二次系统存在双曲线分界线环的充要条件

A Necessary and Sufficient Condition for the Existence of Hyperbolic Separatrix Cycle in Quadratic System

得到二次系统存在双曲线分界线环的充要条件,并将这些分界线环分成类,指出在不同的条件下将会出现不同类型的双曲线分界线环。可验知这些分界线环均不可能是细的,而且它们内部的焦点也不可能是细的。进一步应用这些充要条件于一些特殊类型的二次系统,诸如中心对称二次系统、Ⅱ_(n=0)类系统等,并指出这些类型的二次系统中,除了文[1],[2]、[3]的例子外,均不可能再有任何其它的双曲线分界线环。

In this paper we obtain a necessary and sufficient condition for the existence of hyperbolic separatrix cycle.We arrange these hyperbolic separatrix cycles in classes and point out that under each different condition,there will oppear different type of hyperbolic separatrix cycles.These hyperbolic separatrix cycle can not he weak,while the foci inside of them can not be weak too.Moreover we use the condition upon some particular quadratic systems such as the centre symetricai quadratic system and the Ⅱ_n=0 type system etc.and point out that in each type syslem there are no other hyperbolic separatrix cycles except the exam- ples in ref.[1]、[2]、[3].