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

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.

Ott-Antonsen ansatz truncation of a circular cumulant series

“…We have examined the relevance of the geometric progression allowing for a regular approach to constructing approximations of a prescribed accuracy or with a predefined number of circular cumulants. This progression is always present for wrapped Gaussian, von Mises, and non-Cauchy heavy-tail distributions (38), and is typical in experiments, as demonstrated with data for coupled biological [43] and electrochemical oscillators [45,46]. (A2)…”

“…Asymptotic formula(16) and the exact sum(12) are plotted with lines and symbols, respectively, for a twocumulant truncation for Θ = 0 in panel (a) and finite Θ = 0 in panels (b-d). (a): For the wrapped heavy-tail non-Cauchy distribution(38) with µ = −0.25 (left panel inFig. 3), Θ = 0; σ = 0.1 (blue circles), 0.32 (green squares), 1 (red triangles).…”

Collective in-plane magnetization in a two-dimensional XY macrospin system within the framework of generalized Ott–Antonsen theory

“…In real systems, the form of equations (1.1) is obviously distorted (see [33][34][35], for example), and the generalization of the OA theory to non-ideal situations was a resisting challenge for a decade. A way out has been proposed recently in the form of circular cumulant approach [3,[36][37][38]. This technique allows one to generalize the OA theory and derive closed equation systems for the dynamics of order parameters in the presence of thermal noise (or intrinsic noise) and under other violations of the applicability conditions of the original OA theory.…”

Small and finite inertia in stochastic systems: Moment and cumulant formalisms

“…For the ease of comparison to the formalism of circular cumulants [21,24] we introduce ! n n K n   and recast the latter equation system as…”

“…Recently [21], a circular cumulant approach was introduced for dealing with the systems where the applicability conditions of the Ott-Antonsen theory are violated. Within the framework of the circular cumulant formalism [21][22][23][24], one can consider weak inertia as a perturbation to the Ott-Antonsen properties and construct a low-dimensional description of the macroscopic collective dynamics of populations of phase elements. This task however can be accomplished in many different ways and, therefore, preliminary analysis of the moment and cumulant expansions with respect to a fast variable (velocity) is desirable.…”