【摘要】：正規(guī)形理論的基本思想是：對(duì)一個(gè)給定的非線性微分系統(tǒng)，如何尋找形式簡(jiǎn)單的微分系統(tǒng)，，同時(shí)保持其“本質(zhì)性質(zhì)”不變，也就是所求得的簡(jiǎn)單微分系統(tǒng)與原微分系統(tǒng)是“等價(jià)”的。這里面臨的一個(gè)問題是如何界定兩個(gè)微分系統(tǒng)是等價(jià)的？現(xiàn)有的文獻(xiàn)都把這種等價(jià)描述為所求得的簡(jiǎn)單微分系統(tǒng)與原微分系統(tǒng)具有相同的拓?fù)浣Y(jié)構(gòu)。由于拓?fù)浣Y(jié)構(gòu)與定性結(jié)構(gòu)應(yīng)該是兩個(gè)不同的概念，并且定性結(jié)構(gòu)比拓?fù)浣Y(jié)構(gòu)更能體現(xiàn)一個(gè)非線性微分系統(tǒng)的動(dòng)力學(xué)行為。比如平面非退化線性系統(tǒng)的結(jié)點(diǎn)與焦點(diǎn)具有相同的拓?fù)浣Y(jié)構(gòu)，但顯然它們的動(dòng)力學(xué)行為是完全不同的！然而，到目前為止，在國(guó)內(nèi)外的文獻(xiàn)中還沒有給出平面解析系統(tǒng)定性結(jié)構(gòu)的嚴(yán)格定義。本文將給出平面解析系統(tǒng)定性結(jié)構(gòu)的嚴(yán)格定義，同時(shí)按照已有的拓?fù)浣Y(jié)構(gòu)的定義和我們所給出的定性結(jié)構(gòu)的定義分別對(duì)平面非退化解析系統(tǒng)的奇點(diǎn)進(jìn)行分類。結(jié)果表明：我們的定義是合理的，并且對(duì)于平面非退化解析系統(tǒng)，按定性結(jié)構(gòu)進(jìn)行分類比按拓?fù)浣Y(jié)構(gòu)進(jìn)行分類能更好地刻畫系統(tǒng)的動(dòng)力學(xué)行為。 同一個(gè)非線性微分系統(tǒng)的正規(guī)形一般是不唯一的，因此研究?jī)蓚�(gè)正規(guī)形之間的關(guān)系是有意義的。本文的另一個(gè)工作是利用向量場(chǎng)的內(nèi)積，給出了冪零系統(tǒng)兩種不同正規(guī)形的單項(xiàng)式系數(shù)之間的關(guān)系。 冪零系統(tǒng)是一類具有廣泛應(yīng)用價(jià)值的非線性微分系統(tǒng)，例如在研究偏微分方程行波解的存在性時(shí)，通過一個(gè)行波變換，常常把原來的偏微分方程化為一個(gè)常微分的冪零系統(tǒng)進(jìn)行研究。本文的最后一個(gè)工作是利用正規(guī)形理論及擬齊次極坐標(biāo)Blow up變換研究?jī)缌阆到y(tǒng)奇點(diǎn)的單值性問題。 最后，我們對(duì)全文進(jìn)行了總結(jié)與展望。
[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.
【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)】：O175

【參考文獻(xiàn)】

相關(guān)期刊論文 前2條

1 王鐸;An Introduction to the Normal Form Theory of Ordinary Differential Equations[J];數(shù)學(xué)進(jìn)展;1990年01期

2 姜永,李靜,黃民海;Bogdanov-Takens唯一正規(guī)形的一種情形[J];廈門大學(xué)學(xué)報(bào)(自然科學(xué)版);1999年04期

本文編號(hào)：2273050