Alien limit cycles near a Hamiltonian 2-saddle cycle
type: Item
instance of: scholarly article
title: Alien limit cycles near a Hamiltonian 2-saddle cycle
zbMATH Open document ID: 1083.34025
author: M. Caubergh
author: Freddy Dumortier
author: Robert Roussarie
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2005-06-13T00:00:00Z
review text: The authors consider smooth families of planar vector fields perturbing the corresponding Hamiltonian system whose phase portrait contains a 2-saddle cycle \(\Gamma\) in the boundary of a continuum of periodic orbits. They claim that perturbations from a Hamiltonian 2-saddle cycle \(\Gamma\) can produce limit cycles which are not covered by the Abelian integrals, even when it is generic, and call these limit cycles as alien limit cycles. The authors assume that such phenomenon cannot appear in the case when \(\Gamma\) is a periodic orbit, a nondegenerate singularity, or a saddle loop and present in their small note a way to study this phenomenon in a particular unfolding of a Hamiltonian 2-saddle cycle keeping one connection unbroken at the bifurcation.
reviewed by: Valery A. Gaiko
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C08
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C07
zbMATH DE Number: 2175624
zbMATH Keywords: smooth planar vector field
zbMATH Keywords: limit cycle
zbMATH Keywords: Hamiltonian 2-saddle cycle
zbMATH Keywords: Abelian integral
MaRDI profile type: Publication
full work available at URL: j.crma.2005.03.009
OpenAlex ID: W2000206132
Wikidata QID: Q114952403
cites work: Abelian integrals and limit cycles
cites work: Q4399938
cites work: On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields
cites work: Bifurcation of planar vector fields and Hilbert's sixteenth problem
DOI: 10.1016/J.CRMA.2005.03.009
description: scientific article; zbMATH DE number 2175624