  We study the effects of synchronization and desynchronization in ensembles of
phase oscillators with the global Kuramoto-Sakaguchi coupling under common
noise driving. Since the mechanisms of synchronization by coupling and by
common noise are essentially different, their interplay is of interest. In the
thermodynamic limit of large number of oscillators, employing the Ott-Antonsen
approach, we derive stochastic equations for the order parameters and consider
their dynamics for two cases: (i) identical oscillators and (ii) small natural
frequency mismatch. For identical oscillators, the stability of the perfect
synchrony state is studied; a strong enough common noise is revealed to prevail
over a moderate negative (repelling) coupling and to synchronize the ensemble.
An inequality between the states of maximal asynchrony (zero-value of the order
parameter) and perfect synchrony; the former can be only weakly stable, while
the latter can become adsorbing (the transition to the synchrony becomes
unidirectional). The dependence of the temporal dynamics of the transition on
the system parameters is investigated. For nonidentical oscillators the perfect
synchrony state becomes impossible and an absorbing state disappears; on its
place, only a weakly stable state of imperfect synchrony remains. A nontrivial
effect of the divergence of individual frequencies of oscillators with
different natural frequencies is revealed and studied for moderate repelling
coupling; meanwhile, the order parameter remains non-small for this case. In
Appendix we provide an introduction to the theories of Ott-Antonsen and
Watanabe-Strogatz.
