Quadratic vector fields in \({\mathbb{C}}\mathbb{P}^2\) with two saddle-node type singularities at infinity
type: Item
instance of: scholarly article
title: Quadratic vector fields in \({\mathbb{C}}\mathbb{P}^2\) with two saddle-node type singularities at infinity
zbMATH Open document ID: 0988.37065
DOI: 10.1007/BF02269371
author: Laura Ortiz-Bobadilla
published in: Journal of Dynamical and Control Systems
publication date: 2000-04-13T00:00:00Z
review text: The author studies some group properties of the monodromy transformation of a family of vector fields in the projective space \(\mathbb{C}\mathbb{P}^2\). It is supposed that the fields considered have three different singular points at infinity, two of them saddle-nodes and that they leave the line at infinity invariant. Up to the affine change of coordinates these will be fields written as ODE: \[ \frac{dw}{dz}= \frac{p_{10}z+p_{01}w+zw-w^2} {q_{10}z+q_{01}w} \tag{1} \] and we are interested in the group properties of the monodromy transformation of (1) at infinity. Theorem 1 says that, generically, a necessary and sufficient condition for (1) to have commutative monodromy group at infinity is the invariance of the \(w\)-axis with respect to (1). Theorem 2 gives the conditions of embeddability of monodromy germs for a subfamily of (1). The work is closely related to the works of the Russian school and uses nonlinear Stokes phenomena and groups of germs of conformal mappings (Il'yashenko, Elizarov).
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 32G34
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34M40
zbMATH DE Number: 1431790
zbMATH Keywords: quadratic vector fields
zbMATH Keywords: singularities at infinity
zbMATH Keywords: monodromy transformation
zbMATH Keywords: projective space
zbMATH Keywords: nonlinear Stokes phenomena
zbMATH Keywords: germs of conformal mappings
reviewed by: Zofia Denkowska
MaRDI profile type: Publication
cites work: Groupes d'automorphismes de $({\bbfC},0)$ et équations différentielles $ydy+\cdots =0$
cites work: Q3139741
cites work: Q4888912
cites work: Problèmes de modules pour des équations différentielles non linéaires du premier ordre
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description: scientific article; zbMATH DE number 1431790