Relationships Between the Distribution of Watanabe–Strogatz Variables and Circular Cumulants for Ensembles of Phase Elements

Abstract

The Watanabe–Strogatz (WS) and Ott–Antonsen (OA) theories provided a seminal framework for rigorous and comprehensive studies of collective phenomena in a broad class of paradigmatic models for ensembles of coupled oscillators. Recently, a “circular cumulant” approach was suggested for constructing the perturbation theory for the OA approach. In this paper, we derive the relations between the distribution of WS phases and the circular cumulants of the original phases. These relations are important for the interpretation of the circular cumulant approach in the context of the WS and OA theories. Special attention is paid to the case of hierarchy of circular cumulants, which is generally relevant for constructing perturbation theories for the WS and OA approaches.