Frequency Synchronization and Phase Ordering of Bounded Confidence Oscillators

Limited range of phase measurements motivates the analysis of coupled oscillators that interact if their geodesic distances are within a prescribed bound. Steady state behaviors of this system depend on the natural frequencies of the oscillators and on the underlying graph structure. A necessary and sufficient condition is provided for the graph to remain complete. If the graph is connected over time, it is proved that phase ordering is preserved among the oscillators according to their natural frequencies. The asymptotic convergence to frequency synchronization is proved if the graph is assumed to stay connected and all natural frequencies are the same. A comprehensive analysis of the three-oscillator case shows that phase ordering is not necessary for frequency synchronization, and that graph connectivity, together with an appropriate bound on the range of natural frequencies, ensures frequency synchronization.