Contracting singular cycles
type: Item
instance of: scholarly article
title: Contracting singular cycles
zbMATH Open document ID: 0942.37039
DOI: 10.1016/S0294-1449(98)80004-8
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1998-10-27T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78451
review text: A singular cycle of a vector-field is a finite set of hyperbolic singular and periodic orbits linked in a cyclic way by orbits in the intersections of the stable and unstable manifolds of the singularities and periodic orbits. Simple singular cycles are considered in the paper, i.e. singular cycles for vector-fields in \(\mathbb R^3\) with only one singular point and satisfying some additional properties. Such a cycle is called contracting provided the expanding eigenvalue of the singularity is smaller then the weakest contracting one. In the paper [\textit{M. J. Pacificio} and \textit{A. Rovella}, Ann. Sci. Ec. Norm. Super. (4) 26, 691-700 (1993; Zbl 0802.58036)], it was proved a result on the measure of the bifurcating set for generic one-parameter families through a simple contracting cycle with only one periodic orbit. The paper under review improves that result; in particular it treats the case of several periodic orbits.
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 58K45
Mathematics Subject Classification ID: 37D05
Mathematics Subject Classification ID: 37C29
Mathematics Subject Classification ID: 37C20
zbMATH DE Number: 1199912
zbMATH Keywords: simple singular cycle
zbMATH Keywords: periodic orbit
zbMATH Keywords: singular orbit
author: B. San Martín
reviewed by: Roman Srzednicki
MaRDI profile type: Publication
cites work: The explosion of singular cycles
cites work: Unfolding contracting singular cycles
cites work: The dynamics of perturbations of the contracting Lorenz attractor
cites work: Stable Orbits and Bifurcation of Maps of the Interval
cites work: Erratum. Hyperbolicity, sinks and measure in one dimensional dynamics
description: scientific article; zbMATH DE number 1199912