二次系统存在三次曲线弓形分界线环的充要条件

可证二次系统内含焦点的三次曲线弓形分界线环必由抛物线与直线所围成。定理1 二次系统存在三次曲线弓形分界线环的充要条件是此系统可化为以下形式     

二次系统的三次曲线极限环和分界线环的存在性问题

本文证明二次系统不存在三次曲线极限环,并得到二次系统存在三次曲线分界线环的充要条件;在此基础上又得到Ⅱ类二次系统存在三次曲线分界线环的充要条件,并附带地证明了Ⅰ,Ⅱ_(l=0),Ⅱ_(m=0),Ⅱ_(n=0)类二次系统均不存在三次曲线分界线环。     

中心对称三次系统存在双曲线分界线环的充要条件

本文给出了中心对称三次系统存在双曲线分界线环的充要条件，并证明了此系统还可以至少存在五个极限环。     

三次系统存在一类双纽线分界线环充要条件

给出了中心对称三次系统存在一类双纽线分界线环的充要条件，并举出此系统至少还存在四个极限环的（2．2）分布的例子．还举出了中心对称三次系统至少存在六个极限环作（3.3）分布以及五个极限环，其中一个极限环包围作（2．2）分布的四个极限环的例子．     

具有三次曲线解的中心对称三次系统的极限环的存在性问题

本文证明了具有三次曲线解y＝αx3的中心对称三次系统可以存在极限环，从而纠正了文[1]认为具有三次曲线解的中心对称三次系统不可能存在极限环的错误结论     

三次系统E13存在二角形双曲线分界线环的充要条件

得到了三次系统E1/3存在二角形双曲线分界线环的充要条件．并给出了它们的拓扑分类和各种拓扑结构的参数条件．     

三次系统E_3~1存在二角形双曲线分界线环的充要条件

得到了三次系统 E1 3存在二角形双曲线分界线环的充要条件 .并给出了它们的拓扑分类和各种拓扑结构的参数条件 .     

一类具有二虚不变直线的三次系统的极限环与分支

讨论一类具有二虚平行不变直线的三次系统，求出了奇点O(0,0)的焦点量, 证明了δlmn=0 时系统在O外围至多有一个极限环. 利用分支理论给出了分界线环和半稳 定环分支曲线的分支图，进一步说明了系统至多有二个极限环.     

具有三次曲线解xy2+y=x3的中心对称三次系统极限环的存在性

本文证明了具有三次曲线解ｘｙ²＋ｙ＝ｘ³的中心对称三次系统的极限环存在，而且至少可以存在四个极限环，它们作（２，２）分布．从而纠正了文［１］的结论     

一个群体防卫捕-食系统的定性分析

该文对一个群体防卫捕一食系统进行了较全面的定性分析．讨论了分界线的相对位置，得到了极限环的存在性、唯一性以及分界线环的存在性，首次证明了群体防卫捕一食系统可以至少存在两个或三个极限环．     

A complete bifurcation diagram of nonlocal bifurcations of singular points in the Lorenz system

For the system of Lorenz equations in the parameter space we construct a complete bifurcation diagram of all homoclinic and heteroclinic separatrix contours of singular points that exist in the system. These constructs include the existence surface of a homoclinic butterfly, the existence half-surface of homoclinic loops of saddle-focus separatrices, and the existence curve of a heteroclinic separatrix contour joining a saddle-node with two saddle-foci.     

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.     

Liénard systems for quadratic systems with invariant algebraic curves

We study the character of the friction function f(x) and the restoring force g(x) in the Liénard system to which a quadratic system with an invariant second-order algebraic curve (an ellipse that is a limit cycle, a hyperbola defining two separatrix cycles, or a parabola) or fourth-order algebraic curve with an oval being a limit cycle can be reduced. Invariant curves are constructed for quadratic systems in a five-parameter canonical family, which can readily be reduced to Liénard systems.     

无穷远分界线的稳定性和产生极限环的条件

本文给出了无穷远分界线稳定性的判据以及从无穷远分界线产生极限环的条件，其结果包括了［７］，［８］的主要结果。     

LIMIT CYCLE PROBLEM OF QUADRATIC SYSTEM OF TYPE （ Ⅲ ）m=0, （ Ⅲ ）

To continue the discussion in （Ⅰ ） and （ Ⅱ ）,and finish the study of the limit cycle problem for quadratic system （ Ⅲ ）m=0 in this paper. Since there is at most one limit cycle that may be created from critical point O by Hopf bifurcation,the number of limit cycles depends on the different situations of separatrix cycle to be formed around O. If it is a homoclinic cycle passing through saddle S1 on 1 ＋ax-y = 0,which has the same stability with the limit cycle created by Hopf bifurcation,then the uniqueness of limit cycles in such cases can be proved. If it is a homoclinic cycle passing through saddle N on x= 0,which has the different stability from the limit cycle created by Hopf bifurcation,then it will be a case of two limit cycles. For the case when the separatrix cycle is a heteroclinic cycle passing through two saddles at infinity,the discussion of the paper shows that the number of limit cycles will change from one to two depending on the different values of parameters of system.     

二次系统存在一类四次曲线分界线环的充要条件（续）

本义给出了二次系统存在一类四次曲线分界线环的充要条件，此类分界线环的方程为（ｙ＋ｃｘ￣２）－ｘ￣２（ｘ－ａ）（ｘ－ｂ）＝０。     

关于同宿分支的Leontovich分界线量

由Leontovich定义的鞍点量和分界线量是判断同宿轨道分支出极限环的数目及同宿环稳定性的主要判据．利用Tkachev对多重极限环稳定性判定的方法,对给定的系统,得到了同宿环分支的第三阶分界线量的公式,并对高阶分界线量做了猜测．     

Chaotic dynamics of homogeneous Yang-Mills fields with two degrees of freedom

In the present paper, we consider a scenario of transition to chaotic dynamics in the Hamiltonian system of homogeneous Yang-Mills fields with two degrees of freedom in the case of the Higgs mechanism. We show that in such a system, as well as in other Hamiltonian and conservative systems of equations, the nonlocal effect of multiplication of hyperbolic and elliptic cycles and tori around elliptic cycles in neighborhoods of the separatrix surfaces of hyperbolic cycles plays a key role on the initial stage of transition from a regular motion to a chaotic one. We observe that the new elliptic and hyperbolic cycles of the Hamiltonian system are generated as stable and saddle cycles of the extended dissipative system of equations not only as a result of saddle-node bifurcations but also as a result of fork-type bifurcations.