Condition for Generating Limit Cycles by Bifurcation of the Loop of a Saddle-Point Separatrix

In paper [1] (On the stability of a saddle-point separatrix loop and analytical criterion for itsbifurcation limit cycles Acta Mathematica Sinica Vol.28.No.1,55-70,Bejing China 1985),we consideredthe problem of generating limit cycles by the bifurcation of a stable or an unstable loop of a saddle-pointseparatrix.We gave for the first time a criterion for the stability of the loop as following:Lis stable(unstable) if ∫~∞ (P′+Q′dt<0(>0) where x=(t),y=(t) then a sufficient condition for thebifurcation which generates limit cycles.This paper generalizes the result of [1] to the case where the loopcontains a center or the loop tends to an infinite saddle-point,and removes the restriction that the saddle-point should be an elementary singular point.Applying the results of this paper,the author studies a two-parameter systemx=lx~2+y~2-y+5εxyy=(3l+5)xy+x+εx~2.The results obtained by the author in this in real field coincides with the results given by Prof.Qin Yuanxunby means of the complex qualitative theory in complex field.