Saddle quantities and applications
type: Item
instance of: scholarly article
title: Saddle quantities and applications
zbMATH Open document ID: 0684.34033
DOI: 10.1016/0022-0396(89)90069-7
author: Pierre Joyal
author: Christiane Rousseau
published in: Journal of Differential Equations
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 4120573
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: autonomous differential system
zbMATH Keywords: homoclinic loop bifurcation
zbMATH Keywords: saddle point
zbMATH Keywords: dual Lyapunov constants
zbMATH Keywords: homoclinic bifurcation
zbMATH Keywords: Macsyma program
reviewed by: Yanqian Ye
describes a project that uses: MACSYMA
MaRDI profile type: Publication
full work available at URL: 0022-0396(89)90069-7
OpenAlex ID: W2069228691
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description: scientific article; zbMATH DE number 4120573