Lyapunov spectra behavior for linear discrete-time systems under small perturbations

TL;DR

This paper studies how small perturbations affect the Lyapunov spectrum of linear discrete-time systems, providing verifiable conditions for stability and openness of the spectrum.

Contribution

It introduces the concepts of broken away solutions and splitted systems, offering new necessary and sufficient conditions related to the system's stability and spectrum openness.

Findings

Necessary condition for system stability under perturbations

Sufficient condition for the openness of the Lyapunov spectrum

Examples demonstrating the application of the theoretical results

Abstract

We investigate the behavior of the Lyapunov spectrum of a linear discrete-time system under the action of small perturbations in order to obtain some verifiable conditions for stability and openness of the Lyapunov spectrum. To this end we introduce the concepts of broken away solutions and splitted systems. The main results obtained are a necessary condition for stability and a sufficient condition for the openness of the Lyapunov spectrum, which is given in terms of the system itself. Finally, examples of using the obtained results are presented.