Bifurcation of limit cycles from a 4-dimensional center in \(\mathbb R^m\) in resonance \(1:N\)
type: Item
instance of: scholarly article
title: Bifurcation of limit cycles from a 4-dimensional center in \(\mathbb R^m\) in resonance \(1:N\)
zbMATH Open document ID: 1242.34066
author: Luís Barreira
author: Jaume Llibre
author: Claudia Valls
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-03-16T00:00:00Z
review text: The authors study how many limit cycles emerge from the periodic orbits of a center when we perturb it inside a given class of differential equations in dimension higher than two. More precisely, given \(m \geq 5\) and for every positive integer \(N \geq 2\) they consider the linear differential center \(\dot{x}=A x\) in \(\mathbb{R}^m\) that has at the origin a singular point with eigenvalues \(\pm i\), \(\pm Ni\) and \(0\) with multiplicity \(m-4\). Hence, this singular point has a 4-dimensional center in resonance \(1:N\). The authors are interested in studying how many limit cycles can bifurcate from the periodic orbits of this center when we perturb it in \(\mathbb{R}^m\) with \(m > 4\) inside the class of polynomial vector fields of the form linear plus a homogeneous nonlinearity of degree \(N\), that is, the perturbed system is \[ \dot{x}= Ax + \varepsilon F(x), \] where every component of \(F(x)\) is a linear polynomial plus a homogeneous polynomial of degree \(N\). Some explicit upper bounds for the number of limit cycles are given using the averaging method that improves and extends previous results obtained for \(\mathbb{R}^4\).
reviewed by: Jaume Giné
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C29
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 6015243
zbMATH Keywords: periodic orbit
zbMATH Keywords: averaging method
zbMATH Keywords: limit cycles
zbMATH Keywords: resonance \(1:N\)
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.12.018
OpenAlex ID: W1978148445
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DOI: 10.1016/J.JMAA.2011.12.018
description: scientific article; zbMATH DE number 6015243