QUALITATIVE INVESTIGATION OF A TORAL DIFFERENTIAL EQUATION HAVING CRITICAL POINTS

关于环面上无奇点的动力体系的研究.自从Н.Poincare的开创性工作以后，较早 的研究工作已见于Coddington与Levinson的书中.近期则有秦元勋的工作，他已研究 了具体的微分方程，但仍保持无奇点的假设.近年来国内外又出现了不少研究一般二维 流形上动力体系的拓扑结构或分类的文章，其中考虑了奇点，但却没有具体的微分方程. 本文类比于平面线性定常系统，研究了环面上的微分方程 \[\frac{{dx}}{{dt}} = A\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} x + B\sin {\kern 1pt} {\kern 1pt} y,\frac{{dy}}{{dt}} = C\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} x + D\sin {\kern 1pt} {\kern 1pt} {\kern 1pt} y(AD - BC \ne 0)\](1) 的轨线的全局结构. 在§ 1中假设⑴定义在(x,y)平面上的正方形[{S_1}:0 \le x \le 2\pi ,0 \le y \le 2\pi \]内，然后把 S1的两对对边等同起来，从而得到环面上的解析系统.它有两个初等的非鞍点和两个初 等鞍点,经过分析，得到中心-鞍点，结点-鞍点和焦点-鞍点等三种可能拓扑结构，在最后 一种情况有时能出现极限环，但唯一性未能证明.此外，环面上不存在第二类周期轨线. 在§2中假设⑴定义在正方形\[{S_1}:0 \le x \le 2\pi ,0 \le y \le 2\pi \]内，再把S2的两对对边等同 起来，从而得到环面上的C1系统.此系统只有一个指标为零的奇点，但它的轨线拓扑结 构的可能情况要比§1多一些.环面可以被具有相同旋转数的一族闭轨线所充满，也可 以被一族各态历经的轨线所充满.它可能具有唯一的半稳定极限环，或是一个稳定环和 一个不稳定环，一切其它轨线都从正负向趋向它们.环面还可能被分成一个，两个或三个 单连通域，每一域中充满着具有相同旋转数的奇闭轨线.最后，环面上也可能既存在极限 环，又存在为奇闭轨线所充满的区域.此外，我们还固定(1)式右边的三个系数A,B， 而让C从零变到一∞,以观察方程的全局结构和轨线的旋转数的变化.     

环面上具有一个奇点的微分方程的轨线的拓扑结构

<正> 环面上不具奇点的微分方程的轨线的拓扑结构完全解决了.[11]曾企图解决具有一个奇点的环面上微分方程的轨线的拓扑结构,但由于方法的不合适而未成功,本文的目的是对这种情形作一完全的分类. 我们讨论的微分方程是     

关于Poincaré-Hopf的奇点指数公式

在一般的文献中,Poincaré-Hopf的曲面奇点指数公式的证明需要利用Euler的曲面示性数公式.本文将通过微分方程的定性方法,直接证明Poincaré-Hopf的奇点指数公式,然后作为简单的应用可得到Euler的曲面示性数公式.     

环面上具有有限个奇点的连续流的轨线的拓扑结构

<正> 环面上具有一个奇点的微分方程(连续流)的轨线的拓扑结构已解决.内容是:1.如有不可缩的周期轨线或奇闭轨线,则沿作为分界线的这种轨线将环面切开,成为若干个带域、圆环域和螺旋域,而且可以列举出来.2.如果没有不可缩的周期轨线或奇闭轨线,则必有非平凡的 P~+和 P~-稳定轨线.这时它是诸态备经型或奇异型适当改造而成(加上一个奇点以及可能的一朵花,还可能加上最多二个一端为奇点的带域).     

环面上的微分方程(Ⅰ)

<正> 关于定义在环面上的微分方程,其积分曲线的拓扑研究开始於 H.Poincaré,经过Bohl 及 Denjoy 等的补充瑟发展,对于不具有奇点的情形,在 Kneser 的工作中便基本上完成了.在拓年结构的研究中,旋转数μ的研究是占有决定性的位置的.但是,过去的文献中对于如何由已给的具体方程去算出μ的值是当作一个未解决的难题遗留下來的.如果沒有方法计算μ,也就不能具体地应用上述各文中所得到的完整的拓扑理论.本文及以后各文将对这一问题进行研究和逐步设法解决.     

