Cluster-mediated synchronization dynamics in globally coupled oscillators with inertia

Globally coupled oscillator systems with inertia exhibit complex synchronization patterns, among which the emergence of a couple of secondary synchronized clusters (SCs) in addition to the primary cluster (PC) is especially distinctive. Although previous studies have predominantly focused on the collective properties of the PC, the dynamics of individual clusters and their inter-cluster interactions remain largely unexplored. Here, we demonstrate that multiple clusters emerge and coexist, forming a hierarchical pattern known as the Devil's Staircase. We identify three key findings by investigating individual cluster dynamics and inter-cluster interactions. First, the PC persistently suppresses the formation of SCs during its growth and even after it has fully formed, revealing the significant impact of inter-cluster interactions on cluster formation. Second, once established, SCs induce higher-order clusters exhibiting frequency resonance via inter-cluster interactions, resulting in the Devil's Staircase pattern. Third, sufficiently large SCs can destabilize and fragment the PC, highlighting the bidirectional nature of cluster interactions. We develop a coarse-grained Kuramoto model that treats each cluster as a macroscopic oscillator to capture these inter-cluster dynamics and the resulting phenomena. Our work marks a significant step beyond system-wide averages in the study of inertial oscillator systems, offering new insights into the rich dynamics of cluster formation and synchronization in real-world applications such as power grid networks.