Phase Synchronization and Polarization Ordering of Globally Coupled Oscillators

We introduce a prototype model for globally-coupled oscillators
in which each element is given an oscillation frequency and a preferential
oscillation direction (polarization), both randomly distributed. We found
two collective transitions: to phase synchronization and
to polarization ordering. Introducing a global-phase and a polarization
order parameters, we show that the transition to global-phase
synchrony is found when the coupling overcomes a critical value
and that polarization order enhancement can not take place before
global-phase synchrony. We develop a self-consistent theory to
determine both order parameters in good
agreement with numerical results.