What happens when the paradigmatic Kuramoto model involving interacting oscillators of distributed natural frequencies and showing spontaneous collective synchronization in the stationary state is subject to random and repeated interruptions of its dynamics with a reset to the initial condition? While resetting to a synchronized state, it may happen between two successive resets that the system desynchronizes, which depends on the duration of the random time interval between the two resets. Here, we unveil how such a protocol of stochastic resetting dramatically modifies the phase diagram of the bare model, allowing, in particular, for the emergence of a synchronized phase even in parameter regimes for which the bare model does not support such a phase. Our results are based on an exact analysis invoking the celebrated Ott–Antonsen ansatz for the case of the Lorentzian distribution of natural frequencies and numerical results for Gaussian frequency distribution. Our work provides a simple protocol to induce global synchrony in the system through stochastic resetting.
We numerically study the celebrated Kuramoto model of identical oscillators arranged on the sites of a two-dimensional periodic square lattice and subject to nearest-neighbor interactions and dichotomous noise. In the nonequilibrium stationary state attained after a long time, the model exhibits a Berezinskii–Kosterlitz–Thouless (BKT)-like transition between a phase at a low noise amplitude characterized by quasi long-range order (critically ordered phase) and an algebraic decay of correlations and a phase at a high noise amplitude that is characterized by complete disorder and an exponential decay of correlations. The interplay between the noise amplitude and the noise-correlation time is investigated, and the complete, nonequilibrium stationary-state phase diagram of the model is obtained. We further study the dynamics of a single topological defect for various amplitudes and correlation time of the noise. Our analysis reveals that a finite correlation time promotes vortex excitations, thereby lowering the critical noise amplitude of the transition with an increase in correlation time. In the suitable limit, the resulting phase diagram allows one to estimate the critical temperature of the equilibrium BKT transition, which is consistent with that obtained from the study of the dynamics in the Gaussian white noise limit.
We consider biased random walks on random networks constituted by a random comb comprising a backbone with quenched-disordered random-length branches. The backbone and the branches run in the direction of the bias. For the bare model as also when the model is subject to stochastic resetting, whereby the walkers on the branches reset with a constant rate to the respective backbone sites, we obtain exact stationary-state static and dynamic properties for a given disorder realization of branch lengths sampled following an arbitrary distribution. We derive a criterion to observe in the stationary state a non-zero drift velocity along the backbone. For the bare model, we discuss the occurrence of a drift velocity that is non-monotonic as a function of the bias, becoming zero beyond a threshold bias because of walkers trapped at very long branches. Further, we show that resetting allows the system to escape trapping, resulting in a drift velocity that is finite at any bias.
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