Quadratic centers defining elliptic surfaces
type: Item
instance of: scholarly article
title: Quadratic centers defining elliptic surfaces
zbMATH Open document ID: 1171.34034
author: Sébastien Gautier
published in: Journal of Differential Equations
publication date: 2008-12-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/0704.1948
review text: The purpose of the paper is to classify all the first integrals of planar quadratic differential systems (not necessarily having a center) which define elliptic curves. In the first part, a full classification, up to an affine equivalence, is given of all elliptic foliations, that is those having a center. There are 18 reversible cases, 5 reversible Lotka-Volterra cases and 6 generic Lotka-Volterra cases in the classification. In the second part, all degenerate cases (i.e. without a center) are determined and the topology of their singular fibers is investigated. The results from the first part would be useful to add several new classes of quadratic centers whose limit cycles bifurcating under small perturbations are tractable by the technique (also known as the Abelian integrals method) previously used for the elliptic Hamiltonian cases.
Mathematics Subject Classification ID: 34C99
Mathematics Subject Classification ID: 37C15
Mathematics Subject Classification ID: 34C08
Mathematics Subject Classification ID: 14J27
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34A05
zbMATH DE Number: 5382288
zbMATH Keywords: quadratic centers
zbMATH Keywords: elliptic phase curves
zbMATH Keywords: elliptic surface
zbMATH Keywords: singular surface
zbMATH Keywords: Kodaira's classification
zbMATH Keywords: birational mapping blowing up
reviewed by: Iliya D. Iliev
MaRDI profile type: Publication
OpenAlex ID: W1966869635
arXiv ID: 0704.1948
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DOI: 10.1016/J.JDE.2008.06.033
description: scientific article