Existence of 121 limit cycles in a perturbed planar polynomial Hamiltonian vector field of degree 11
type: Item
instance of: scholarly article
title: Existence of 121 limit cycles in a perturbed planar polynomial Hamiltonian vector field of degree 11
zbMATH Open document ID: 1158.34022
published in: Chaos, Solitons and Fractals
publication date: 2008-08-22T00:00:00Z
review text: The paper explores a systematic route to find the maximal number of limit cycles of perturbed planar polynomial \(Z_q\)-equivariant Hamiltonian system. Thus, the paper is devoted to the investigation of the weakened 16-th Hilbert's problem proposed by Arnold. The first step in searching limit cycles is to define an optimized \(Z_q\)-equivariant Hamiltonian polynomial system with the most possible number and type of closed orbits, compound cycles and oval schemes. The equations determining the coefficients of the polynomial Hamiltonian function for the desired level curve structure have been constructed. Taking special consideration of \(Z_{12}\)-equivariant vector fields of degree \(11\), \(99\) closed orbits are obtained. Computing the global bifurcation of limit cycles gives rise to the maximum of \(121\) limit cycles. Two conjecture are proposed regarding the maximal number of closed orbits for equivariant Hamiltonian polynomial vector fields and the maximal number of limit cycles bifurcating from well defined Hamiltonian vector fields.
reviewed by: Alexander Grin
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 37C27
Mathematics Subject Classification ID: 34C07
zbMATH DE Number: 5315347
zbMATH Keywords: \(Z_q-\)equivariant Hamiltonian polynomial system
zbMATH Keywords: cyclicity
zbMATH Keywords: limit cycle
zbMATH Keywords: Melnikov function
zbMATH Keywords: bifurcation
zbMATH Keywords: 16-th Hilbert's problem
author: Sharon Wang
author: P. Yu
MaRDI profile type: Publication
full work available at URL: j.chaos.2005.12.016
OpenAlex ID: W2141310733
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DOI: 10.1016/J.CHAOS.2005.12.016
description: scientific article; zbMATH DE number 5315347