  The optical bistability (OB) in a two-mode optomechanical system with a
Bose-Einstein condensate (BEC) is studied. By investigating the behavior of
steady state solutions, we show that how OB develops in the system for a
certain range of cavity-pump detunings and pump amplitudes. We then investigate
the effects of the decay rate of the cavity photons and coupling strength
between the cavity and the BEC as well as the pump-atom detuning on the optical
behaviour of the system. We find that one can control the OB threshold and
width of the bistability curve via adjusting properly the decay rate, coupling
strength and the detuning. By applying Routh-Hurwitz criterion, we then derive
stability conditions for different branches of the OB curve. Moreover, by
introducing an effective potential for the system, a simple physical
interpretation is obtained.
