  We predict the limit cycle solution for a ring-shape bosonic condensate of
exciton-polaritons confined in an optically induced rotating trap. The limit
cycle manifests itself with polarization oscillations on a characteristic
timescale of tens of picoseconds. The effect arises due to the interplay
between orbital motion and the polarization degree of freedom. It is specific
to spinor bosonic condensates and would be absent in a scalar case, where a
bi-stability of stationary solutions would be observed instead. This work
offers a tool of initialisation and control of qubits based on superpositions
of polariton condensates characterised by different topologic charges.
