一类四次椭圆Hamilton向量场在三次多项式下的扰动

本文研究一类四次椭圆Hamilton向量场在所有三次多项式下的扰动,证明了如下结论: (1)除全局中心外,围绕一个中心定义的Abel积分的孤立零点的个数不超过12; (2)存在一个三次系统,它在扰动前属于一个鞍点环的情形,而在扰动后至少存在3个极限环.     

一类具有尖点环的三次Hamilton向量场的Abel积分

文进一步完善了文 [6]的工作 ,证明了 Abel积分I(h) =∮Γh(α βx γx2 ) ydx的零点个数上界 B(3)满足不等式 4≤B(3)≤ 6,这里 Γh是代数曲线 H(x,y) =12 y2 13x3 14 x4=h的连通闭分支 ,h∈ (- 1 / 1 2 ,0 )∪ (0 , ∞ )     

极限环和拟旋转向量场

本文定义了平面上的拟旋转向量场,研究在拟旋转向量场中极限环随参数的变化情况,证明它具有和旋转向量场完全族类似的性质.     

一类三次系统的极限环分枝

本研究一类对称的三次系统dx/dt=x(a 2y cy^2 dx^2) dy/dt=1/4-x^2-y^2(d≠0)，给出了系统不存在极限环的参数区域，并在有环的参数区域内给出了它的局部极艰环分枝图，证明了不存在Poincaé分枝。     

一类平面三次拟齐次向量场的全局拓扑结构

利用中心投影变换的思想证明一类平面三次拟齐次向量场的几何性质依赖于它的切向量场和诱导向量场.讨论了该系统的拓扑结构,并进行了分类;证明了该系统具有25类不同类的拓扑结构相图.     

一类三次系统的奇闭轨与极限环

本文运用旋转向量场理论，推广了文［１］的一个结论；证明了系统ｄｘｄｔ＝－ｕｘ＋２ｙ＋ｖｘ３＋ｗｘｙ２－２ｙ３，ｄｙｄｔ＝ｘ＋ｘ３在一定条件下具有“８字形奇闭轨”，并且它的外面至少还有一环；适当选取ｕ，ｖ，ｗ的值，系统至少有五个环     

R3中一类二次系统的5个极限环

建立了联系R3中一类二次系统与二维流形球面S2的切向量场之间的桥梁,证明了R3中一类二次系统存在5个极限环,而且这5个极限环分别位于该系统的5个不变闭锥上.     

一类生化反应方程的分枝

文[1]建立了一类生化反应方程E_(α,β),该文借助于计算机作了极限环的一些数值计算。[2]讨论了E_(σ,β)的极限环的不存在性,存在性和唯一性。本文弄清楚了E_(α,β)(α≥0,β>0)的各种分枝,包括高阶奇点分枝,各阶Hopf分枝,同宿轨道分枝,半稳定环分枝。如果还假定至多只有2个极限环,刚半稳定环分枝还是唯一的(详见定理A)。     

多项式系统的类旋转向量场

构造了一类依赖于某一参数δ的多项式系统,位于此系统的向量场中的多个相邻的单重极限环可以随δ的单调变化而同时扩大(或缩小),不过这时极限环的扩大(或缩小)不一定是单调的.由于这种向量场类似于旋转向量,故称此系统的这些极限环关于δ形成“类旋转向量场”,它们可以作为研究重环和分界线环分支的一种有效工具.     

Bogdanov-Takens系统的三次齐次扰动

对Bogdanov-Takens向量场的三次齐次扰动系统进行了讨论，得到了当其前二阶Melnikov函数恒为0时，则其后各阶Melnikov函数一定为0，且对于小的扰动参数，此系统为可积的或为Hamilton的；并对M1（h)≠0和M1(h)≡0，M2(h)≠0两种情形该系统的分岔结构给出了较全面的结论。     

向量场的积分曲线分类和链群与有向同伦不变性

利用积分曲线的极限集,对紧流形M上向量场X所确定的积分曲线进行了分类,在分类的基础上定义了向量场X的链群、极限集闭链群和极限集边缘链群,以及两个同类群,最后还引入了向量场正向同伦和负向同伦的概念,并证明了极限集是正向同伦和负向同伦不变的,链群、极限集闭链群和极限集边缘链群以及两个同类群在向量场双向同伦的情况下是同构的.     

