  The stroboscopic evolution of a time-periodically driven isolated quantum
system can always be described by an effective time-independent Hamiltonian.
Whether this concept can be generalized to open Floquet systems, described by a
Markovian master equation with time-periodic Lindbladian generator, remains an
open question. By using a two level system as a model, we explicitly show the
existence of two well-defined parameter regions. In one region the stroboscopic
evolution can be described by a Markovian master equation with a
time-independent Floquet Lindbladian. In the other it cannot; but here the
one-cycle evolution operator can be reproduced with an effective non-Markovian
master equation that is homogeneous but non-local in time. Interestingly, we
find that the boundary between the phases depends on when the evolution is
stroboscopically monitored. This reveals the non-trivial role played by the
micromotion in the dynamics of open Floquet systems.
