Marginally unstable discrete-time linear switched systems with highly irregular trajectory growth

We investigate the uniform stability properties of discrete-time linear switched systems subject to arbitrary switching, focusing on the "marginally unstable " regime in which the system is not Lyapunov stable but in which trajectories cannot escape to infinity at exponential speed. For a discrete linear system of this type without switching the fastest-growing trajectory must grow as an exact polynomial function of time, and a significant body of prior research has focused on investigating how far this intuitive picture can be extended from systems without switching to cases where switching is present. In this note we give an example of a family of discrete linear switched systems in three dimensions, with two switching states, for which this intuition fails badly: for a generic member of this family the maximal rate of uniform growth of escaping trajectories can be made arbitrarily slow along one subsequence of times and yet also faster than any prescribed slower-than-linear function along a complementary subsequence of times. Using this construction we give new counterexamples to a conjecture of Chitour, Mason and Sigalotti and obtain a negative answer to a related question of Jungers, Protasov and Blondel. Our examples have the additional feature that marginal stability and marginal instability are densely intermingled in the same parameter space. (c) 2022 The Author(s). Published by Elsevier B.V.