  Recent experimental advances in the field of cold-atom cavity QED provide a
powerful tool for exploring non-equilibrium correlated quantum phenomena beyond
conventional condensed-matter scenarios. We present the dynamical phase diagram
of a driven Bose-Einstein condensate coupled with the light field of a cavity,
with a transverse driving field red-detuned from atomic resonance. We identify
regions in parameter space showing dynamical instabilities in the form of limit
cycles, which evolve into chaotic behavior in the strong driving limit. Such
limit cycles originate from the interplay between cavity dissipation and
atom-induced resonance frequency shift, which modifies the phase of cavity mode
and gives excessive negative feedback on the atomic density modulation, leading
to instabilities of the superradiant scattering. We find interesting merging of
the limit cycles related by a $Z_2$ symmetry, and identify a new type of limit
cycle formed by purely atomic excitations. The effects of quantum fluctuations
and atomic interactions are also investigated.
