Solutions of \(\dot x(t)=-g(x(t-1))\) approach the Kaplan-Yorke orbits for odd sigmoid \(g\)
type: Item
instance of: scholarly article
title: Solutions of \(\dot x(t)=-g(x(t-1))\) approach the Kaplan-Yorke orbits for odd sigmoid \(g\)
zbMATH Open document ID: 0823.34068
author: Andreas Herz
published in: Journal of Differential Equations
publication date: 1995-05-30T00:00:00Z
review text: If \(g\) is an odd and differentiable function with positive derivative, then every bounded solution of (1) \(x'(t) = - g(x(t - 1))\) satisfies \(\lim_{t \to \infty} [x(t) + x(t - 2)] = 0\) and all periodic solutions of (1) are of Kaplan-Yorke type. Solutions of this type are oscillatory periodic solutions of period 4 and they satisfy \(x'(t) = - g(y(t))\), \(y'(t) = g(x(t))\) where \(y(t) = x(t - 1)\). Further the convergence of the trajectories of (1) to the orbit of a Kaplan-Yorke solution is discussed and the stability of Kaplan-Yorke solutions is investigated.
Mathematics Subject Classification ID: 34K99
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34K20
zbMATH DE Number: 759460
zbMATH Keywords: delay differential equation
zbMATH Keywords: oscillatory periodic solutions
zbMATH Keywords: convergence
zbMATH Keywords: Kaplan-Yorke solution
zbMATH Keywords: stability
MaRDI profile type: Publication
full work available at URL: jdeq.1995.1066
OpenAlex ID: W2006124040
DOI: 10.1006/JDEQ.1995.1066
description: scientific article; zbMATH DE number 759460