Non-Hermitian time evolution: From static to parametric instability

Eigenmode coalescence imparts remarkable properties to non-hermitian time evolution, culminating in a purely spectral degeneracy known as an exceptional point (EP). Here, we revisit evolution at the EP and classify two-level Hamiltonians terms of M\"obius group. We then leverage that classification study dynamical encircling, by applying it periodically-modulated (Floquet) Hamiltonians. This reveals Floquet systems exhibit rich physics whose complexity is not captured EP-encircling rule. For example, EPs can occur without encircling vice-versa. Instead, show elaborate interplay between modulation instabilities better understood through lens parametric resonance.