Avalanches in networks of weakly coupled phase-shifting rotators

Abstract

The networks of phase-shifting rotators interacting through exchange of weak ±-kicks are considered. Such a rotator consists of a particle rotating on a circle which at some discrete mo- ments receives some ±-kicks. We assume that the kicks are not of mechanical character: they change a particle's position but not the rate. A comparison of the rotator networks with the BTW model of self-organized criticality, Burridge-Knopoff's model in seismicity, Herz-Hopfield neural networks, and the Turing-Smale system in biological cells is presented. This work studies the avalanches in rotator networks — the events when a number of rotators almost simultaneously hit some threshold levels. The asymptotic relations linking distribution of avalanches with the architecture of a network are proved. The equivalence of two well-known power-law conjectures, in lattice models of statistical physics and in interacting threshold systems, is established.