Origin of frequency clusters and self-organized triplet locking in the Kuramoto model with inertia

We investigate the origin of frequency clusters – states in which multiple groups of oscillators with different mean frequencies coexist – in the globally coupled Kuramoto model with inertia and identical oscillators. Two frequency clusters are studied in the thermodynamic limit, three frequency clusters with a system of seven oscillators. Using bifurcation analysis, we demonstrate that in both cases the frequency clusters emerge through homoclinic bifurcations. In the case of three frequency clusters, this necessarily entails the formation of a triplet locked state, characterized by rational relations among the mean frequency differences. The individual clusters may lose phase-synchrony via either transcritical or period-doubling bifurcations. Finally, we establish that Hopf bifurcations cannot generate frequency clusters in phase oscillator systems. Instead, they can only arise through global bifurcations.