  We develop and study an atom-only description of the Dicke model with
time-periodic couplings between atoms and a dissipative cavity mode. The cavity
mode is eliminated giving rise to effective atom-atom interactions and
dissipation. We use this effective description to analyze the dynamics of the
atoms that undergo a transition to a dynamical superradiant phase with
macroscopic coherences in the atomic medium and the light field. Using Floquet
theory in combination with the atom-only description we provide a precise
determination of the phase boundaries and of the dynamical response of the
atoms. From this we can predict the existence of dissipative time crystals that
show a subharmonic response with respect to the driving frequency. We show that
the atom-only theory can describe the relaxation into such a dissipative time
crystal and that the damping rate can be understood in terms of a cooling
mechanism.
