不变代数曲线与一类三次系统的中心判定问题

对于一类具有一条抛物线解、两条直线解和中心-焦点型奇点的三次系统,证明它以原点为中心的充要条件是它的第一阶焦点量为零.系统在原点的中心条件是通过不变代数曲线构造积分因子或利用Poincare对称原理得以证明.     

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     

一类三次系统的极限环个数与奇点分支

给出二次系统I的一类相伴系统在奇点O(0,0)的焦点量公式,证明了O至多为2阶细焦点,δlmn=0时系统在O外围至多有一个极限环,从而说明了系统在细焦点外围至多有一个极限环。最后给出了各个奇点的分支情况及几何特征。     

一类多项式微分系统无穷远点的中心-焦点判定

本文研究了一类七次系统无穷远点的中心-焦点判定问题.通过将实系统转化为复系统研究,给出了计算无穷远点奇点量的递推公式,并在计算机上用Mathematica推导出该系统无穷远点前十二个奇点量,进一步导出了无穷远点成为中心的条件和分别成为七阶、九阶、十二阶细焦点的条件.     

一类具细焦点的三次系统极限环的唯一性

继续相关文献的工作,给出与二次系统Ⅰ相伴的一类三次系统在奇点N(0,1/n)的焦点量公式,证明了系统在细焦点N外围至多有一个极限环,同时证明了当N或O为细焦点时,系统在另一个焦点外围无极限环,结合相关文献的结论,说明了具有细焦点的该系统在全平面至多有一个极限环.     

一类特殊五次系统的奇点量与可积性条件的研究

本文采用代数运算方法研究了一类五次系统的中心一焦点判定问题，给出了系统的13个基本如不变量，得到了直接用系统的系数表示的奇点量公式与可积性条件。     

一类单中心Hamilton系统在三次扰动下的Poincare分岔

使用一阶Mel‘nikov函数讨论了一类具有以抛物线与直线为边界的周期环域的单中心二次Hamilton系统的三次扰动下的Poincare分岔,得到其Poincare分岔最多可以产生两个极限环。     

平面n次微分自治系统的焦点量与几类极限环分枝

本文讨论系统 E_n(μ),其主要结果是:1.把同心圆的 Poincare分枝,高阶Hopf分枝以及中心点的极限环分枝诸问题的解决统一地归结为焦点量公式的推导和代数方程的求根,由此具体地给出了分枝出多个极限环的充分条件;2.在复域中对存在周期轨道时参数μ的取值范围作了估计;3.作为应用,对系统E_2(μ)统一给出了由上述分枝问题产生三个极限环的实例。     

具有全局中心的三次Hamilton系统的Poincaré分支

本文讨论一类具有全局中心的三次:Hamilton系统的Poincare分支,证明了 其Poincare分支最多可以产生两个极限环,而且可以产生两个极限环.     

中心条件推导的间接方法

对于一类多项式系统,给出两类对称条件的推导算法,具体讨论了一类三次系统的中心条件;对于Poincare型系统,给出一类等时中心的充分条件．     

Green's function for a third-order generalized right focal problem

We determine the Green's function for the third-order three-point generalized right focal boundary value problem

     

Comparative index for solutions of symplectic difference systems

We introduce the comparative index of two conjoined bases of a symplectic difference system, which generalizes difference analogs of canonical systems of differential equations. We consider the main properties of the comparative index and its relation to the number of focal points of a conjoined basis of the symplectic system. We prove a formula relating the number of focal points (including multiplicities) of two bases in the interval (i, i + 1] and corollaries of this formula such as an estimate for the difference of the numbers of focal points of two conjoined bases in the interval (0, N + 1], the equality of the numbers of focal points of principal solutions for the primal and reciprocal systems, sufficient conditions for the solvability of the Riccati equation for a disconjugate symplectic system, etc.     

Multiple fixed-sign solutions for a system of generalized right focal problems with deviating arguments

We consider the following system of generalized right focal boundary value problems

     

Limit cycles and singular point quantities for a 3D Lotka-Volterra system

Four limit cycles are constructed for a three dimensional Lotka-Volterra system. This gives a good example to the cyclicity of 3D Lotka-Volterra systems. A recursion formula for computation of the singular point quantities is given for the corresponding Hopf bifurcation equation. What is worth mentioning is that the expressions of focal values are simpler, and the formula is readily done with using computer symbol operation system such as Mathematica due to its linearity.     

Isoparametric hypersurfaces in Finsler space forms

In this paper, we study isoparametric hypersurfaces in Finsler space forms by investigating focal points, tubes and parallel hypersurfaces of submanifolds. We prove that the focal submanifolds of isoparametric hypersurfaces are anisotropic-minimal and obtain a general Cartan-type formula in a Finsler space form with vanishing reversible torsion, from which we give some classifications on the number of distinct principal curvatures or their multiplicities.     

实Clifford分析中的拟Bochner Martinelli型高阶奇异 积分

该文借助于高阶奇异积分的Hadmard主值思想以及归纳法思想讨论了实Clifford分析中拟Bochner Martinelli型高阶奇异积分Hadmard主值的存在性、递推公式、计算公式,以及在Hadamard主值意义下的微分公式.     

Research on the composition center of a class of rigid differential systems

In this paper, we answer the question: under what conditions a class of rigid differential systems have a composition center. We give the sufficient and necessary conditions for these systems to have a center at origin point. At the same time, we give the formula of focal values and the highest order of fine focus.     

实Clifford分析中含两个奇点的拟Bochner-Martinelli型高阶奇异积分

在实C lifford分析中讨论了带有两个奇点的拟Bochner-M artinelli型高阶奇异积分的归纳定义,H adm ard主值的存在性,递推公式,计算公式,以及在H adam ard主值意义下的微分公式.     

Comparison theorems for symplectic systems of difference equations

In the present paper, we prove comparison theorems for symplectic systems of difference equations, which generalize difference analogs of canonical systems of differential equations. We obtain general relations between the number of focal points of conjoined bases of two symplectic systems with matrices W _i and $ \hat W_i $ \hat W_i as well as their corollaries, which generalize well-known comparison theorems for Hamiltonian difference systems. We consider applications of comparison theorems to spectral theory and in the theory of transformations. We obtain a formula for the number of eigenvalues λ of a symplectic boundary value problem on the interval (λ ₁, λ ₂]. For an arbitrary symplectic transformation, we prove a relationship between the numbers of focal points of the conjoined bases of the original and transformed systems. In the case of a constant transformation, we prove a theorem that generalizes the well-known reciprocity principle for discrete Hamiltonian systems.