Synchronization of coupled active rotators by common noise

Abstract

We study the effect of common noise on coupled active rotators. While such a noise always facilitates synchrony, coupling may be attractive/synchronizing or repulsive/desynchronizing. We develop an analytical approach based on a transformation to approximate angle-action variables and averaging over fast rotations. For identical rotators, we describe a transition from full to partial synchrony at a critical value of repulsive coupling. For nonidentical rotators, the most nontrivial effect occurs at moderate repulsive coupling, where a juxtaposition of phase locking with frequency repulsion (anti-entrainment) is observed. We show that the frequency repulsion obeys a nontrivial power law.

Interplay of the mechanisms of synchronization by common noise and global coupling for a general class of limit-cycle oscillators

“…3 and 4, one can see that f and h for significantly nonlinear oscillators are very close to their harmonic approximations (red circles) and undistinguishable from the OA approximations (blue squares), even though B(ϕ) is far from a sinusoidal shape. Since the effect of frequency entrainment/anti-entrainment, as it can be seen from the analytical theory, are determined dominantly by f (θ) and h(θ), one should expect the results for the Van der Pol and Van der Pol-Duffing oscillators to be similar to the results previously derived for the systems admitting the OA approach [20,21,22]. Thus, the results derived with the Ott-Antonsen approach turn out to be even more valuable than one could initially expect, since they simultaneously allow one to describe the role of the imperfectness of synchrony accurately and are equivalent to the results in the cases, where this approach is not applicable.…”

“…The analytical result (22) as well as the asymptotic laws (23) and (26) can be observed with the results of numerical simulation for example systems in the next subsection.…”

“…Further in this subsection we derive the asymptotic laws for the dependence of θ on ω given by Eq. (22). For Eq.…”

“…To understand this shift, let us remind the results for the Kuramoto and Kuramoto-Sakaguchi ensembles [20,21], where it was possible to take the synchrony imperfectness into account rigorously. For the perfect synchrony, only the power law of form (23) was observed at ω → 0 (this law was analytically derived in Appendix B of [22]). The transition to a linear dependence for extremely small ω was observed only as a result of the synchrony imperfectness.…”

“…Although the works [20,21,22] yielded the basic understanding of the interplay between two generic mechanisms of synchronization and, in particular, revealed the phenomenon of the frequency repulsion accompanying synchronization, the theoretical study remains restricted to a quite specific class of OA systems. In these systems, the clustering dynamics is forbidden (the distribution of elements between clusters is frozen).…”