О связи между распределением фаз Ватанабэ–Строгаца и круговыми кумулянтами

Abstract

The theories of Watanabe–Strogatz and Ott–Antonsen served as the basis for rigorous and comprehensive investigations of collective phenomena in a broad class of paradigmatic models of en-sables of coupled oscillators. Recently, the “circular cumulant” approach was suggested for constructing a perturbation theory for the Ott–Anthonsen approach. In this paper we derived the relationship between the distribution of Watanabe–Strogatz phases and the circular cumulants of the original phases. These relationships are important for interpreting the approach of circular cumulants in the context of the Watanabe–Strogatz and Ott–Anthonsen theories. Particular attention is devoted to the case of the hierarchy of circular cumulants; this case is typical when constructing perturbation theories on top of the Watanabe–Strogats and Ott–Anthonsen theories.