The bifurcation of limit cycles in \(Z_n\)-equivariant vector fields
type: Item
instance of: scholarly article
title: The bifurcation of limit cycles in \(Z_n\)-equivariant vector fields
zbMATH Open document ID: 1208.34047
author: Haibo Chen
author: Chaoxiong Du
author: Yirong Liu
published in: Applied Mathematics and Computation
publication date: 2010-11-18T00:00:00Z
review text: The authors study bifurcation of limit cycles from several diverse (weak) foci for polynomial dynamical systems in the plane. In the case of \(G\)-equivariant dynamical systems (\(G\) a Lie group), singular points appear in \(G\)-orbits. The authors are specially interested in the case \(G=\mathbb Z_n\), and, in particular, focus their attention on a certain \(\mathbb Z_8\)-equivariant system \[ dx/dt = f(x,y),\quad dy/dy=g(x,y) \] with \(f\) and \(g\) polynomials of degree 7 depending on nine parameters (with four relations between these, so five free parameters), and a varying parameter \(\delta\). They show that, for \(\delta = 0\), this specific system has eight foci (sitting on the unit circle) with the same topological structure. As the parameters are varied, and, in particular, if \(\delta\) becomes positive (of order ten in the perturbation parameter \(\varepsilon\)), then the system undergoes bifurcations; in particular, the author prove that, by a suitable choice of the perturbation, the (weak foci of the) system can bifurcate 40 limit cycles, and there are 24 stable limit cycles among them. These, as well as the perturbed and unperturbed parameters, are identified by completely explicit formulas. A short reference list is given including previous work by the authors dealing with the same problem of bifurcation of limit cycles from \(\mathbb Z_n\)-equivariant dynamical systems of degree \(k\), with \(n\) and \(k\) smaller than in the present note.
reviewed by: Giuseppe Gaeta
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 37G40
zbMATH DE Number: 5817291
zbMATH Keywords: dynamical systems
zbMATH Keywords: bifurcation of limit cycles
zbMATH Keywords: equivariant dynamics
MaRDI profile type: Publication
full work available at URL: j.amc.2010.07.005
OpenAlex ID: W2037429668
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DOI: 10.1016/J.AMC.2010.07.005
description: scientific article; zbMATH DE number 5817291