  Classical optomechanical systems feature self-sustained oscillations, where
multiple periodic orbits at different amplitudes coexist. We study how this
multistability is realized in the quantum regime, where new dynamical patterns
appear because quantum trajectories can move between different classical
orbits. We explain the resulting quantum dynamics from the phase space point of
view, and provide a quantitative description in terms of autocorrelation
functions. In this way we can identify clear dynamical signatures of the
crossover from classical to quantum mechanics in experimentally accessible
quantities. Finally, we discuss a possible interpretation of our results in the
sense that quantum mechanics protects optomechanical systems against the
chaotic dynamics realized in the classical limit.
