  We study the instability of a ring Bose-Einstein condensate under a periodic
modulation of inter-atomic interactions. The condensate exhibits temporal and
spatial patterns induced by the parametric resonance, which can be
characterized by Bogoliubov quasi-particle excitations in the Floquet basis. As
the ring geometry significantly limits the number of excitable Bogoliubov
modes, we are able to capture the non-linear dynamics of the condensate using a
three-mode model. We further demonstrate the robustness of the temporal and
spatial patterns against disorder, which are attributed to the mode-locking
mechanism under the ring geometry. Our results can be observed in cold atomic
systems and are also relevant to physical systems described by the non-linear
Schrodinger's equation.
