On the relaxation dynamics of the Kuramoto oscillators with small inertia

Abstract

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)10.1016/j.physd.2011.11.011; Choi et al. “Complete synchronization of Kuramoto oscillators with finite inertia,” Physica D 240, 32–44 (2011)]10.1016/j.physd.2010.08.004