Synchronous harmony in an ensemble of Hamiltonian mean-field oscillators and inertial Kuramoto oscillators.

We study a dynamic interplay between Hamiltonian mean-field oscillators and inertial Kuramoto oscillators. We present several sufficient frameworks leading to asymptotic complete synchronization for the mixed ensemble. For a two-oscillator system with the same natural frequencies, we prove that the mixed ensemble exhibits asymptotic complete synchronization for any initial data, whereas we also show that the two-oscillator system tends to asymptotic complete synchronization under an assumption on the uniform boundedness on the phase diameter. For the many-body system, we show that asymptotic complete frequency synchronization occurs for Kuramoto oscillators with inertia if the oscillators have the same natural frequencies. Moreover, we show that overall phase concentration can be controlled by increasing the coupling strengths. We also provide several numerical experiments and compare them with analytical results.