On the number of limit cycles of a \(Z_{4}\)-equivariant quintic polynomial system
type: Item
instance of: scholarly article
title: On the number of limit cycles of a \(Z_{4}\)-equivariant quintic polynomial system
zbMATH Open document ID: 1201.34053
author: Weijiao Xu
author: Maoan Han
published in: Applied Mathematics and Computation
publication date: 2010-07-19T00:00:00Z
review text: The authors study limit cycles of the near-Hamiltonian \(\mathbb{Z}_4\)-equivariant polynomial system \[ \dot x=H_y+\epsilon P_5(x,y), \quad \dot y=-H_x+\epsilon Q_5(x,y) \tag{1} \] with the Hamiltonian \(H=-2 (x^2+y^2)+(x^4+y^4)\) and \(P_5, Q_5\) being some polynomials of degree five. Using the methods of Hopf and heteroclinic bifurcation, it is shown that system (1) can have 13 limit cycles.
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 5757564
zbMATH Keywords: limit cycle
zbMATH Keywords: polynomial system
zbMATH Keywords: heteroclinic loop
zbMATH Keywords: \(\mathbb{Z}_4\)-equivariance
zbMATH Keywords: Hopf bifurcation
MaRDI profile type: Publication
full work available at URL: j.amc.2010.04.014
OpenAlex ID: W2058179926
cites work: Mathematical problems for the next century
cites work: Bifurcation of small limit cycles in \(Z_{5}\)-equivariant planar vector fields of order 5
cites work: BIFURCATIONS OF LIMIT CYCLES IN A Z3-EQUIVARIANT VECTOR FIELD OF DEGREE 5
cites work: INVESTIGATIONS OF BIFURCATIONS OF LIMIT CYCLES IN Z2-EQUIVARIANT PLANAR VECTOR FIELDS OF DEGREE 5
cites work: Small limit cycles bifurcating from fine focus points in cubic order \(Z_{2}\)-equivariant vector fields
cites work: Analysis on limit cycles of \(Z_{q}\)-equivariant polynomial vector fields with degree 3 or 4
cites work: On the study of limit cycles of a cubic polynomial system under \(\mathbb Z_{4}\)-equivariant quintic perturba\-tion
cites work: Bifurcations of limit cycles for a cubic Hamiltonian system under quartic perturbations
cites work: Bifurcations of limit cycles in a \(Z_6\)-equivariant planar vector field of degree 5
cites work: Bifurcation set and distribution of limit cycles for a class of cubic Hamiltonian system with higher-order perturbed terms
cites work: Limit cycles near homoclinic and heteroclinic loops
cites work: Q3500469
cites work: HILBERT'S 16TH PROBLEM AND BIFURCATIONS OF PLANAR POLYNOMIAL VECTOR FIELDS
cites work: Bifurcation set and limit cycles forming compound eyes in a perturbed Hamiltonian system
cites work: Cyclicity 1 and 2 conditions for a 2-polycycle of integrable systems on the plane
DOI: 10.1016/J.AMC.2010.04.014
description: scientific article; zbMATH DE number 5757564