Limit cycles of a Liénard cubic system with quadratic friction function
type: Item
instance of: scholarly article
title: Limit cycles of a Liénard cubic system with quadratic friction function
zbMATH Open document ID: 1364.34036
DOI: 10.1134/S0012266108020109
published in: Differential Equations
publication date: 2010-01-15T00:00:00Z
review text: Consider the planar autonomous system \[ {dx\over dt}= y,\;{dy\over dt}= x[1-(1+ L)x+Lx^2]- \varepsilon \sum^3_{j=1} a_j x^{j-1}y,\tag{\(*\)} \] where \(-1<L<0\), \(\varepsilon>0\). System \((*)\) has three equilibria: \(E_1=(1/L,0)\), \(E_2= (1,0)\), \(S=(0,0)\), where \(S\) is a saddle, \(E_1\) and \(E_2\) are anti-saddles. Using the notation \((i_{E_1}, i_{E_2}, i_S)\) which denotes that \((*)\) has \(i_{E_1}(i_{E_2})\) limit cycles surrounding the equilibrium \(E_1(E_2)\), and \(i_S\) limit cycles surrounding all three equilibria, the authors describe a method to construct systems \((*)\) with the following distribution of limit cycles: \((2,0,0)\), \((0,2,0)\), \((1,1,1)\), \((1,1,0)\), \((1,0,1)\), \((0,1,1)\), \((0,0,2)\), \((0,1,2)\), \((1,0,2)\).
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 5659344
zbMATH Keywords: distribution of limit cycles
author: Leonid Cherkas
author: I. N. Sidorenko
MaRDI profile type: Publication
cites work: Perturbation from an elliptic Hamiltonian of degree four. IV: Figure eight-loop.
cites work: Abelian integrals in non-symmetric perturbation of symmetric Hamiltonian vector field
cites work: Limit cycles of quadratic systems in the plane
cites work: Mathematical problems for the next century
full work available at URL: s0012266108020109
OpenAlex ID: W2044724862
description: scientific article; zbMATH DE number 5659344