Relationships between the Distribution of Watanabe-Strogatz Variables and Circular Cumulants for Ensembles of Phase Elements

TL;DR

This paper establishes the mathematical relations between Watanabe-Strogatz phases and circular cumulants, enhancing the understanding of collective dynamics in coupled oscillator ensembles and aiding perturbation theory development.

Contribution

It derives the relations between Watanabe-Strogatz phases and circular cumulants, clarifying their connection within the theories of coupled oscillators.

Findings

Derived relations between Watanabe-Strogatz phases and circular cumulants.

Highlighted the importance of hierarchical cumulant structures for perturbation theories.

Provided insights into interpreting circular cumulants in the context of collective oscillator dynamics.

Abstract

The Watanabe-Strogatz and Ott-Antonsen 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 Ott-Antonsen approach. In this paper, we derive the relations between the distribution of Watanabe-Strogatz 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 Watanabe-Strogatz and Ott-Antonsen theories. Special attention is paid to the case of hierarchy of circular cumulants, which is generally relevant for constructing perturbation theories for the Watanabe-Strogatz and Ott-Antonsen approaches.