多項(xiàng)式系統(tǒng)的擬齊次分解與單值性問(wèn)題
發(fā)布時(shí)間：2018-08-05 20:28
【摘要】：在平面微分系統(tǒng)定性理論研究中，重要課題之一是對(duì)系統(tǒng)的孤立奇點(diǎn)進(jìn)行分類，并建立各類奇點(diǎn)的判別準(zhǔn)則，其中一個(gè)經(jīng)典的問(wèn)題是判定何時(shí)它是單值的（即判定一個(gè)奇點(diǎn)是不是中心-焦點(diǎn)類型）。根據(jù)平面解析系統(tǒng)如果有軌線進(jìn)入系統(tǒng)的孤立奇點(diǎn)，則它只能螺旋形地進(jìn)入或沿固定方向進(jìn)入的事實(shí)可知：平面解析微分系統(tǒng)的孤立奇點(diǎn)是焦點(diǎn)-中心類型當(dāng)且僅當(dāng)沒(méi)有軌線沿固定方向離開(kāi)（進(jìn)入）。當(dāng)奇點(diǎn)是非強(qiáng)退化（即系統(tǒng)在奇點(diǎn)的線性化矩陣非零）時(shí)，單值性問(wèn)題已基本上解決，而對(duì)強(qiáng)退化的情形，即使是解析系統(tǒng)，，截止到目前仍然是一個(gè)沒(méi)有得到完全解決的經(jīng)典難題。 在大部分微分方程定性理論的經(jīng)典專著中，通常都是把解析系統(tǒng)進(jìn)行齊次分解，再根據(jù)特殊方向作出典型域來(lái)研究孤立強(qiáng)退化奇點(diǎn)鄰域內(nèi)軌線的行為。但是這種方法計(jì)算十分麻煩，有時(shí)需要進(jìn)行無(wú)窮次計(jì)算從而使得問(wèn)題實(shí)際上是難以解決的。近年來(lái)，許多數(shù)學(xué)家開(kāi)始著手于利用解析系統(tǒng)的牛頓圖的有界邊把它進(jìn)行擬齊次分解來(lái)研究孤立強(qiáng)退化奇點(diǎn)鄰域內(nèi)軌線的行為。 本文的第一個(gè)工作是基于固定權(quán)向量（即牛頓圖的某條有界邊）的擬齊次多項(xiàng)式與擬齊次多項(xiàng)式系統(tǒng)在通常的加法與數(shù)乘意義下都構(gòu)成線性空間這個(gè)事實(shí)，通過(guò)研究這樣的線性空間的維數(shù)與基底，給出解析系統(tǒng)的比較直觀且容易計(jì)算的擬齊次分解式，并用幾個(gè)具體的實(shí)例來(lái)實(shí)現(xiàn)這樣的分解式。 本文的另外一個(gè)工作是在這樣的擬齊次分解式基礎(chǔ)上，把微分方程定性理論中通過(guò)把解析系統(tǒng)進(jìn)行齊次分解來(lái)定性分析孤立強(qiáng)退化奇點(diǎn)的經(jīng)典問(wèn)題而引進(jìn)的示性方程、特征方向或特殊方向、特征軌線、典型域及其性質(zhì)等推廣到擬齊次系統(tǒng)的情形，給出擬齊次示性方程、擬特征方向、擬典型域及其性質(zhì)，特別是給出了擬特征方向個(gè)數(shù)的估計(jì)。同時(shí)利用這些知識(shí)研究了解析系統(tǒng)的孤立強(qiáng)退化奇點(diǎn)附近軌線的定性行為。 最后，對(duì)全文進(jìn)行了總結(jié)與展望。
[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.
【學(xué)位授予單位】：浙江理工大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O174.14
本文編號(hào)：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.
【學(xué)位授予單位】：浙江理工大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O174.14
【參考文獻(xiàn)】
相關(guān)期刊論文 前1條
1 杜飛飛;黃土森;;牛頓圖的性質(zhì)與擬齊次多項(xiàng)式系統(tǒng)的中心問(wèn)題[J];浙江理工大學(xué)學(xué)報(bào);2013年01期
本文編號(hào)：2166927
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