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We report finite size numerical investigations and mean field analysis of aKuramoto model with inertia for fully coupled and diluted systems. Inparticular, we examine for a Gaussian distribution of the frequencies thetransition from incoherence to coherence for increasingly large system size andinertia. For sufficiently large inertia the transition is hysteretic and withinthe hysteretic region clusters of locked oscillators of various sizes anddifferent levels of synchronization coexist. A modification of the mean fieldtheory developed by Tanaka, Lichtenberg, and Oishi [Physica D, 100 (1997) 279]allows to derive the synchronization profile associated to each of theseclusters. We have also investigated numerically the limits of existence of thecoherent and of the incoherent solutions. The minimal coupling required toobserve the coherent state is largely independent of the system size and itsaturates to a constant value already for moderately large inertia values. Theincoherent state is observable up to a critical coupling whose value saturatesfor large inertia and for finite system sizes, while in the thermodinamic limitthis critical value diverges proportionally to the mass. By increasing theinertia the transition becomes more complex, and the synchronization occurs viathe emergence of clusters of whirling oscillators. The presence of these groupsof coherently drifting oscillators induces oscillations in the order parameter.We have shown that the transition remains hysteretic even for randomly dilutednetworks up to a level of connectivity corresponding to few links peroscillator. Finally, an application to the Italian high-voltage power grid isreported, which reveals the emergence of quasi-periodic oscillations in theorder parameter due to the simultaneous presence of many competing whirlingclusters.

Hysteretic transitions in the Kuramoto model with inertia