Lower bounds for the local cyclicity for families of centers
type: Item
instance of: scholarly article
title: Lower bounds for the local cyclicity for families of centers
zbMATH Open document ID: 1467.34034
author: Jaume Giné
author: Luiz F. S. Gouveia
author: Joan Torregrosa
published in: Journal of Differential Equations
publication date: 2021-01-08T00:00:00Z
review text: This paper considers the local cyclicity of a family of centers depending on parameters. The authors extend the results of Christopher for linear and higher-order studies when the center has parameters. They prove that the number of small amplitude limit cycles bifurcating from an equilibrium of monodromic type in the class of polynomial vector fields of degrees 3 is \(M(3)\geq12\) by studying the center \(CD^{12}_{31}\). Besides, they also prove that the number of small amplitude limit cycles bifurcating from the one-parametric cubic reversible center \(CR_{17}^{12}\) and two-parameter holomorphic cubic center is 11, and that the number of small amplitude limit cycles bifurcating from the two-parameter cubic center \(CD^{11}_{10}\) is 12. For polynomial vector fields of degrees 4, they prove that the number of small amplitude limit cycles bifurcating from the center is \(M(4)\geq21\) by investigating the bifurcation diagram for a two-parameter quartic center. Moreover, they give a partial study on the bifurcation diagram for a four-parameter quartic holomorphic center.
reviewed by: Xiuli Cen
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 7291340
zbMATH Keywords: small-amplitude limit cycles
zbMATH Keywords: polynomial vector fields
zbMATH Keywords: center cyclicity
zbMATH Keywords: Lyapunov constants
zbMATH Keywords: higher-order developments and parallelization
MaRDI profile type: Publication
full work available at URL: j.jde.2020.11.035
OpenAlex ID: W3111486174
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DOI: 10.1016/J.JDE.2020.11.035
description: scientific article; zbMATH DE number 7291340