  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 (1997) 279]
allows to derive the synchronization profile 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 few links per
oscillator. Finally, an application to the Italian high-voltage power grid is
reported, which reveals the emergence of quasi-periodic oscillations in the
order parameter due to the simultaneous presence of many competing whirling
clusters.
