Limit cycle bifurcations by perturbing a class of planar quintic vector fields
type: Item
instance of: scholarly article
title: Limit cycle bifurcations by perturbing a class of planar quintic vector fields
zbMATH Open document ID: 1454.34057
author: Yanqin Xiong
author: Maoan Han
published in: Journal of Differential Equations
publication date: 2020-10-13T00:00:00Z
review text: In the qualitative theory of differential systems the main problems are to obtain the phase portraits, the singular points and to determine the maximum number of limits cycles that these systems can have. The problem of establishing an upper bound for the maximum number of limits cycles has been the subject of many papers but it is still open. Due to the difficulty of this problem several researchers study the lower or upper bounds for the maximum number of limit cycles of particular differential systems. In this paper, the authors consider a concrete class of planar quintic vector fields and determine all the phase portraits. Moreover, they perturbe this vector field with polynomials of degree \(n\), \(n\in{\mathbb{N}}\) up to first order in \(\epsilon\). Using the corresponding abelian integrals they prove that \(3\left[ \tfrac{n-1}{2}\right]\), \(n\geq5\), is a lower bound for the maximum number of limit cycles bifurcating from the periodic orbits.
reviewed by: Jeidy Johana Jimenez
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C08
zbMATH DE Number: 7260127
zbMATH Keywords: phase portrait
zbMATH Keywords: singular point
zbMATH Keywords: limit cycle
MaRDI profile type: Publication
full work available at URL: j.jde.2020.07.004
OpenAlex ID: W3088722364
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DOI: 10.1016/J.JDE.2020.07.004
description: scientific article; zbMATH DE number 7260127