Kuramoto model subject to subsystem resetting: How resetting a part of the system may synchronize the whole of it

Abstract

We introduce and investigate the effects of a new class of stochastic resetting protocol called subsystem resetting, whereby a subset of the system constituents in a many-body interacting system undergoes bare evolution interspersed with simultaneous resets at random times, while the remaining constituents evolve solely under the bare dynamics. We pursue our investigation within the ambit of the well-known Kuramoto model of coupled phase-only oscillators of distributed natural frequencies. Here, the reset protocol corresponds to a chosen set of oscillators being reset to a synchronized state at random times. We find that the mean $\omega_0$ of the natural frequencies plays a defining role in determining the long-time state of the system. For $\omega_0=0$, the system reaches a synchronized stationary state at long times, characterized by a time-independent non-zero value of the synchronization order parameter. Moreover, we find that resetting even an infinitesimal fraction of the total number of oscillators has the drastic effect of synchronizing the entire system, even when the bare evolution does not support synchrony. By contrast, for $\omega_0 \ne 0$, the dynamics allows at long times either a synchronized stationary state or an oscillatory synchronized state, with the latter characterized by an oscillatory behavior as a function of time of the order parameter, with a non-zero time-independent time average. Our results thus imply that the non-reset subsystem always gets synchronized at long times through the act of resetting of the reset subsystem. Our results, analytical using the Ott-Antonsen ansatz as well as those based on numerical simulations, are obtained for two representative oscillator frequency distributions, namely, a Lorentzian and a Gaussian. We discuss how subsystem resetting may be employed as an efficient mechanism to control attainment of global synchrony.