推广后继函数法研究第二临界情况下同宿环的稳定性

THE CRITERION FOR DETERMINING THE STABILITYOF A HOMOCLINIC CYCLE FOR THE SECOND CRITICAL CASE AND ITS APPLICATION

本文通过灵活选取参照闭曲线,推广了研究闭轨线的后继函数法.通过计算后继函数,本文首先获得了二重极限环的半稳定性判据.在此基础上,运用推广的后继函数法,获得了第二临界情况下同宿环的内稳定性判据,事实上,推广的后继函数法可对以往的结果和本文的结果用统一的方法给予证明,并可向更高临界情况推广.最后本文证明了二重极限环及第二临界情况下的同宿环在一定条件下分支出极限环的唯二性.

In this paper, we first obtained the criterion for determining the stability of a semi-stable multiple limit cycle by computing the successor function. Then by generalizing the criterion, we obtained the criterion for determining the stability of a homoclinic cycle for the second critical case. Appling the criterion we determined the stability of a homoclinic cycle with the second critical case. The method used in this paper can give a unified treatment for the results given in the past. Finally, we study the number of limit cycles by bifurcation from a multiple limit cycle of a homoclinic cycle for the second critical case.