A stability dichotomy for discrete-time linear switching systems in dimension two

TL;DR

This paper establishes a clear stability dichotomy for two-dimensional discrete-time linear switching systems, providing an algebraic criterion to determine whether such systems are stable or have trajectories diverging to infinity.

Contribution

It extends the stability dichotomy to complex two-dimensional systems and offers a practical algebraic criterion for stability assessment.

Findings

Systems are either Lyapunov stable or have diverging trajectories.

A checkable algebraic criterion distinguishes the two cases.

The dichotomy holds specifically in two complex dimensions.

Abstract

We prove that for every discrete-time linear switching system in two complex variables and with finitely many switching states, either the system is Lyapunov stable or there exists a trajectory which escapes to infinity with at least linear speed. We also give a checkable algebraic criterion to distinguish these two cases. This dichotomy was previously known to hold for systems in two real variables, but is known to be false in higher dimensions and for systems with infinitely many switching states.