The same distribution of limit cycles in a Hamiltonian system with nine seven-order perturbed terms
type: Item
instance of: scholarly article
title: The same distribution of limit cycles in a Hamiltonian system with nine seven-order perturbed terms
zbMATH Open document ID: 1144.34022
author: Tao Jiang
author: Zhiyan Yang
published in: Acta Mathematicae Applicatae Sinica. English Series
publication date: 2008-05-26T00:00:00Z
review text: The authors investigate the number, distribution and accurate positions of limit cycles of nine perturbed planar Hamiltonian systems \[ \dot x = y(1+x^2 -ay^2) + \varepsilon p_i(x,y), \quad \dot y = -x(1-cx^2 +y^2) + \varepsilon q_i(x,y), \] where \(a>c>0\), \(ac>1\), \(0 < \varepsilon \ll 1\), the terms \(p_i(x,y)\) and \(q_i(x,y)\) are some polynomials in the ring \(\mathbb{R}[x,y]\) of degree \(7\) with coefficients depending on real parameters \(l,m,\lambda\), and \(i=1, \dots, 9\). Using qualitative analysis (e.g., methods of detection functions) and numerical simulation, the authors show that the above nine perturbed systems have the same bifurcation diagrams of limit cycles and have \(13\) limit cycles for some parameters. They illustrate the positions of \(13\) limit cycles and detection curves for some concrete parameters.
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 37J45
zbMATH DE Number: 5278782
zbMATH Keywords: perturbed Hamiltonian system
zbMATH Keywords: limit cycle
zbMATH Keywords: bifurcation
zbMATH Keywords: detection function
reviewed by: Alexey O. Remizov
describes a project that uses: Dynamics
MaRDI profile type: Publication
full work available at URL: s10255-007-7015-8
OpenAlex ID: W1966926592
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DOI: 10.1007/S10255-007-7015-8
description: scientific article; zbMATH DE number 5278782