  A periodically kicked ring of a Bose-Einstein condensate is considered as a
nonlinear generalization of the quantum kicked rotor, where the nonlinearity
stems from the mean field interactions between the condensed atoms. For weak
interactions, periodic motion (anti-resonance) becomes quasiperiodic (quantum
beating) but remains stable. There exists a critical strength of interactions
beyond which quasiperiodic motion becomes chaotic, resulting in an instability
of the condensate manifested by exponential growth in the number of
noncondensed atoms. In the stable regime, the system remains predominantly in
the two lowest energy states and may be mapped onto a spin model, from which we
obtain an analytic expression for the beat frequency and discuss the route to
instability. We numerically explore parameter regime for the occurrence of
instability and reveal the characteristic density profile for both condensed
and noncondensed atoms. The Arnold diffusion to higher energy levels is found
to be responsible for the transition to instability. Similar behavior is
observed for dynamically localized states (essentially quasiperiodic motions),
where stability remains for weak interactions but is destroyed by strong
interactions.
