Resonance and marginal instability of switching systems

We analyze the so-called Marginal Instability of linear switching systems, both in continuous and discrete time. This is a phenomenon of unboundedness of trajectories when the Lyapunov exponent is zero. We disprove two recent conjectures of Chitour, Mason and Sigalotti (2012) stating that for generic systems, the resonance is sufficient for marginal instability and for polynomial growth of the trajectories. The concept of resonance originated with the same authors is modified. A characterization of marginal instability under some mild assumptions on the system is provided. These assumptions can be verified algorithmically and are believed to be generic. Finally, we analyze possible types of fastest asymptotic growth of trajectories. An example of a marginally unstable pair of matrices with non-polynomial growth is given.