一类五次多项式系统的奇点量与极限环分支

该文研究一类五次多项式微分系统在高次奇点与无穷远点的极限环分支问题. 该系统的原点是高次奇点, 赤道环上没有实奇点. 首先推导出计算高次奇点与无穷远点奇点量的代数递推公式,并用之计算系统原点、无穷远点的奇点量，然后分别讨论了系统原点、无穷远点中心判据. 给出了多项式系统在高次奇点分支出5个极限环同时在无穷远点分支出2个极限环的实例. 这是首次在同步扰动的条件下讨论高次奇点与无穷远点分支出极限环的问题.     

具有十个大振幅极限环的五次多项式系统

本文研究一类五次平面多项式系统赤道极限环分支问题.运用奇点量方法,首次证明了五次多项式系统可在赤道分支出十个极限环.     

平面五次微分系统高次奇点的拓扑分类

该文讨论了平面五次微分系统高次奇点的拓扑分类，并给出按右端多项式系数的判断准则．     

一类带有15阶结点的平面五次系统的定性分析

讨论一类带有一个15阶结点的平面五次多项式系统的全局结构,利用Poincare变换得到系统的奇点性质,并给出极限环存在的若干条件.     

在有限部分具有四个奇点二次系统的无穷远奇点

为了研究具有四个奇点二次系统的结构,本文研究了这类系统的无穷远奇点。 一般来说,二次系统的无穷远奇点是比较复杂的。但是在有限范围内具有四个奇点的次系统,它的无穷远奇点则具有某些特殊性质。为了方便,下面以E_2~4表示这种系统。     

在有限部分具有三个奇点二次系统的无穷远奇点

<正> 文[1]讨论了在有限部分具有四个奇点二次系统(记作 E_2~4)的无穷远奇点.本文进而讨论在有限部分具有三个奇点二次系统(记作 E_2~3)的无穷远奇点.一般说来,二次系统(E_2)在有限部分奇点个数越少,无穷远奇点情况越复杂.     

一类五次系统的奇点量和可积性条件

本文采用代数运算方法研究了一类五次系统的原点奇点量和可积性条件，并给出了该系统的15个基本Lie-不变量。     

全局半稳定的三次系统

当每个解都正向趋于零时,本文证明,虽然二次系统的奇点必为渐近稳定,但是在同样的条件下,某些三次系统的奇点能够是不稳定的.     

一类平面奇次系统的全局结构

通过研究一类平面奇次系统的有限远奇点和无穷远奇点的类型,对其进行定性分析,得到系统的全局结构及其相图.     

具有星形结点的一类平面四次系统的全局结构（Ⅰ）

该文，证明了前文中的Ⅰ－Ⅳ型区域同样适合于原点是星形结点和有五个无穷远奇点的一类平面四次系统．应用它得到上述系统的三十九种可能的全局结构．     

Isomonodromic confluence of singular points

We consider the problem of confluence of singular points under isomonodromic deformations of linear systems. We prove that a system with irregular singular points is a result of isomonodromic confluence of singular points with minimal Poincaré ranks, i.e., of singular points whose Poincaré rank does not decrease under gauge transformations.     

Computing eigenfunctions of singular points in nonlinear parametrized two-point BVPs

The iterative computation of singular points in parametrized nonlinear BVPs by so-called extended systems requires good starting values for the singular point itself and the corresponding eigenfunction. Using path-following techniques such starting values for the singular points should be generated automatically. However, path-following does not provide approximations for the eigenfunction if the singularity is a bifurcation point. We propose a new modification of this standard technique delivering such starting values. It is based on an extended system by which singular as well as nonsingular points can be determined.     

Quantities at infinity in translational polynomial vector fields

In this paper, we study quantities at infinity and the appearance of limit cycles from the equator in polynomial vector fields with no singular points at infinity. We start by proving the algebraic equivalence of the corresponding quantities at infinity (also focal values at infinity) for the system and its translational system, then we obtain that the maximum number of limit cycles that can appear at infinity is invariant for the systems by translational transformation. Finally, we compute the singular point quantities of a class of cubic polynomial system and its translational system, reach with relative ease expressions of the first five quantities at infinity of the two systems, then we prove that the two cubic vector fields perturbed identically can have five limit cycles simultaneously in the neighborhood of infinity and construct two systems that allow the appearance of five limit cycles respectively. The positions of these limit cycles can be pointed out exactly without constructing Poincaré cycle fields. The technique employed in this work is essentially different from more usual ones, The calculation can be readily done with using computer symbol operation system such as Mathematics.     

