  We analyze in detail an open cavity array using mean-field description, where
each cavity field is coupled to a number of three-level atoms. Such system is
highly tunable and can be described by a Jaynes-Cummings like Hamiltonian with
additional non-linear terms. In the single cavity case we provide simple
analytic solutions and show, that the system features a bistable region. The
extra non-linear term gives rise to a rich dynamical behaviour including
occurrence of limit cycles through Hopf bifurcations. In the limit of large
non-linearity, the system exhibits an Ising like phase transition as the
coupling between light and matter is varied. We then discuss how these results
extend to the two-dimensional case.
