  The Kuramoto model of coupled phase oscillators with inertia on Erdos-Renyi
graphs is analyzed in this work. For a system with intrinsic frequencies
sampled from a bimodal distribution we identify a variety of two cluster
patterns and study their stability. To this end, we decompose the description
of the cluster dynamics into two systems: one governing the (macro) dynamics of
the centers of mass of the two clusters and the second governing the (micro)
dynamics of individual oscillators inside each cluster. The former is a
low-dimensional ODE whereas the latter is a system of two coupled Vlasov PDEs.
Stability of the cluster dynamics depends on the stability of the
low-dimensional group motion and on coherence of the oscillators in each group.
We show that the loss of coherence in one of the clusters leads to the loss of
stability of a two-cluster state and to formation of chimera states. The
analysis of this paper can be generalized to cover states with more than two
clusters and to coupled systems on W-random graphs. Our results apply to a
model of a power grid with fluctuating sources.
