SYMMETRIES AND THE CALCULATIONS OF DEGREE

This paper considers the calculations of Loray-Schaudor degree of equivariant compact operators undor any compact Lie group actions. The main results include two parts. One is a local Leray-Schauder index formula on regular zero orbits. The other is a generalized Borsuk theorem.     

非Chetaev型变质量非完整系统的Lie对称性与Noether对称性

研究非Chetaev型变质量非完整系统的Lie对称性与Noether对称性以及其间的 关系,给出Lie对称性导致Noether对称性以及Noether对称性导致Lie对称性的条件.     

几类非线性差分方程的对称和精确解

本文将微分方程的Lie变换群方法推广到差分方程,给出了三类非线性差分方程的不变变换,利用这种变换由差分方程的平凡解得到非平凡的单参数解族。     

极限环的表达式和计算

本文提出一种解析法和数值法相结合的方法，用来计算多项式微分系统的极限环，极限环表示为x=∑k≥0(ak cos kφ bk sin kφ),y=∑k≥0(ck cos kφ dk sin kφ)，先用解析法求出小参数时极限环的初始表达式，然后用增量法和迭代法求出任意参数时极限环满足给定精度的表达式，半稳定极限环和分叉值也可以计算。     

CALIBRATED SUBMANIFOLDS OF R7 AND R8 WITH SYMMETRIES

The principal theory of this paper comprises a technique forconstructing associative, coassociative and Cayley submanifoldsof Euclidean space with symmetries, using first-order ordinarydifferential equations. Explicit examples of U(1)-invariantassociative cones in ⁷ and SU(2)-invariant Cayley 4-folds in⁸ are then produced using this method. Further examples of associative3-folds are presented, which are ruled, and other systems ofdifferential equations defining calibrated submanifolds in ⁷and ⁸ are given.     

一类Hamilton算子,遗传对称及可积系

本文构造了一类矩阵微分Hamilton算子并且生成了新的遗传对称和相应的可积系,进一步提出了一个新可积族的双Hamilton结构和公共遗传强对称算子。     

NONLOCAL SYMMETRIES AND NONLOCAL RECURSION OPERATORS

1 IntroductionTl1ere is a well developed theory for local symnwtries with partial differential equations(ref[1-2]). However this theory does not apply to many systems of integrable equations, such asthe iuterlllediate wave equation whicl1 involves integrals in their definition and so are essentiallynonlocal. Oll the otl1er hand, wlien investigating differelltial equations, we often use OPeratorssuch as integrthdtherential recursion operators, which, in general, are in sonle inverse to differ-…     

Abel方程极限环的个数及应用

本文研究Ａｂｅｌ方程极限环的唯一性和唯二性。首先将〔１〕的定理Ｃ推广到一类平面多项式系统。其次，〔１〕的定理Ａ和Ｂ仅讨论了Ａｂｅｌ方程当Ａ（θ）或Ｂ（θ）之一为定号时其极限环的个数问题，本文将它推广到Ａ（θ）和Ｂ（θ）都可以是不完定号的情形，给出了一类平面多项式系统至多存在一个极限环的充分条件。     

PROPERTIES OF CHARACTERISTIC FUNCTIONS AND EXISTENCE OF LIMIT CYCLES OF LIENARD''S EQUATION

In this paper, the author considers Lienard''s equation, studies the properties of so-called characteristic functions and gives three theorems which ensure that the equation has at least one limit cycle. The theorems generlize Filippov''s Theorem which is a repesentative result, Dragilev''s Theorem, Theorem 1 2 of [6], Theorem 9 of [8] and Theorems 1, 2 of [9] respectively.     

有向路和有向圈的控制集数

1引言设G=(V,E)为无向图．子集D (?)V(G)是无向图G的控制集,如果对于任意的y,∈V(G)-D,都存在x∈D,使xy∈E(G)．G的控制集D是G的分裂控制集,如果G中由V(G)-D导出的子图G〈V(G)-D〉是不连通的．G的一个控制集D是G的一个强(弱)控制集,若dG(x)≥d_G(y)(d_G(x)≤d_G(y)),其中d_G(x)表示G中与点x关联的边数．对于有向图H=(V,A),子集D(?)V(H)称为H的控制集,如果对于任意的y∈     

A NECESSARY AND SUFFICIENT CONDITION FOR THE COEXISTENCE OF A CLASS OF CUBIC CURVE SEPARATRIX CYCLES AND LIMIT CYCLES TO THE CUBIC SYSTEM

In this paper, we give the necessary and sufficient condition for the coexistence of a class of cubic curve separatrix cycles and limit cycles to the cubic system, and study their topological structures.     

