  We experimentally and theoretically study a driven hybrid circuit quantum
electrodynamics (cQED) system beyond the dispersive coupling regime. Treating
the cavity as part of the driven system, we develop a theory applicable to such
strongly coupled and to multi-qubit systems. The fringes measured for a single
driven double quantum dot (DQD)-cavity setting and the enlarged splittings of
the hybrid Floquet states in the presence of a second DQD are well reproduced
with our model. This opens a path to study Floquet states of multi-qubit
systems with arbitrarily strong coupling and reveals a new perspective for
understanding strongly driven hybrid systems.
