Uncovering the microscopic mechanism of the devil's staircase in the Kuramoto model with inertia

We originally propose a phase diagram of the synchronization transitions of the second order Kuramoto oscillator system. Emergence of the secondary clusters and the devil's staircase is highlighted and focused. The staircase consists of one giant cluster, a pair of secondary frequency clusters, and higher order clusters. We uncover the microscopic mechanism underlying the formation of the devil's staircase; applying a modified Melnikov's method to the secondary sync clusters, we set up a couple of self-consistent relations to determine its step widths and step heights of the secondary sync clusters. Furthermore, higher order steps are derived from the n-to-m mode locking to the secondary cluster pulsing around the giant cluster at a nontrivial angular speed. As a consequence, each stair step is characterized by a collection of oscillators locked into the rational multiple of the angular speed of the secondary cluster in the long time average. This microscopic results may provide useful insights in stabilizing power grids.