In a ring cavity filled with an atomic condensate, self-bunching of atoms due to the cavity pump mode produce an inversion that re-emits into the cavity probe mode with an exponential gain, forming atomic recoil lasing. An optomechanical ring cavity is formed when one of the reflective mirrors is mounted on a mechanical vibrating beam. In this paper, we extend studies on the stability of linear optomechanical cavities to such ring cavities with two counter-propagating cavity modes, especially when the forward propagating pump mode is taken to its weak coupling limit. We find that when the atomic recoil is in action, stable states of the mechanical mode of the mirror converge into branch cuts, where the gain produced by the recoiling strikes balance with the multiple decay sources, such as cavity leakage in the optomechanical system. This balance is obtained when the propagation delay in the dispersive atomic medium matches in a periodic pattern to the frequencies and linewidths of the cavity mode and the collective bosonic mode of the atoms. Weshow an input-output hysteresis cycle between the atomic mode and the cavity mode to verify the multi-valuation of the stable states after branching at the weak coupling limit.