  We study the emergence of dynamical phases of a Bose-Einstein condensate that
is optomechanically coupled to a dissipative cavity mode and transversally
driven by a laser. We focus on the regime close to the optomechanical
resonance, where the atoms' refractive index shifts the cavity into resonance,
assuming fast cavity relaxation. We derive an effective model for the atomic
motion, where the cavity degrees of freedom are eliminated using perturbation
theory in the atom-cavity coupling and benchmark its predictions using
numerical simulations based on the full model. Away from the optomechanical
resonance, perturbation theory in the lowest order (adiabatic elimination)
reliably describes the dynamics and predicts chaotic phases with unstable
oscillations. Interestingly, the dynamics close to the optomechanical resonance
are qualitatively captured only by including the corrections to next order
(non-adiabatic corrections). In this regime we find limit cycle phases that
describe stable oscillations of the density with a well defined frequency. We
further show that such limit cycle solutions are metastable configurations of
the adiabatic model. Our work sheds light on the mechanisms that are required
to observe dynamical phases and predict their existence in atom-cavity systems
where a substantial timescale separation is present.
