Limit cycles of polynomial differential systems bifurcating from the periodic orbits of a linear differential system in \(\mathbb R^d\)
type: Item
instance of: scholarly article
title: Limit cycles of polynomial differential systems bifurcating from the periodic orbits of a linear differential system in \(\mathbb R^d\)
zbMATH Open document ID: 1181.34042
author: Jaume Llibre
author: Amar Makhlouf
published in: Bulletin des Sciences Mathématiques
publication date: 2009-09-29T00:00:00Z
review text: The authors investigate the number of limit cycles of the following polynomial system \[ x_1' = -x_2 + \varepsilon P_1(x_1, \dots, x_d) + \varepsilon^2Q_1(x_1, \dots, x_d), \] \[ x_2' = x_1 + \varepsilon P_2(x_1, \dots, x_d) +\varepsilon^2Q_2(x_1, \dots, x_d), \] \[ x_k' = \varepsilon P_k(x_1, \ldots, x_d) + \varepsilon^2Q_k(x_1, \ldots, x_d) \] for \(k = 3,\dots, d\), where the maximum degree of \(P_k\) and \(Q_k\) is \(n_k\). When \(d \geq 2\) and \(\max\{n_1, n_2\}\geq 2\), by applying the first order averaging method to the above system, it is proved that at most \([(m-1)/2]n_3n_4\cdots n_d\) limit cycles bifurcate from the periodic orbits of the linear system with \(\varepsilon =0\). Further, by applying the second order averaging method to the above system, it is shown that at most \((m-1)(2n_3-1)(n_3+n_4-1)\cdots(n_3+n_d-1)\) limit cycles bifurcate from the periodic orbits of the linear system with \(\varepsilon = 0\). Although there are plenty of research publications on limit cycles of planar polynomial systems, not many have dealt with limit cycles of higher dimensional polynomial systems. This is an interesting paper.
reviewed by: Zhanyuan Hou
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C29
zbMATH DE Number: 5608810
zbMATH Keywords: limit cycles
zbMATH Keywords: polynomial vector fields
zbMATH Keywords: averaging methods
zbMATH Keywords: bifurcation
MaRDI profile type: Publication
full work available at URL: j.bulsci.2009.04.004
OpenAlex ID: W2094165126
cites work: Averaging methods for finding periodic orbits via Brouwer degree.
cites work: Limit cycles of differential equations
cites work: Limit cycles of some polynomial differential systems in dimension 2, 3 and 4, via averaging theory
cites work: On the nonexistence, existence and uniqueness of limit cycles
cites work: Q4771357
DOI: 10.1016/J.BULSCI.2009.04.004
description: scientific article; zbMATH DE number 5608810