Transient amplification in Floquet media: the Mathieu oscillator example

The Mathieu equation occurs naturally in the description of non linear vibrations or by considering the propagation of a wave in an infinite medium with time-periodic refractive index. It is known to lead to parametric instability since it supports unstable solutions in some regions of the parameter space. However, even in the stable region the matrix that propagates the initial conditions forward in time is non-normal and therefore it can result in transient amplification. By optimizing over initial conditions as well as initial time we show that significant transient amplifications can be obtained, going beyond the one simply stemming from adiabatic invariance. Moreover, we explore the monodromy matrix in more depth, by studying its $\epsilon$-pseudospectra and Petermann factors, demonstrating that is the degree of non-normality of this matrix that determines the global amplifying features.