几类一阶微分方程的反射函数与可积性等价性研究

发布时间：2018-05-20 05:08

  本文选题：反射积分 + 逆积分因子　； 参考：《扬州大学》2017年硕士论文

【摘要】：研究微分系统x'= X(t,x)的解的性态,不仅推动着微分方程理论的发展,同时对研究客观世界中物体的运动规律也具有很大的实际应用价值.当微分系统为自治系统,对于它的解的性态的研究成果已有很多.而对于非自治系统的研究成果就相对有限了.我们知道,对于周期时变系统的研究,可以借助于Poincare映射和Lyapunov变换[1-4],但有时寻找这些变换是很困难的.上世纪八十年代,Mironenko[5]创建了反射函数理论.利用反射函数,我们可以建立周期时变系统（？）的Poincare映射,借助它能研究该系统的解的定性性态.我们称具有相同反射函数的两个微分系统类是等价的,而等价的周期系统的周期解的性态是相同的.所以当研究一类复杂的非自治微分系统解的性态时,只需研究与该系统等价的简单系统或自治系统解的性态即可.Mironenko在[5-6]中研究了微分系统（？）与x' = Y(t,x)(2)的等价性,得出(2)等价于(1),当且仅当(2)可表示为（？）.(3)这里F(t,x)为(1)的反射函数.但是对于一般微分系统要求出其反射函数是相当困难的.那如何在反射函数未知的情况下,判定(1)与(2)等价?于是Mironenko在[7]中给出,若△(t,x)满足（？）时,（？）与(1)等价,这里α(t)为t的奇的纯量函数,由此并推出x' = X(t,x)+ ∑αi(t)△,(t,x)(6)也与(1)等价,这里αi(t)为奇的纯量函数,△i.(t,x)为(4)的解.由此可看出,求出(4)的解△(t,x)即反射积分,对判定两个微分系统的等价性尤为重要.Belskii 在[26]中给出 Riccati 方程（？）和 Abel 方程（？）及一般多项式方程x'=∑i-0nai(t)xi的反射积分的结构形式,及这些方程具有这些反射积分的充分条件.Veresovich[19],Varenikova[25]研究了一个平面多项式微分系统与其线性部分等价的判定准则.在本文中,本人主要研究了几类一阶非自治有理分式型微分方程的反射积分及逆积分因子.通过它们建立了与这些方程等价的一阶微分方程类,利用逆积分因子研究了这些方程的可积性及其解的定性性态.其次还研究了两个非自治线性方程组的等价性,并给出了若干判定的准则.在这篇文章的第三章中,本人研究了一次有理分式方程具有各种类型的反射积分的充分条件,建立了与(7)等价的微分方程类.并利用这些反射积分讨论了微分系统的逆积分因子、首次积分及其解的定性性态.其次研究了二次有理分式方程(?)具有二次有理分式形式的反射积分的充分条件.建立了与(9)等价的微分方程类,并利用反射积分研究了微分系统(?)的逆积分因子及可积性问题及其解的定性性态.在第四章中,研究了两个非自治线性微分系统（？）等价性,并给出当它们等价时,其系数矩阵A(t),B(t)所满足的必要条件,以及它们等价的若干判定准则.特别地,还讨论了（？）等价时,(这里φ(t)为纯量函数C为常数矩阵),φ(t),C所具有的特征性质.
[Abstract]:The study of the behavior of the solution of the differential system xn = X t X) not only promotes the development of the theory of differential equations, but also has great practical application value for the study of the law of motion of objects in the objective world. When the differential system is an autonomous system, many researches have been done on the behavior of its solution. However, the research results of non-autonomous systems are relatively limited. We know that the study of periodic time-varying systems can be done by means of Poincare maps and Lyapunov transformations [1-4], but sometimes it is difficult to find these transformations. Mironenko [5] founded the theory of reflection function in the 1980s. Using the reflection function, we can establish the periodic time-varying system. By means of Poincare mapping, the qualitative behavior of the solution of the system can be studied. We say that two classes of differential systems with the same reflection function are equivalent, and the behavior of the periodic solutions of the equivalent periodic systems is the same. Therefore, when we study the behavior of solutions of a class of complex nonautonomous differential systems, we only need to study the properties of solutions of simple systems or autonomous systems equivalent to the system. Mironenko has studied the differential systems in [5-6]. The equivalence with x'= Y ~ (t) ~ (X ~ (+) ~ (2) is obtained, which is equivalent to ~ (1), if and only if ~ (2) can be expressed as the reflection function of ~ (1), where F _ (t ~ (X) is a ~ (1). However, it is very difficult for the general differential system to require its reflection function. How can we determine the equivalence of 1) and 2) when the reflection function is unknown? So Mironenko gives in [7], if TX) satisfies the ) It is equivalent to 1), where 伪 t) is an odd scalar function of t, from which we derive the solution of x'= XTX) 鈭，

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