On the relaxation dynamics of the Kuramoto oscillators with small inertia

For the Kuramoto oscillators with small inertia, we present several quantitative estimates on the relaxation dynamics and formational structure of a phase-locked state (PLS) for some classes of initial configurations. In a super-critical regime where the coupling strength is strictly larger than the diameter of natural frequencies, we present quantitative relaxation dynamics on the collision numbers and the structure of PLS. In a critical coupling regime where the coupling strength is exactly the diameter of natural frequencies, we provide a sufficient condition for an asymptotically PLS solution. In particular, we show the existence of slow relaxation to a PLS, when there are exactly two natural frequencies. This generalizes the earlier results of Choi et al. ["Asymptotic formation and orbital stability of phase locked states for the Kuramoto model," Physica D 241, 735-754 (2012); "Complete synchronization of Kuramoto oscillators with finite inertia," Physica D 240, 32-44 (2011)] (C) 2013 AIP Publishing LLC.