向量场的旋度判定平面（空间）微分系统闭轨的不存在性

利用向量场的旋度，我们可判定平面、空间微分系统闭轨的不存在性，得到类似Bendixson，Dulac判断法的一种新判定法。同时还得到平面微分系统闭轨存在时的相对位置。     

一类具两条不连续相交线的平面系统的闭轨

研究了一类具两条不连续相交线的平面系统的闭轨.利用微分包含理论和点变换的方法,获得了一些有趣的结果,包括滑模解,同宿解和闭轨的存在性.同时给出了闭轨存在的必要条件.     

E3系统的积分直线与闭轨线

董金柱等对存在积分直线的E_2系统的定性分析,得到了很好的结果。本文得到:当m=2,3时,若m次微分系统存在闭轨线和n_m条积分直线,则0≤n_m≤m 1。     

在混合扰动下从闭轨族分支的极限环

本文讨论了对确定微分方程组的向量场和分析其轨线穿过方向的参考闭曲线族同时进行扰动分支极限环的方法，并给出了一个平面二次微分系统在混合扰动下分支出三个极限环的例子     

一类带有三次项的平面五次微分系统的全局拓扑结构分析

一类带有三次项的平面五次微分系统在Poincare变换下可以讨论系统的无穷远奇点的性质,进而得到奇点附近轨线的拓扑结构,并利用判断函数给出极限环存在与否的条件,补充完善了五次系统的定性分析.     

平面自治系统闭轨的存在性

本文研究一般平面自治系统ｄｘ/ｄｔ＝Ｐ（ｘ，ｙ），ｄｙ/ｄｔ＝Ｑ（ｘ，ｙ）的闭轨的存在性，所得结果不仅较大地推广和改进了已有的结果，而且还可以初步估计闭轨的存在位置．     

Nagumo方程行波系统的定性分析及其有界行波解

本文对Nagumo方程的行波系统进行了定性分析,该系统存在一端连接鞍点的有界异宿轨,进而选择奇点为鞍点的平面线性自治系统,利用该平面自治系统轨线向径的斜率,根据齐次平衡原则,构造出了Nagumo方程行波系统的行波解.其次Nagumo方程的行波系统还存在着对应中心周围闭轨的周期解,因而提出新的CPP解法,求出了对应的周期解.     

具有二阶或三阶细鞍点的二次系统

本文证明具有二阶或三阶细鞍点的二次系统在全平面不存在任何闭轨和分界线环。     

有界噪声扰动下平面微分系统周期轨的分支

本文研究有界噪声扰动下平面微分系统周期轨的分支问题.首先介绍平面微分系统周期轨的三类经典分支:叉型分支、鞍结分支和跨临界分支,并给出其分支方向以及分支出的周期轨的稳定性.然后讨论有界噪声扰动下平面微分系统周期轨的分支现象,考虑该扰动系统极小正向不变集的个数随参数值变化而变化的规律,证明该系统在一定条件下发生所谓的硬分支现象,并给出分支参数值的阶数估计.最后给出具体的例子并进行数值模拟,形象直观地说明本文结果.     

几类非线性微分系统极限环的存在性和唯一性

给出系统(E1),(E2)和(E3)等非线性微分系统存在闭轨的一些新的判定条件,推广了非线性微分系统极限环的存在性和唯一性许多这方面研究的结果,并大大改进了它们的某些条件．在这个基础上,还给出了系统(E1)和(E2)恰有一个极限环的一组充分条件．     

THE CRITERION FOR THE STABILITY OF A MULTIPLE LIMIT CYCLE UNDER HIGHER DEGENERATIONS

In this paper, we obtain the exact computation formulae to determine the stability of a multiple limit cycle with the third or fourth order degenerations. We employ the method of computing the expansion of the Poincar′e map around the closed orbit using "normal bundle" coordinates parameterized by time variable in a neighborhood of the closed orbit. An example is given to show the feasibility of our results.     

