  We explore the dissipative dynamics of nonlinearly driven oscillator systems
tuned to resonances where multiple excitations are generated. Such systems are
readily realised in circuit QED systems combining Josephson junctions with a
microwave cavity and a drive achieved either through flux or voltage bias. For
resonances involving 3 or more photons the system undergoes a sequence of two
closely spaced dynamical transitions (the first one discontinuous and the
second continuous) as the driving is increased leading to steady states that
form complex periodic structures in phase space. In the vicinity of the
transitions the system displays interesting bistable behaviour: we find that
coherent effects can lead to surprising oscillations in the weight of the
different dynamical states in the steady state of the system with increasing
drive. We show that the dynamics is well-described by a simple effective rate
model with transitions between states localised at different points in the
phase space crystal. The oscillations in the weights of the dynamical states is
reflected in corresponding oscillations in a time-scale that describes
transitions between the states.
