PP-graphics with a nilpotent elliptic singularity in quadratic systems and Hilbert's 16th problem.
type: Item
instance of: scholarly article
title: PP-graphics with a nilpotent elliptic singularity in quadratic systems and Hilbert's 16th problem.
zbMATH Open document ID: 1046.34055
DOI: 10.1016/S0022-0396(03)00119-0
published in: Journal of Differential Equations
publication date: 2004-01-26T00:00:00Z
review text: This paper is a part of the program to prove the finiteness part of Hilbert's 16th problem for quadratic systems which consists in proving that 121~graphics of such systems have finite cyclicity. The authors show that any pp-graphic through a multiplicity~3 nilpotent singularity of elliptic type which does not surround a center has finite cyclicity. Altogether, the authors show the finite cyclicity of 15~graphics. In particular, they prove the finite cyclicity of a pp-graphic with a nilpotent elliptic singular point together with a hyperbolic saddle with hyperbolicity different from~1 which appears in generic 3-parameter families of vector fields and hence belongs to the ``zoo'' of Kotova and Stanzo.
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 37G15
zbMATH DE Number: 2033006
zbMATH Keywords: Hilbert's 16th problem
zbMATH Keywords: quadratic system
zbMATH Keywords: nilpotent elliptic singularity
zbMATH Keywords: PP-graphic
author: Christiane Rousseau
author: Huanping Zhu
reviewed by: Valery A. Gaiko
MaRDI profile type: Publication
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full work available at URL: s0022-0396(03)00119-0
OpenAlex ID: W2014760113
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description: scientific article; zbMATH DE number 2033006