Stability properties of periodic solutions of a Duffing equation in the presence of lower and upper solutions.
type: Item
instance of: scholarly article
title: Stability properties of periodic solutions of a Duffing equation in the presence of lower and upper solutions.
zbMATH Open document ID: 1034.34060
DOI: 10.1016/S0096-3003(02)00062-0
author: Franic Ikechukwu Njoku
author: Pierpaolo Omari
published in: Applied Mathematics and Computation
publication date: 2003-01-28T00:00:00Z
review text: The paper is concerned with the perturbed Duffing equation \[ x''+cx'+g(t,x)=h(t). \tag{1} \] It is proved that a periodic solution to equation (1) is asymptotically stable if and only if it is bracketed by a lower solution \(\alpha\) and an upper solution \(\beta\) satisfying \(\alpha(t)>\beta(t)\) for every \(t\), provided that the derivative of \(g\) with respect to \(x\) is not too large. Furthermore, the asymptotic stability of the \(T\)-periodic solutions to equation (1) is characterized in terms of the order stability: under certain assumptions, a \(T\)-periodic solution is asymptotically stable if and only if it is isolated and order stable. As an application, the existence of stable and unstable periodic solutions for Duffing equations with singular and oscillating nonlinearities is discussed.
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34D10
zbMATH DE Number: 1861359
zbMATH Keywords: periodic solutions
zbMATH Keywords: Duffing equation
zbMATH Keywords: lower and upper solutions
zbMATH Keywords: asymptotic stability
zbMATH Keywords: topological degree
zbMATH Keywords: antimaximum principle
reviewed by: Svitlana P. Rogovchenko
MaRDI profile type: Publication
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full work available at URL: s0096-3003(02)00062-0
OpenAlex ID: W2171339743
description: scientific article; zbMATH DE number 1861359