Self-organized criticality with synchrony and self-breaking phenomena

Self-organizing and spontaneous breaking are seemingly opposite phenomena, and hardly captured in a single model. We develop a second order Kuramoto model \footnote{F. D orfler and F. Bullo, \textit{On the critical coupling for Kuramoto oscillators} , May. 2011. Available at https://arxiv.org/pdf/1011.3878 .} \footnote{Y.-P. Choi, S.-Y. Ha, and S.-B. Yun, \textit{Complete synchronization of Kuramoto oscillators with finite inertia}, Physica D, {\bf 240}, 32-44 (2011)} with relative damping(friction) which shows frequency locking together with spontaneous synchrony breaking. As the oscillators are synchronizing in frequency, the relative friction decreases accordingly, eventually making the system marginally stable. In the regime that the interacting force and the damping ratio are of same order, the dynamic behaviors of the oscillators alternate irregularly through the process between synchronization, formation-breaking, and reorganization. Especially when the oscillators are maintaining frequency locking, the system’s reaction against a random external perturbation shows a power-law distribution, which is another evidence of self-organized criticality\footnote{Steven H. Strogatz. \textit{Exploring complex networks}, Nature {\bf410}, 268-276 (2001).} inherited in the system