The effect of inversely unstable solutions on the attractor of the forced pendulum equation with friction
type: Item
instance of: scholarly article
title: The effect of inversely unstable solutions on the attractor of the forced pendulum equation with friction
zbMATH Open document ID: 1074.34055
published in: Journal of Differential Equations
publication date: 2005-06-01T00:00:00Z
review text: This paper deals with an equation of pendulum type having friction and periodic forcing. When the unknown is interpreted as an angle, the equation becomes a dissipative system on the cylinder. In this situation the attractor, understood as the set of bounded solutions, is compact and nonempty. It is known, that this attractor is homeomorphic to a circle (Jordan curve) when the friction is large. In this paper, the author obtains a complementary result: the existence of an inversely unstable periodic solution implies that the attractor is not a Jordan curve. The notion of inversely unstable periodic solution was introduced by Levinson and had an analytical character. Namely, this notion referred to a hyperbolic solution with negative Floquet multipliers. The new notion introduced in this paper is based on topological degree and extends Levinson's definition.
reviewed by: Rafael Ortega
Mathematics Subject Classification ID: 34D45
Mathematics Subject Classification ID: 37C70
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C15
zbMATH DE Number: 2173048
zbMATH Keywords: pendulum
zbMATH Keywords: attractor
zbMATH Keywords: periodic solution
author: Rogério Ferreira Martins
MaRDI profile type: Publication
full work available at URL: j.jde.2004.07.012
OpenAlex ID: W2020759332
cites work: The index and the asymptotic stability of fixed points in two dimensions: A counterexample
cites work: Q4840149
cites work: Q5509059
cites work: Propriétés des attracteurs de Birkhoff
cites work: Transformation theory of non-linear differential equations of the second order
cites work: Global behavior in the dynamical equation
cites work: Q4204461
cites work: The First Interval of Stability of a Periodic Equation of Duffing Type
cites work: Q4860451
cites work: Bounded homeomorphisms of the open annulus
DOI: 10.1016/J.JDE.2004.07.012
description: scientific article; zbMATH DE number 2173048