Contracting return maps for monotone delayed feedback.
type: Item
instance of: scholarly article
title: Contracting return maps for monotone delayed feedback.
zbMATH Open document ID: 1034.34086
author: Hans-Otto Walther
published in: Discrete and Continuous Dynamical Systems
publication date: 2003-03-16T00:00:00Z
review text: The existence and stability of slowly oscillating periodic solutions to the delay differential equation \[ \dot x(t)=-\mu x(t)+f(x(t-1)) \] is derived for nonlinearities \(f\) which satisfy the standard negative feedback condition, \(xf(x)<0, x\neq0\), and are close to the step function \(f(x)=-\text{sign\,}(x)=-1\) for \(x>0\), \(f(x)=\text{sign\,} (x)=+1\) for \(x<0\), outside a small neighborhood \((-\beta,\beta)\) of \(0\). A crucial assumption is that \(f(x)\) is convex on an interval \((0,\varepsilon)\) for some \(0<\varepsilon<\beta\) [respectively, \(f(x)\) is concave in \((-\varepsilon,0)\)]. The proof involves the construction of a return map on a subset of initial functions \(C([-1,0], {\mathbb R})\), which is Lipschitz continuous and a contraction. Applications include \(f(x)=\text{arctan}(\gamma x)\) and \(f(x)=\text{tanh}(\gamma x)\) among others.
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 37C25
Mathematics Subject Classification ID: 34K20
zbMATH DE Number: 1882975
zbMATH Keywords: Scalar differential delay equations
zbMATH Keywords: negative feedback
zbMATH Keywords: slowly oscillating periodic solutions
zbMATH Keywords: existence and stability
reviewed by: Anatoli F. Ivanov
MaRDI profile type: Publication
full work available at URL: dcds.2001.7.259
OpenAlex ID: W2050884371
DOI: 10.3934/DCDS.2001.7.259
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description: scientific article; zbMATH DE number 1882975