具不变集的平面多项式系统

本文证明了以y＝xm为不变集的平面n次多项式系统（m＞n）不会有极限环，但可以存在奇闭轨．     

Hamiltonian flows on the space of star-shaped curves

We provide a geometric interpretation of the KdV equation as an evolution equation on the space of closed curves in the centroaffine plane. There is a natural symplectic structure on this space and the KdV-flow is generated by a Hamiltonian given by the total centroaffine curvature. In this way we obtain another example for a soliton equation coming naturally from a differential geometric problem [1]. Furthermore, we present a group action of the diffeomorphism group of the circle on the space of closed centroaffine curves.     

On invariant closed curves for one-step methods

Summary We show that a one-step method as applied to a dynamical system with a hyperbolic periodic orbit, exhibits an invariant closed curve for sufficiently small step size. This invariant curve converges to the periodic orbit with the order of the method and it inherits the stability of the periodic orbit. The dynamics of the one-step method on the invariant curve can be described by the rotation number for which we derive an asymptotic expression. Our results complement those of [2, 3] where one-step methods were shown to create invariant curves if the dynamical system has a periodic orbit which is stable in either time direction or if the system undergoes a Hopf bifurcation.     

Normal forms and unfoldings of linear systems in eigenspaces of (anti)-automorphisms of order two

In this article we classify normal forms and unfoldings of linear maps in eigenspaces of (anti)-automorphisms of order two. Our main motivation is provided by applications to linear systems of ordinary differential equations, general and Hamiltonian, which have both time-preserving and time-reversing symmetries. However, the theory gives a uniform method to obtain normal forms and unfoldings for a wide variety of linear differential equations with additional structure. We give several examples and include a discussion of the phenomenon of orbit splitting. As a consequence of orbit splitting we observe passing and splitting of eigenvalues in unfoldings.     

A GENERAL PROPERTY OF THE QUADRATIC DIFFERENTIAL SYSTEMS

In this paper, it is proved that the quadratic differential systems with a weak saddle of order 2 or 3 have no closed or singular closed orbit. Then by the results of [3], it follows that the greatest order of the homoclinic loop bifurcation of a quadratic differential system is between 2 and 3 It means a homoclinic loop can be split into at most three limit cycles.     

逐步删除法确定复杂系统主导结构及其应用

在流率基本入树建模法、反馈环行列式计算法和系统基模X-0-1行列式计算法的基础上,结合系统行为模拟分析的变化曲线,提出了确定系统关键变量主导结构的逐步删除法,为复杂系统的动态结构和行为模式的分析提供有效的方法.运用此方法,对"公司+农户"规模经营模式的系统关键变量主导结构进行了分析,并提出相应的管理方针,实现了用系统科学方法分析"三农"问题的目的.     

一个微生物生态模型分岔的周期轨

本文研究一类与厌氧消化过程微生物生态模型有关的微分方程组在非双曲情形下的扰动。在其一维不变流形上讨论自治扰动下中心附近发生的Pionceare分岔,所涉及的闭轨存在性问题是许多熟知结果^[1-5]不能包含的情形。进而,利用Melnikov函数方法,给出周期扰动下发生次调和分岔的参数条件。     

On the Nonrepresentability of Functions of Several Variables by Quadratures

The paper completes the construction of a multidimensional topological version of differential Galois theory. We construct a rich class of germs of functions of several variables which is closed under superpositions and other natural operations. The main theorem describes the behavior of the monodromy groups of such germs under the natural operations. As a result, we obtain topological obstructions to the representability of functions by quadratures, which give the strongest known statements about unsolvability of equations in closed form.     

On the tangent flow of a stochastic differential equation with fast drift

We investigate the behavior of the tangent flow of a stochastic differential equation with a fast drift. The state space of the stochastic differential equation is the two-dimensional cylinder. The fast drift has closed orbits, and we assume that the orbit times vary nontrivially with the axial coordinate. Under a nondegeneracy assumption, we find the rate of growth of the tangent flow. The calculations involve a transformation introduced by Pinsky and Wihstutz.