Hysteretic transitions in the Kuramoto model with inertia

Abstract

We report finite-size numerical investigations and mean-field analysis of a Kuramoto model with inertia for fully coupled and diluted systems. In particular, we examine, for a gaussian distribution of the frequencies, the transition from incoherence to coherence for increasingly large system size and inertia. For sufficiently large inertia the transition is hysteretic, and within the hysteretic region clusters of locked oscillators of various sizes and different levels of synchronization coexist. A modification of the mean-field theory developed by Tanaka, Lichtenberg, and Oishi [Physica D 100, 279 (1997)] allows us to derive the synchronization curve associated to each of these clusters. We have also investigated numerically the limits of existence of the coherent and of the incoherent solutions. The minimal coupling required to observe the coherent state is largely independent of the system size, and it saturates to a constant value already for moderately large inertia values. The incoherent state is observable up to a critical coupling whose value saturates for large inertia and for finite system sizes, while in the thermodinamic limit this critical value diverges proportionally to the mass. By increasing the inertia the transition becomes more complex, and the synchronization occurs via the emergence of clusters of whirling oscillators. The presence of these groups of coherently drifting oscillators induces oscillations in the order parameter. We have shown that the transition remains hysteretic even for randomly diluted networks up to a level of connectivity corresponding to a few links per oscillator. Finally, an application to the Italian high-voltage power grid is reported, which reveals the emergence of quasiperiodic oscillations in the order parameter due to the simultaneous presence of many competing whirling clusters.

The Kuramoto model in complex networks

“…m [301]. During this process, a secondary synchronization of drifting oscillators is observed for larger m. This phenomenon was confirmed in [307], where the synchronized motions were validated by comparing the evolution of the instantaneous frequency ν i (t) =θ i of the secondary synchronized oscillators in random networks and also in the Italian high-voltage power grid.…”

Chimera states in coupled Kuramoto oscillators with inertia

“…In particular, the introduction of inertia allows the oscillators to synchronize via the adaptation of their own frequencies, in analogy with the mechanism observed in the firefly Pteroptix malaccae [3]. The modification of the classical Kuramoto model with the addition of an inertial term results in first order synchronization transitions and complex hysteretic phenomena [4][5][6][7][8][9]. Furthermore, networks of rotators have recently found applications in different technological contexts, including disordered arrays of Josephson junctions [10] and electrical power grids [11][12][13][14] and they could also be relevant for micro-electromechanical systems and optomechanical crystals, where chimeras and other partially disordered states likely play an important role with far reaching ramifications.…”

Kuramoto Phase Model with Inertia: Bifurcations Leading to the Loss of Synchrony and to the Emergence of Chaos

“…When the frequency difference ∆ decreases from large values corresponding to the rotation mode at the bifurcation of homoclinic orbit of the saddle |∆| = aγ h ((βa) −1/2 ) the reverse transition to the complete synchronization due to Statement 2 occurs only from the asynchronous mode of the peripheral oscillators. Note that this hysteretic behaviour being similar to the transitions in the Josephson junction model [25] was discussed in the recent paper [26].…”