On the number of limit cycles of a \(\mathbb{Z}_4\)-equivariant quintic near-Hamiltonian system
type: Item
instance of: scholarly article
title: On the number of limit cycles of a \(\mathbb{Z}_4\)-equivariant quintic near-Hamiltonian system
zbMATH Open document ID: 1351.34035
author: Xianbo Sun
author: Maoan Han
publication date: 2015-11-06T00:00:00Z
review text: Consider the planar polynomial system \[ {dx\over dt}= H_t+\varepsilon P_5(x,y),\quad {dy\over dt}= -H_x+\varepsilon Q_5(x,y),\tag{\(*\)} \] where \(\varepsilon\) is a small parameter, \(\deg P_5= \deg Q_5= 5\), and \[ H(x,y):= x^2+ y^2-{3\over 4} (x^4+ y^4)+{1\over 6} (x^6+ y^6). \] Under the assumption that \((*)\) is \(\mathbb{Z}_4\)-equivariant, the authors prove that \((*)\) can have 28 limit cycles, the corresponding phase portraits are described.
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C14
zbMATH DE Number: 6505626
zbMATH Keywords: limit cycle
zbMATH Keywords: near-Hamiltonian system
zbMATH Keywords: heteroclinic loop
zbMATH Keywords: \(\mathbb{Z}_4\)-equivariance
zbMATH Keywords: Hopf bifurcation
MaRDI profile type: Publication
full work available at URL: s10114-015-2117-3
OpenAlex ID: W2183441907
cites work: Q3871220
cites work: Lower bounds for the Hilbert number of polynomial systems
cites work: BIFURCATION OF LIMIT CYCLES IN A FOURTH-ORDER NEAR-HAMILTONIAN SYSTEM
cites work: A new cubic system having eleven limit cycles
cites work: Limit cycles near homoclinic and heteroclinic loops
cites work: Q3500469
cites work: ON THE NUMBER AND DISTRIBUTION OF LIMIT CYCLES IN A CUBIC SYSTEM
cites work: Centennial History of Hilbert's 16th Problem
cites work: AN IMPROVED LOWER BOUND ON THE NUMBER OF LIMIT CYCLES BIFURCATING FROM A HAMILTONIAN PLANAR VECTOR FIELD OF DEGREE 7
cites work: A cubic system with thirteen limit cycles
cites work: HILBERT'S 16TH PROBLEM AND BIFURCATIONS OF PLANAR POLYNOMIAL VECTOR FIELDS
cites work: Bifurcations of limit cycles in a \(Z_6\)-equivariant planar vector field of degree 5
cites work: Q3701788
cites work: New results on the study of \(Z_q\)-equivariant planar polynomial vector fields
cites work: BIFURCATIONS OF LIMIT CYCLES IN A Z2-EQUIVARIANT PLANAR POLYNOMIAL VECTOR FIELD OF DEGREE 7
cites work: Q4280774
cites work: Q3868118
cites work: Bifurcation of limit cycles in a quintic Hamiltonian system under a sixth-order perturbation
cites work: Existence of 121 limit cycles in a perturbed planar polynomial Hamiltonian vector field of degree 11
cites work: BIFURCATION OF LIMIT CYCLES IN Z10-EQUIVARIANT VECTOR FIELDS OF DEGREE 9
cites work: ON THE NUMBER AND DISTRIBUTIONS OF LIMIT CYCLES IN A QUINTIC PLANAR VECTOR FIELD
cites work: Bifurcations of limit cycles in a \(Z_{4}\)-equivariant quintic planar vector field
cites work: Twelve limit cycles in a cubic order planar system with \(Z_2\) symmetry
cites work: TWELVE LIMIT CYCLES IN A CUBIC CASE OF THE 16th HILBERT PROBLEM
cites work: The number and distributions of limit cycles for a class of quintic near-Hamiltonian systems
cites work: Bifurcations of limit cycles for a cubic Hamiltonian system under quartic perturbations
cites work: Bifurcations of limit cycles in equivariant quintic planar vector fields
DOI: 10.1007/S10114-015-2117-3
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published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6505626