Geodesic
Synchronization in Kuramoto-Sakaguchi ensembles with competing influence of common noise and global coupling
Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 25 (2017) no. 6, pp. 5-37.

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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 is found; 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.
Keywords: Synchronization, stochastic processes, Kuramoto-Sakaguchi ensemble, Ott-Antonsen ansatz. 
@article{IVP_2017_25_6_a1,
     author = {D. S. Goldobin and A. S. Pikovsky and A. V. Dolmatova and M. G. Rozenbljum},
     title = {Synchronization in {Kuramoto-Sakaguchi} ensembles with competing influence of common noise and global coupling},
     journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
     pages = {5--37},
     publisher = {mathdoc},
     volume = {25},
     number = {6},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVP_2017_25_6_a1/}
}
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D. S. Goldobin; A. S. Pikovsky; A. V. Dolmatova; M. G. Rozenbljum. Synchronization in Kuramoto-Sakaguchi ensembles with competing influence of common noise and global coupling. Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 25 (2017) no. 6, pp. 5-37. http://geodesic.mathdoc.fr/item/IVP_2017_25_6_a1/