Inertial Kuramoto model. The inertial Kuramoto model is a well-known synchronization model for \(N\) oscillators with phases \(\theta\), each of which has its own natural frequency \(\nu_i\). When considering an all-to-all network connection, this model is described by the following differential equations: for each \(i=1,\ldots,N\), \[ \label{eq:example2} m\ddot{\theta}_i(t) + \gamma\dot{\theta}_i(t) = \nu_i + \frac{\kappa}{N}\sum_{j=1}^N \sin(\theta_j(t) - \theta_i(t)),\quad t>0, \] where \(m\) is the inertia and \(\gamma\) is the damping coefficient. This model has been applied to describe synchronization phenomena in biological systems (e.g., fireflies) and to study transient stability in power systems.

From numerical simulations, one can easily observe asymptotic phase-locking, meaning that \(\exists\lim_{t\to\infty} (\theta_i(t)-\theta_j(t))\) for each \((i,j).\) However, proving this is challenging, especially when the initial phases are randomly distributed on the circle. We provide mathematical analysis for the long time behavior of the system \(\eqref{eq:example2}\) with generic initial configurations (Work I and II).