[Authors] Kryzhevich, S. G.; Nazarov, A. I. [Title] Stability by linear approximation for time scale dynamical systems [AMS Subj-class] 34D05 Asymptotic properties 34D20 Stability [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 the timescale and eigenvalues of a constant matrix we formulate conditions, sufficient for stability by linear approximation. We demonstrate that, like in classical cases, those conditions are close to necessary ones. For non-constant matrices and/or 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 theorem on conditional instability are proved. [Keywords] time-scale system, linearization, Lyapunov function, stability [Comments] LaTeX, English, 27 pp. [Contact e-mail] kryzhevicz@gmail.com