  Rigorous conditions for cluster synchronization of Kuramoto oscillators are
presented. The method of averaging plays an important role in stability
analysis, but the standard Lyapunov's second method is not applicable due to
the lack of uniform continuity. This paper contributes to overcoming this
difficulty with the help of nonmonotonic Lyapunov functions. Our extensions of
averaging in stability theory are key to derive the two interrelated cluster
synchronization conditions: (i) the coupling strengths between clusters are
sufficiently weak and/or (ii) the natural frequencies are largely different
between clusters. Cluster phase cohesiveness in the absence of network
partitions ensuring the existence of invariant manifolds is also investigated.
Moreover, we apply our theoretical findings to brain networks and exhibit
certain relations among network parameters and functional connectivity.
