Stable periodic motion of a delayed spring
type: Item
instance of: scholarly article
title: Stable periodic motion of a delayed spring
zbMATH Open document ID: 1039.34062
DOI: 10.12775/TMNA.2003.001
author: Hans-Otto Walther
published in: Topological Methods in Nonlinear Analysis
publication date: 2004-03-08T00:00:00Z
review text: The paper considers the system \(\dot x(t)=v(t)\), \(\dot v(t)=-\mu v(t)+f(x(t-1))\), where \(\mu>0\) and \(f\) is an odd bounded Lipschitz continuous real function satisfying \(\xi f(\xi)<0\) if \(| \xi| \) is large. The system stands for a damped spring with a delayed position-dependent restoring force. The author proves that there exists a stable periodic orbit with strong attraction properties for sufficiently large \(\mu\) and under certain conditions on \(f\). Moreover, if \(f\) is continuously differentiable, the periodic orbit is stable and hyperbolic. The proof extends a method introduced by the author in [Discrete Contin. Dyn. Syst. 7, 259--274 (2001; Zbl 1034.34086)].
reviewed by: Jan Sieber
Mathematics Subject Classification ID: 34K13
zbMATH DE Number: 2052548
zbMATH Keywords: autonomous delay differential equation
zbMATH Keywords: stable periodic orbit
zbMATH Keywords: asymptotic phase
zbMATH Keywords: damped spring
zbMATH Keywords: delayed restoring force
MaRDI profile type: Publication
full work available at URL: tmna.2003.001
OpenAlex ID: W2547604738
description: scientific article; zbMATH DE number 2052548