  We study dynamical phases of a driven Bose-Einstein condensate coupled to the
light field of a high-Q optical cavity. For high field seeking atoms at red
detuning the system is known to show a transition from a spatially homogeneous
steady-state to a self-organized regular lattice exhibiting super-radiant
scattering into the cavity. For blue atom pump detuning the particles are
repelled from the maxima of the light-induced optical potential suppressing
scattering. We show that this generates a new dynamical instability of the
self-ordered phase, leading to the appearance of self-ordered stable
limit-cycles characterized by large amplitude self-sustained oscillations of
both the condensate density and cavity field. The limit-cycles evolve into
chaotic behavior by period doubling. Large amplitude oscillations of the
condensate are accompanied by phase-slippage through soliton nucleation at a
rate which increases by orders of magnitude in the chaotic regime. Different
from a superfluid in a closed setup, this driven dissipative superfluid is not
destroyed by the proliferation of solitons since kinetic energy is removed
through cavity losses.
