Influence of parametric perturbations on Lyapunov exponents of discrete linear time-varying systems

Abstract

In this paper we investigate the problem of influence of parametric perturbations on the Lyapunov spectrum of the discrete linear time-varying system. The main result of the paper is that for any two sequences of positive real numbers and any rate of convergence there exist a discrete linear time-varying system and a perturbation tending to zero with the given rate of convergence such that the spectra of the perturbed and unperturbed systems coincide with the a priori given sequences. Moreover, we show that this phenomenon is possible even when the perturbations are different from zero, rarely, in a certain sense. Finally, as a conclusion from the main result, we obtain that the separation type and the index of exponential stability may vary arbitrarily under the influence of exponentially decreasing perturbations.