We investigate the nonlinear dynamics of two coupled annular Bose-Einstein condensates (BECs). For certain values of the coupling strength the nonrotating ground state is unstable with respect to fluctuations in the higher angular momentum modes. The two branched Bogoliubov spectrum exhibits distinct regions of instability enabling one to selectively occupy certain angular momentum modes. For sufficiently long evolution times angular momentum Josephson oscillations spontaneously appear, breaking the initial chiral symmetry of the BECs.