Bifurcation and Isochronicity at Infinity in a Class of Cubic Polynomial Vector Fields

In this paper, we study the appearance of limit cycles from the equator and isochronicity of infinity in polynomial vector fields with no singular points at infinity. We give a recursive formula to compute the singular point quantities of a class of cubic polynomial systems, which is used to calculate the first seven singular point quantities. Further, we prove that such a cubic vector field can have maximal seven limit cycles in the neighborhood of infinity. We actually and construct a system that has seven limit cycles. The positions of these limit cycles can be given exactly without constructing the Poincare cycle fields. The technique employed in this work is essentially different from the previously widely used ones. Finally, the isochronous center conditions at infinity are given.     

一个三维Chemostat竞争系统的Hopf分支和周期解

本文研究了一个三维Chemostat竞争系统的解的结构,分析了平衡点的稳定性和当系统的某一微生物物种处于竞争劣势趋于灭绝时另一微生物物种和养料的二维流形上极限环的存在性,以及系统的Hopf分支问题.文中用Friedrich方法得到了系统存在Hopf分支的条件,并判定了周期解的稳定性.     

向量场的Nielsen数

对于紧致流形M上的任意一个向量场X，定义了一个由向量场X确定的自映射fX：M→M，使得向量场X的奇异点均为fX的不动点．证明了向量场的Nielsen数是不依赖于向量场选取的量．     

Bifurcations and distribution of limit cycles for near-Hamiltonian polynomial systems

This paper concerns with limit cycles through Hopf and homoclinic bifurcations for near-Hamiltonian systems. By using the coefficients appeared in Melnikov functions at the centers and homoclinic loops, some sufficient conditions are obtained to find limit cycles.     

Number and Location of Limit Cycles in a Class of Perturbed Polynomial Systems

In this paper,we investigate the number,location and stability of limit cycles in a class of perturbedpolynomial systems with (2n 1) or (2n 2)-degree by constructing detection function and using qualitativeanalysis.We show that there are at most n limit cycles in the perturbed polynomial system,which is similar tothe result of Perko in [8] by using Melnikov method.For n=2,we establish the general conditions dependingon polynomial's coefficients for the bifurcation,location and stability of limit cycles.The bifurcation parametervalue of limit cycles in [5] is also improved by us.When n=3 the sufficient and necessary conditions for theappearance of 3 limit cycles are given.Two numerical examples for the location and stability of limit cycles areused to demonstrate our theoretical results.     

一类三次系统的中心判定问题

对于一类具有三次衄线解x~2(x-1)-y~2-1=0,通过点(1,0)的直线解和中心-焦点型奇点的三次系统,证明了它以原点为中心的充要条件是它的前五阶焦点量全为零.这些中心条件是通过构造积分因子得以验证的.     

The Closed Orbits of a Class of Cubic Vector Fields in R3

In this paper, we investigate the isolated closed orbits of two types of cubic vector fields in R₃ by using the idea of central projection transformation, which sets up a bridge connecting the vector field X(x) in R₃ with the planar vector fields. We have proved that the cubic vector field in R₃ can have two isolated closed orbits or one closed orbit on the invariant cone. As an application of this result, we have shown that a class of 3-dimensional cubic system has at least 10 isolated closed orbits located on 5 invariant cones, and another type of 3-dimensional cubic system has at least 26 isolated closed orbits located on 13 invariant cones or 26 invariant cones.     

一类平面三次微分系统极限环的存在性与唯一性

蛤出三次系统{dx/dt=-y(ax^2 bx 1) δx-lx^2 dy/dt=x(ax^2 bx 1)极限环的不存在性，存在性及唯一性的一些充分条件．     

Pointwise Well-Posedness of Perturbed Vector Optimization Problems in a Vector-Valued Variational Principle

In this note, we point out and correct some errors in Ref. 1. Another type of pointwise well-posedness and strong pointwise well-posedness of vector optimization problems is introduced. Sufficient conditions to guarantee this type of well-posedness are provided for perturbed vector optimization problems in connection with the vector-valued Ekeland variational principle.