Stability by linear approximation for time scale dynamical systems

TL;DR

This paper develops stability criteria for linear and near-linear systems on general time scales, extending classical stability results from differential equations to more general dynamic systems.

Contribution

It formulates sufficient stability conditions based on eigenvalues and time scale properties, and adapts Lyapunov exponent techniques and Chetaev's theorem to time scale systems.

Findings

Derived stability conditions close to necessary for linear systems on time scales.

Extended Lyapunov exponent techniques to non-periodic time scales.

Proved time scale versions of Chetaev's theorem on conditional instability.

Abstract

We study systems on time scales that are generalizations of classical differential or difference equations. In this paper we consider linear systems and their small nonlinear perturbations. In terms of time scales and of eigenvalues of matrices we formulate conditions, sufficient for stability by linear approximation. We demonstrate that those conditions are close to necessary ones. For non-periodic time scales we use techniques of central upper Lyapunov exponents (a common tool of the theory of linear ODEs) to study stability of solutions. Also, time scale versions of the famous Chetaev's theorem on conditional instability are proved.