Analytical approach to synchronous states of globally coupled noisy rotators

TL;DR

This paper develops analytical methods to describe the transition to synchronization in populations of globally coupled noisy rotators, including cases with inertia and parameter dispersion, validated by numerical simulations.

Contribution

The paper introduces new analytical approaches for describing synchronous states and transition criteria in noisy rotator populations, extending to cases with inertia and parameter heterogeneity.

Findings

Analytical solutions for synchronous states with constant order parameter.

Criteria for distinguishing supercritical and subcritical transitions.

Validation of analytical results through numerical simulations and Fokker-Planck solutions.

Abstract

We study populations of globally coupled noisy rotators (oscillators with inertia) allowing a nonequilibrium transition from a desynchronized state to a synchronous one (with the non-vanishing order parameter). The newly developed analytical approaches resulted in solutions describing the synchronous state with constant order parameter for weakly inertial rotators, including the case of zero inertia, when the model is reduced to the Kuramoto model of coupled noise oscillators. These approaches provide also analytical criteria distinguishing supercritical and subcritical transitions to the desynchronized state. All the obtained analytical results are confirmed by the numerical ones, both by direct simulations of the large ensembles and by the solution of the associated Fokker-Planck equation. We also propose generalizations of the developed approaches for setups where different rotators parameters (natural frequencies, masses, noise intensities, strengths and phase shifts in coupling) are dispersed.