关于Poincare—Hopf的奇点指数公式

在一般的文献中，Poincare-Hopf的曲面奇点指数公式的证明需要利用Euler的曲面示性数公式。本文将通过微分方程的定性方法，直接证明了Poincare-Hopf的奇点指数公式，然后作为简单的应用可得到Euler的曲面示性数公式。     

k维环面上坐标自映射下拓扑熵的一个下界

给出了ｋ维环面上坐标自映射下拓扑熵的一个下界，最后，还指出了ｋ维环面上渐近Ｒｅｉｄｅｍｅｉｓｔｅｒ数严格大于渐近Ｎｉｅｌｓｅｎ数的情形，并说明了文（３）（或文（４）中引理１为该文的一个特例。     

环面上一个推广的C~r封闭引理

<正> 在微分动力系统理论中,对推广的C~r(r≥1)封闭引理([5])的验证是十分重要的难题.对r=1,文[1]已经证明;对任r≥1,已经得到:在环面上无奇点时([3]、[4])以及在射影平面和Klein瓶上([5]该引理是成立的.本文在环面T~2和紧致二维流形M~2上对任r≥1情形得到如下结论(本文未阐明的记号的含意见[5]).     

极限环问题

1.研究极限环的重要性所谓极限环就是平面定常系的孤立闭轨线,它附近的轨线当 t→∞或-∞时都以螺旋状方式向它无限接近。H.Poincaré首先发现极限环是非线性系统所特有的一种轨线,并找到研究极限环的三种重要方法,即地形系法,后继函数法和小参数法。的确,就平面定性理论的观点看来,要搞清楚不可积分的一阶非线性方程的积分曲线的全局结构,那末研究极限环问题是有着非常重要的意义的。因为研究积分线的全局结构,无非就是要解决下面三个问题:1)奇点附     

n维空间奇点的拓扑分类

在n维相空间内,奇点邻域内的微分方程积分曲线分布的拓扑分类问题,是B.B。在1952年提出来的。其实,这个问题,在H.Poincare的名著[2]中,就已经看到了。大家知道,H.Poincare把主要的平面奇点,分为结点、鞍点、焦点和中心四类;而在空间奇点中,主要增加了鞍焦点一类。他并且指出,四维空间的奇点,主要有八类;五维空间,则有十类。H.Poincare并且把三维空间的非孤立寄点分为结点弧、鞍点弧     

On certain modified boundary value problems for ordinary differential equations with irregular singularity

We consider a linear second-order differential equation with irregularly singular point at the beginning of the interval. For the corresponding homogeneous differential equation, we obtain the asymptotics of the solutions and their derivatives near the singular point. Using some modified Green functions and taking into account the asymptotics, we consider three boundary value problems with various boundary conditions (including a weighted one) at the singular point, proving theorems on the existence and uniqueness of the solutions and giving their structure. Lithuanian Mathematical Journal, Vol. 49, No. 1, 2009, pp. 109–121     

Radicals of 0-regular algebras

Summary We consider a generalisation of the Kurosh--Amitsur radical theory for rings (and more generally multi-operator groups) which applies to 0-regular varieties in which all operations preserve 0. We obtain results for subvarieties, quasivarieties and element-wise equationally defined classes. A number of examples of radical and semisimple classes in particular varieties are given, including hoops, loops and similar structures. In the first section, we introduce 0-normal varieties (0-regular varieties in which all operations preserve 0), and show that a key isomorphism theorem holds in a 0-normal variety if it is subtractive, a property more general than congruence permutability. We then define our notion of a radical class in the second section. A number of basic results and characterisations of radical and semisimple classes are then obtained, largely based on the more general categorical framework of L. M\'arki, R. Mlitz and R. Wiegandt as in [13]. We consider the subtractive case separately. In the third section, we obtain results concerning subvarieties and quasivarieties based on the results of the previous section, and also generalise to subtractive varieties some results for multi-operator group radicals defined by simple equational rules. Several examples of radical and semisimple classes are given for a range of fairly natural 0-normal varieties of algebras, most of which are subtractive.     

