Saddle quantities and applications
The authors enumerate in the first three sections many known results about the normal forms of a 2-dimensional autonomous differential system and their relation with the generalized homoclinic loop bifurcation, strengthing the superiority of the normal forms got by one of them, both from theoretical and practical point of views. In {\S} 4 the definition of the saddle quantities is given through the coefficients in the asymptotic expansion of the map defined by the flow near the saddle point. In {\S} 5 definitions and properties of the dual Lyapunov constants are given, dualities between a weak focus and a weak saddle are shown in Table I. In {\S} 6 three saddle quantities in the case of quadratic system are given in another form, together with a theorem concerning their important properties, all of these had been already obtained by Zhu Deming, etc. some years ago. {\S} 7 deals with generalized homoclinic bifurcation of order 5, the method used is similar to that in a previous paper of C. Z. Li and C. Rousseau. In {\S} 8, saddle quantities of a symmetric cubic system are obtained by using a Macsyma program, and it is shown that \(V^*_ 1=...=V^*_ 5=0\) implies all \(V^*_ i=0\) by transforming the system into that of K. S. Sibirskij through complex transformation and then using his result.
scientific article; zbMATH DE number 4120573
scientific article