  We study the effect of a strong electron-electron (e-e) interaction in a
system of two concentric one-dimensional rings with incommensurate areas A_1
and A_2, coupled by a tunnel amplitude. For noninteracting particles the
magnetic moment (persistent current) m of the many-body ground state and first
excited states is an irregular function of the external magnetic field. In
contrast, we show that when strong e-e interactions are present the magnetic
field dependence of m becomes periodic. In such a strongly correlated system
disorder can only be caused by inter-ring charge fluctuations, controllable by
a gate voltage. The oscillation period of m is proportional to 1/(A_1 + A_2) if
fluctuations are suppressed. Coherent inter-ring tunneling doubles the period
when charge fluctuations are allowed.
