  We show that the dynamics of a driven quantum system weakly coupled to the
environment can exhibit two distinct regimes. While the relaxation basis is
usually determined by the system+drive Hamiltonian (system-governed dynamics),
we find that under certain conditions it is determined by specific features of
the environment, such as, the form of the coupling operator
(environment-governed dynamics). We provide an effective coupling parameter
describing the transition between the two regimes and discuss how to observe
the transition in a superconducting charge pump.
