是一个没有得到完全解决的经典难题。 在大部分微分方程定性理论的经典专著中，通常都是把解析系统进行齐次分解，再根据特殊方向作出典型域来研究孤立强退化奇点邻域内轨线的行为。但是这种方法计算十分麻烦，有时需要进行无穷次计算从而使得问题实际上是难以解决的。近年来，许多数学家开始着手于利用解析系统的牛顿图的有界边把它进行拟齐次分解来研究孤立强退化奇点邻域内轨线的行为。 本文的第一个工作是基于固定权向量（即牛顿图的某条有界边）的拟齐次多项式与拟齐次多项式系统在通常的加法与数乘意义下都构成线性空间这个事实，通过研究这样的线性空间的维数与基底，给出解析系统的比较直观且容易计算的拟齐次分解式，并用几个具体的实例来实现这样的分解式。 本文的另外一个工作是在这样的拟齐次分解式基础上，把微分方程定性理论中通过把解析系统进行齐次分解来定性分析孤立强退化奇点的经典问题而引进的示性方程、特征方向或特殊方向、特征轨线、典型域及其性质等推广到拟齐次系统的情形，给出拟齐次示性方程、拟特征方向、拟典型域及其性质，特别是给出了拟特征方向个数的估计。同时利用这些知识研究了解析系统的孤立强退化奇点附近轨线的定性行为。 最后，对全文进行了总结与展望。
[Abstract]:In the study of qualitative theory of planar differential systems, one of the important tasks is to classify the isolated singularities of the systems, and to establish the criteria of the singularities. One of the classic questions is to determine when it is single-valued (that is, to determine whether a singularity is a center-focus type). According to the plane analytic system if the track line enters the isolated singularity of the system, The fact that it can only enter in a spiral or along a fixed direction shows that the isolated singularity of the plane analytic differential system is a focus-center type if and only if there is no orbit leaving (entering) along the fixed direction. When the singularity is non-strongly degenerate (that is, the linearized matrix of the system at the singularity is nonzero), the singularities problem is basically solved, and for the strongly degenerate case, even the analytic system, Up to now, it is still a classical problem that has not been completely solved. In most classical monographs of qualitative theory of differential equations, the analytic system is usually decomposed homogeneous, and then a typical domain is made according to the special direction to study the behavior of the orbit in the neighborhood of isolated strongly degenerate singularities. But this method is very troublesome, sometimes it needs infinite computation to make the problem difficult to solve. In recent years, many mathematicians have begun to use the bounded edges of Newton graphs of analytic systems to decompose it to study the behavior of the inner orbits in the neighborhood of isolated strongly degenerate singularities. The first work of this paper is based on the fact that the quasi-homogeneous polynomial and the quasi-homogeneous polynomial system of a fixed weight vector (that is, a bounded edge of a Newtonian graph) constitute a linear space in the general sense of addition and multiplication. By studying the dimension and base of such linear space, the quasi-homogeneous decomposition formula of analytic system is given, which is more intuitive and easy to calculate, and it is realized by several examples. Another work of this paper is to qualitatively analyze the classical problem of isolated strongly degenerate singularities in the qualitative theory of differential equations based on the quasi homogeneous decomposition, which is used to qualitatively analyze the classical problems of isolated strongly degenerate singularities in the qualitative theory of differential equations. The characteristic direction or special direction, characteristic orbit, canonical field and its properties are extended to the case of quasi homogeneous system. The quasi homogeneous representation equation, quasi characteristic direction, quasi canonical field and their properties are given, especially the estimate of the number of quasi characteristic directions is given. The qualitative behavior of orbit near isolated strongly degenerate singularities of analytic systems is also studied by using these knowledge. Finally, the full text is summarized and prospected.
【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】：O174.14
本文编号：2166927
[Abstract]:In the study of qualitative theory of planar differential systems, one of the important tasks is to classify the isolated singularities of the systems, and to establish the criteria of the singularities. One of the classic questions is to determine when it is single-valued (that is, to determine whether a singularity is a center-focus type). According to the plane analytic system if the track line enters the isolated singularity of the system, The fact that it can only enter in a spiral or along a fixed direction shows that the isolated singularity of the plane analytic differential system is a focus-center type if and only if there is no orbit leaving (entering) along the fixed direction. When the singularity is non-strongly degenerate (that is, the linearized matrix of the system at the singularity is nonzero), the singularities problem is basically solved, and for the strongly degenerate case, even the analytic system, Up to now, it is still a classical problem that has not been completely solved. In most classical monographs of qualitative theory of differential equations, the analytic system is usually decomposed homogeneous, and then a typical domain is made according to the special direction to study the behavior of the orbit in the neighborhood of isolated strongly degenerate singularities. But this method is very troublesome, sometimes it needs infinite computation to make the problem difficult to solve. In recent years, many mathematicians have begun to use the bounded edges of Newton graphs of analytic systems to decompose it to study the behavior of the inner orbits in the neighborhood of isolated strongly degenerate singularities. The first work of this paper is based on the fact that the quasi-homogeneous polynomial and the quasi-homogeneous polynomial system of a fixed weight vector (that is, a bounded edge of a Newtonian graph) constitute a linear space in the general sense of addition and multiplication. By studying the dimension and base of such linear space, the quasi-homogeneous decomposition formula of analytic system is given, which is more intuitive and easy to calculate, and it is realized by several examples. Another work of this paper is to qualitatively analyze the classical problem of isolated strongly degenerate singularities in the qualitative theory of differential equations based on the quasi homogeneous decomposition, which is used to qualitatively analyze the classical problems of isolated strongly degenerate singularities in the qualitative theory of differential equations. The characteristic direction or special direction, characteristic orbit, canonical field and its properties are extended to the case of quasi homogeneous system. The quasi homogeneous representation equation, quasi characteristic direction, quasi canonical field and their properties are given, especially the estimate of the number of quasi characteristic directions is given. The qualitative behavior of orbit near isolated strongly degenerate singularities of analytic systems is also studied by using these knowledge. Finally, the full text is summarized and prospected.
【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】：O174.14
【参考文献】
相关期刊论文 前1条
1 杜飞飞;黄土森;;牛顿图的性质与拟齐次多项式系统的中心问题[J];浙江理工大学学报;2013年01期
本文编号：2166927
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