The number of limit cycles of a quintic polynomial system with center
type: Item
instance of: scholarly article
title: The number of limit cycles of a quintic polynomial system with center
zbMATH Open document ID: 1181.34040
author: Ali Atabaigi
author: N. Nyamoradi
author: Hamid R. Z. Zangeneh
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-07-24T00:00:00Z
review text: The bifurcation of limit cycles in the system \[ \dot{x}=y(1+x^4),\;\dot{y}=-x(1+x^4)+\varepsilon P(x)y^{2m-1} \] is studied for small \(\varepsilon\). It is assumed that \(P(x)\) is a real polynomial of degree \(2n+2\) or \(2n+3\) and \(m,n\) are natural numbers. The main result states that an upper bound for the number of limit cycles bifurcating from the periodic orbits of the initial system (\(\varepsilon=0\)) is given by \(\min\{N_1, N_2, N_3\}\) where \[ \begin{cases} N_1=4m+2n-2+\sin^2(n\pi/2)+[\frac14(m+n-3+\sin^2(n\pi/2))],\\ N_2=3m+2n-2-\sin^2(n\pi/2)+[\frac12(m+n-1)],\\ N_3=5m+2n-2+\sin^2(n\pi/2).\end{cases} \] Moreover, there are systems with at least \(3m+n-2\) limit cycles. The proof follows from an estimation of the number of positive zeros of the integral \(\Phi(h)=\oint_{x^2+y^2=h}P(x)y^{2m}dx/(1+x^4)\) which is elementary and is calculated explicitly in the paper. (Reviewer's remark). The integral \(\Phi(h)\), as taken by the authors, is identically zero. One should consider a similar integral with \(y^{2m-1}\) instead of \(y^{2m}\) in order to obtain information about the limit cycles in the perturbed system.
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C08
zbMATH DE Number: 5586835
zbMATH Keywords: Limit cycles
zbMATH Keywords: small perturbation
zbMATH Keywords: Hilbert's 16th problem
zbMATH Keywords: argument principle
reviewed by: Iliya D. Iliev
MaRDI profile type: Publication
full work available at URL: j.na.2009.01.213
OpenAlex ID: W2041853839
cites work: Q4280773
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DOI: 10.1016/J.NA.2009.01.213
description: scientific article; zbMATH DE number 5586835