Instability results for slowly time varying linear dynamic systems on time scales
type: Item
instance of: scholarly article
title: Instability results for slowly time varying linear dynamic systems on time scales
zbMATH Open document ID: 1114.34039
author: Jeffrey J. DaCunha
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-02-12T00:00:00Z
review text: This paper provides instability criteria for linear dynamic equations on time scales. More detailed, linear systems of the form \[ x^\Delta=A(t)x \] are considered, where \(A:{\mathbb T}\to{\mathbb R}^{n\times n}\) is an \(rd\)-continuously differentiable function. Special cases of this class include ODEs and difference equations. In the time-invariant case \(A(t)\equiv A\) it is well known that stability properties depend on the spectrum of \(A\) (cf.~the reviewer, \textit{S. Siegmund} and \textit{F. Wirth} [Discrete Contin. Dyn. Syst. 9, No. 5, 1223--1241 (2003; Zbl 1054.34086)] for a general time scale approach). The present paper develops eigenvalue criteria under which a ``slowly time varying'' linear dynamic equation is unstable. ``Slowly varying'' means that the derivative \(A^\Delta\) has to have a small global bound. The corresponding tools include a quadratic Lyapunov function. As usual in stability theory for dynamic equations, the time scale is assumed to have bounded graininess \(\mu\); beyond that, however, also \(\mu^\Delta\) is supposed to exist as a bounded function. Remark: Related results for linear dynamic equations having an exponential dichotomy can be found in [the reviewer, J. Math. Anal. Appl. 289, No. 1, 317--335 (2004, Zbl 1046.34076)].
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 39A13
Mathematics Subject Classification ID: 39A14
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34A30
zbMATH DE Number: 5124032
zbMATH Keywords: stability
zbMATH Keywords: time scale
zbMATH Keywords: slowly varying
zbMATH Keywords: linear system
reviewed by: Christian Pötzsche
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.06.045
OpenAlex ID: W2007562422
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DOI: 10.1016/J.JMAA.2006.06.045
description: scientific article; zbMATH DE number 5124032