UNIQUENESS AND DISTRIBUTION OF LIMIT CYCLES FOR BOUNDED QUADRATIC SYSTEM

1 IntroductionAs we know, any given quadratic system which may have limit cycle (LC,fOr abbreviation) can be written in the fOllowing fOrm (see [1] 512)where 6, l, m, n, a, 6 are all real parameters.If all trajectories of a quadratic system remain bounded fOr t 2 0, we saythat the system is bounded, and fOr abbreviation denote by BQS in this paper.The research work for BQS begin with Dickson-Perko [3]. And then, in [4],they made use of the conclusions of [51 to give a detailed classifica…     

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。     

二次系统极限环的相对位置与个数

<正> 中的P_2(x,y)与Q_2(x,y)为x,y的二次多项式.文[1].曾指出,系统(1)最多有三个指标为+1的奇点,且极限环只可能在两个指标为+1的奇点附近同时出现.如果方程(1)的极限环只可能分布在一个奇点外围,我们就说此系统的极限环是集中分布的.本文主要研究具非粗焦点的方程(1)的极限环的集中分布问题,和极限环的最多个数问题.文[2]-[5]曾证明,当方程(1)有非粗焦点与直线解或有两个非粗焦点或有非粗焦点与具特征根模相等的鞍点时。方程(1)无极限环.本文给出方程(1)具非粗焦点时,极限环集     

一类三次系统极限环的存在唯一性

本文得到三次系统x=-y(1-ax)(1-ax) δx-lx3,y=x(1-ax)(1-bx)极限环的存在性、唯一性及不存在性的完整结果．     

EXISTENCE AND UNIQUENESS OF LIMIT CYCLES FOR A CLASS OF CUBIC DIFFERENTIAL SYSTEM

In this paper we consider the existence, uniqueness and nonexistence of limit cycles for the class of planar cubic system x=-y+δx+a2xy+a3x2+a7x3, y=x, where a7≠0.     

BOUNDEDNESS OF SOLUTIONS AND EXISTENCEOF LIMIT CYCLES FOR A NONLINEAR SYSTEMOF DIFFERENTIAL EQUATIONS

IntroductionTheautonomousplanarsystemoftheformwasfirststudiedbyLevinsonandSmithintheclassicalpaper[2]astheequivalelltsystellloftheequationofLienardtypeandlateronsomeauthorshavecontributedtotiletheoryofthissystemwithrespecttoqualitativebehaviorofsolutionsll--9].InthepresentpaperwestudyasystemOfaslightlymoregeneraltype,namely,wherethefllnctionsp(y),q(y),g(x)andh(x,y)arecontilluousforallvalueoftheirarguments,and3'resubjecttotheconditionswhichensurethattheexistenceofuniquesolutiontotheinitialvalu…     

THE PROLONGATION STRUCTURES AND NONLOCAL SYMMETRIES FOR MODIFIED BOUSSINESQ SYSTEM

The Wahlquist-Estabrook （WE） prolongation structures of modified Boussi-nesq （MB） system are studied from the coverings point of view. The realizations and classifications of one-dimensional coverings of this system are obtained completely. More-over the sufficient and necessary conditions for a vector field to be a nonlocal symmetry of this system are also demonstrated in the WE prolongation structures.     

一类四次系统极限环的个数与分布

本文研究一类四次系统的极限环,通过计算四次系统鞍点分界线之间的有向距离,计算一阶焦点量及二阶焦点量,判别同宿轨内外的稳定性,利用分支理论与定性分析技巧发现这类系统有六个极限环,并给出了它们的分布.     

一类三次系统极限环的个数与分布

本文研究一类三次系统的极限环,利用分支理论与定性分析技巧发现这类系统有四个极限环,并给出了他们的分布。