  Recent progress in the study of dynamical phase transitions has been made
with a large-deviation approach to study trajectories of stochastic jumps using
a thermodynamic formalism. We study this method applied to an open quantum
system consisting of a superconducting single-electron transistor, near the
Josephson quasiparticle resonance, coupled to a resonator. We find that the
dynamical behavior shown in rare trajectories can be rich even when the mean
dynamical activity is small and thus the formalism gives insights into the form
of fluctuations. The structure of the dynamical phase diagram found from the
quantum-jump trajectories of the resonator is studied and we see that sharp
transitions in the dynamical activity may be related to the appearance and
disappearance of bistabilities in the state of the resonator as system
parameters are changed. We also demonstrate that for a fast resonator, the
trajectories of quasiparticles are similar to the resonator trajectories.
