Abelian integrals of quadratic Hamiltonian vector fields with an invariant straight line
type: Item
instance of: scholarly article
title: Abelian integrals of quadratic Hamiltonian vector fields with an invariant straight line
zbMATH Open document ID: 0856.58033
DOI: 10.5565/PUBLMAT_39295_11
author: Chengzhi Li
author: Jaume Llibre
author: Zhifen Zhang
published in: Publicacions Matemàtiques
publication date: 1997-03-03T00:00:00Z
full work available at URL: https://eudml.org/doc/41230
review text: The authors consider a special case of the weakened 16th Hilbert problem, that is to find the number \(N(n, m)\) of isolated zeros of the Abelian integrals \[ I(h)= \int_{\Gamma_h} Q(x, y) dx- P(x, y) dy, \] where \(\Gamma_h\) varies in the compact components of \(H^{- 1}(h)\), and \(H(x, y)\) is a real polynomial of degree \(n+ 1\), \(Q(x, y)\), \(P(x, y)\) are real polynomials of degree at most \(m\). The weakened 16th Hilbert problem is closely related to the problem of determining an upper bound of the number of limit cycles for the perturbed Hamiltonian system \[ \dot x= {\partial H\over \partial y}+ \varepsilon P(x, y),\quad \dot y= - {\partial H\over \partial x}+ \varepsilon Q(x, y). \] The authors prove that \(N(2, 2)= 1\) for the quadratic perturbation of quadratic Hamiltonian vector fields having an invariant straight line. They also give a classification of all phase portraits of such systems.
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 889855
zbMATH Keywords: quadratic Hamiltonian system
zbMATH Keywords: bifurcation
zbMATH Keywords: weakened 16th Hilbert problem
zbMATH Keywords: perturbation
reviewed by: Duo Wang
MaRDI profile type: Publication
OpenAlex ID: W2170306519
description: scientific article; zbMATH DE number 889855