Synchronization clusters emerge as the result of a global coupling among classical phase oscillators

When large ensembles of phase oscillators interact globally, and when bimodal frequency distributions are chosen for the natural frequencies of the oscillators themselves, Bellerophon states are generically observed at intermediate values of the coupling strength. These are multi-clustered states emerging in symmetric pairs. Oscillators belonging to a given cluster are not locked in their instantaneous phases or frequencies, rather they display the same long-Time average frequency (a sort of effective global frequency). Moreover, Bellerophon states feature quantized traits, in that such average frequencies are all odd multiples (±(2n-1), n = 1, 2..) of a fundamental frequency Ω₁. We identify and investigate (analytically and numerically) several typical bifurcation paths to synchronization, including first-order and second-order-like. Linear stability analysis allows to successfully solve the critical transition point for synchronization. Our results highlight that the spontaneous setting of higher order forms of coherence could be achieved in classical Kuramoto model.