Cardinality of Collisions in Asymptotic Phase-Locking for the Kuramoto Model with Inertia

We study the cardinality of collisions between Kuramoto oscillators in the (asymptotic) phase-locking process in the presence of inertia. In the absence of inertia, it has been known that the finiteness of collisions between oscillators is equivalent to the emergence of phase-locking. Thus, a natural question is whether this finiteness result is still valid for the Kuramoto model with inertia or not. In a small inertia regime, we show that the finiteness of collisions is also equivalent to phase-locking like the Kuramoto model. In contrast, in a large inertia regime, we show that a homogeneous Kuramoto ensemble with the same natural frequency can exhibit phase-locking, while there are a countable number of collisions between Kuramoto oscillators. This is the contrasted effect of a large inertia in the phase-locking process.