Instability of rapidly-oscillating periodic solutions for discontinuous differential delay equations
type: Item
instance of: scholarly article
title: Instability of rapidly-oscillating periodic solutions for discontinuous differential delay equations
zbMATH Open document ID: 1021.34055
author: Marianne Akian
author: Sophie Bismuth
published in: Differential and Integral Equations
publication date: 2003-10-09T00:00:00Z
review text: Consider the scalar delay differential equation \[ x'(t)=-h(x(t-1))+f(x(t)), \tag{1} \] with a continuous initial condition \(x_0:[-1,0]\to \mathbb{R}\) having finitely many zeros, the function \(h\) is odd, takes the value \(a\) for \(0<x<c\), and \(b\) for \(x\geq c\), where \(a,b,c\) are positive constants, \(a>b\). The function \(f\) is odd, Lebesgue integrable, with essential supremum in \(\mathbb{R}\) less than \(b\). A solution \(x:[-1,\infty)\to \mathbb{R}\) to (1) is said a rapidly oscillating symmetric 2-phase periodic solution (ROS2PP solution) if \(x\) is \(T\)-periodic, \(x(t+T/2)=-x(t)\) for all \(t\), there exists \(t_1\geq -1\) such that \(x\) changes signs exactly once between \(t_1\) and \(t_1+T\), and \(x\) has two different zeros in some interval of length less than one. In the main result of the paper, the authors characterize the ROS2PP solutions to (1) with \(\|x\|_{\infty}>c\), and they show that all are unstable.
reviewed by: Eduardo Liz
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 34K20
zbMATH DE Number: 1836163
zbMATH Keywords: scalar delay differential equations
zbMATH Keywords: instability
zbMATH Keywords: rapidly-oscillating periodic solutions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1836163