Bifurcations of limit cycles from cubic Hamiltonian systems with a center and a homoclinic saddle-loop
type: Item
instance of: scholarly article
title: Bifurcations of limit cycles from cubic Hamiltonian systems with a center and a homoclinic saddle-loop
zbMATH Open document ID: 0962.34026
DOI: 10.5565/PUBLMAT_44100_08
author: Yulin Zhao
author: Zhifen Zhang
published in: Publicacions Matemàtiques
publication date: 2001-06-14T00:00:00Z
full work available at URL: https://eudml.org/doc/41390
review text: This paper is concerned with the maximum number of limit cycles of the system \[ \dot x= y,\quad\dot y= kx- (k+1) x^2+ x^3+ \varepsilon(\alpha+ \beta x+\gamma x^2) y, \] when \(k> {11+ \sqrt{33}\over 4}\), \(0< |\varepsilon|\ll 1\), \(|\alpha|+|\beta|+ |\gamma|> 0\). Such a problem is related to the weak form of Hilbert's 16th problem, as proposed by Arnold. The system is considered as a perturbation of the Hamiltonian system obtained for \(\varepsilon= 0\). The authors estimate the number of zeroes of a suitable Abelian integral and prove that for small values of \(\varepsilon\) there exist no more than 2 limit cycles. This extends previous results about limit cycles of perturbations of cubic Hamiltonian systems.
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 1505972
zbMATH Keywords: limit cycles
zbMATH Keywords: Hilbert's 16th problem
zbMATH Keywords: cubic Hamiltonian systems
reviewed by: Marco Sabatini
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1505972