THE ROTATION NUMBER AND DIRECTION OF A CONTINUOUS FLOW ON THE TORUS

In this paper the rotation direction r_f and rotation number p_f for any continuous flowon the torus are defined by applying Weil's theorem proved by Markley.The followingresults are obtained:(i)r_f=0 iff all orbits of f are proper and each limit set of f is homotopic to zeo onT~2;(ii)if r_f≠0,then p_f is irrational iff f has at least one non-trivial P stable orbit,p_f isrational iff f has at least one non-zero-homotopic closed or singular closed orbit.Then a method of computing the rotation number of certain flows is given.     

The Stokes phenomenon in the confluence of the hypergeometric equation using Riccati equation

In this paper we study the confluence of two regular singular points of the hypergeometric equation into an irregular one. We study the consequence of the divergence of solutions at the irregular singular point for the unfolded system. Our study covers a full neighborhood of the origin in the confluence parameter space. In particular, we show how the divergence of solutions at the irregular singular point explains the presence of logarithmic terms in the solutions at a regular singular point of the unfolded system. For this study, we consider values of the confluence parameter taken in two sectors covering the complex plane. In each sector, we study the monodromy of a first integral of a Riccati system related to the hypergeometric equation. Then, on each sector, we include the presence of logarithmic terms into a continuous phenomenon and view a Stokes multiplier related to a 1-summable solution as the limit of an obstruction that prevents a pair of eigenvectors of the monodromy operators, one at each singular point, to coincide.     

Fredholm theory for pairs of closed subspaces of a Banach space

We study the Fredholm theory for pairs of closed subspaces of a Banach space developed by Kato. We define the strictly singular and the strictly cosingular pairs of subspaces, and we show that some of the results of operator theory can be extended to this context. However, there are some notable differences. On the one hand, the perturbation classes problem has a positive answer in this context: the upper and lower semi-Fredholm pairs are stable under strictly singular and strictly cosingular perturbations, respectively, and this stability characterizes the strictly singular and the strictly cosingular pairs. Note that it has been proved recently that the perturbation classes problem for continuous semi-Fredholm operators has a negative answer. On the other hand, unlike in the case of operators, the Fredholm pairs are not stable under perturbation by strictly singular or strictly cosingular pairs. We also show the stability under composition of the compact, the strictly singular and the strictly cosingular pairs of subspaces.     

Irreducible triangulations of the torus

By Steinitz' Theorem all triangulations of a sphere are generated from one triangulation with four vertices by certain sequences of operations called vertex splittings. A theorem of Barnette asserts that all triangulations of the projective plane can be generated from two irreducible triangulations. In the present work we obtain an analogous result for the torus: we show that all triangulations of the torus are generated by 21 irreducible triangulations (they are found explicitly) by applying the same vertex splitting operations. Two tables, one figure.Translated from Ukrainskií Geometricheskií Sbornik, No. 30, 1987, pp. 52–62.     

Series solutions of coupled differential equations with one regular singular point

We consider two linear second-order ordinary differential equations. r=0 is a regular singular point of these equations. Applying the classical Method of Frobenius, we do not obtain any indicial equation and therefore no solution, because the differential equations are coupled.

In this paper, we present an extended Method of Frobenius on a coupled system of two ordinary differential equations. These equations come from the micropolar theory, which is one of the three kinds of the new 3M physics.     

平面自治系统同宿轨族的存在唯一性

本文研究一般平面自治系统     

Some cases of the solvability in a closed form of singular integral equations and two-dimensional characteristic systems

We consider the characteristic system of singular integral equations for two unknown functions. We obtain certain correlations connecting coefficients of the system which allow one to solve the latter in a closed form. Based on the obtained results, we consider cases of the solvability in a closed form for one class of singular integral equations.