  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.
