Instability results for slowly time varying linear dynamic systems on time scales
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)].
scientific article; zbMATH DE number 5124032