Center, limit cycles and isochronous center of a Z 4-equivariant quintic system

In this paper, we study the limit cycles bifurcations of four fine focuses in Z4-equivariant vector fields and the problems that its four singular points can be centers and isochronous centers at the same time. By computing the Liapunov constants and periodic constants carefully, we show that for a certain Z4-equivariant quintic systems, there are four fine focuses of five order and five limit cycles can bifurcate from each, we also find conditions of center and isochronous center for this system. The process of proof is algebraic and symbolic by using common computer algebra soft such as Mathematica, the expressions after being simplified in this paper are simple relatively. Moreover, what is worth mentioning is that the result of 20 small limit cycles bifurcating from several fine focuses is good for Z4-equivariant quintic system and the results where multiple singular points become isochronous centers at the same time are less in published references.     

Global structure for a class of dynamical systems

In this paper, we shall consider the global structure of positive bounded systems on the plane which have m singular points, but not any closed orbits and singular closed orbits. We shall prove that these systems have at least m−1 connecting orbits; and all the connecting orbits, homoclinic orbits and singular points constitute a compact simply connected set. Each of other orbits tends to a singular point as t→+∞, and approaches to the infinity as t→−∞.     

Use of an interval global optimization tool for exploring feasibility of batch extractive distillation

An interval arithmetic based branch-and-bound optimizer is applied to find the singular points and bifurcations in studying feasibility of batch extractive distillation. This kind of study is an important step in synthesizing economic industrial processes applied to separate liquid mixtures of azeotrope-forming chemical components. The feasibility check methodology includes computation and analysis of phase plots of differential algebraic equation systems (DAEs). Singular points and bifurcations play an essential role in judging feasibility. The feasible domain of parameters can be estimated by tracing the paths of the singular points in the phase plane; bifurcations indicate the border of this domain. Since the algebraic part of the DAE cannot be transformed to an explicit form, implicit function theorem is applied in formulating the criterion of bifurcation points. The singular points of the maps at specified process parameters are found with interval methodology. Limiting values of the parameters are determined by searching for points satisfying bifurcation criteria.     

Singular points of solutions of elliptic systems in the plane

We study the structure of solutions for some important classes of singular elliptic systems in the plane. In particular, it is proved that the solutions of such systems have principally nonanalytic behavior in neighborhoods of fixed singular points. These results enable one to correctly state certain boundary-value problems and make their complete analysis.     

Multidimensional Poincaré construction and singularities of lifted fields for implicit differential equations

This paper is devoted to singular points of the so-called lifted vector fields, which arise in studying systems of implicit differential equations by using the method of lifting the equation to a surface, a generalization of the construction used by Poincaré for a single implicit equation. The author studies the phase portraits and renormal forms of such fields in a neighborhood of their singular points. In conclusion, this paper considers the lifted vectors fields generated by Euler-Lagrange and Euler-Poisson equations and fast-slow systems. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 19, Optimal Control, 2006.     

Symplectic Geometry of Manifolds with Almost Nowhere Vanishing Closed 2-Form

We study local geometric properties of manifolds equipped with a closed 2-form nondegenerate at all points of a dense proper subset. We introduce the natural notion of tame singular point, at which the matrix of the 2-form degenerates in a regular way. We find a condition for Hamiltonian dynamical systems to be extended smoothly to tame singular points, generalize the Darboux theorem about the local reduction of the matrix of the 2-form to canonical form, and study the singular behavior of directional gradients.     

On cascades of bifurcations leading to chaos in several nonlinear dissipative systems of ODEs

This paper considers several nonlinear dissipative systems of ordinary differential equations. The studied systems undergo a full analysis of corresponding singular points on a whole set of parameters’ values variation. Specifically, types of singular points, boarders of stability regions, as well as presented local bifurcations, are determined. By using numerical methods a consideration of scenarios of transition to chaos in these systems with one bifurcation parameter variation is held. The aim of this research is a confirmation of a Feigenbaum–Sharkovskii–Magnitskii mechanism of transition to chaos unique for all dissipative systems of ODEs. As the result of analysis of one of the systems the lack of any chaotic behavior is shown with the help of Poincare sections.