Theory of phase reduction from hypergraphs to simplicial complexes: a general route to higher-order Kuramoto models

Abstract

Kuramoto-type models are paradigmatic models in the study of coupled oscillators, since they provide a simple explanation of the collective synchronization transition. Their universality comes from the fact that they are derived through phase reduction. Recent evidence highlighting the importance of higher-order (many-body) interactions in the description of real-world systems has led to extensions of Kuramoto-type models to include such frameworks. However, most of these extensions were obtained by adding higher-order terms instead of performing systematic phase reduction, leaving the questions of which higher-order couplings naturally emerge in the phase model. In this paper, we fill this gap by presenting a theory of phase reduction for coupled oscillators on hypergraphs, i.e., the most general case of higher-order interactions. We show that, although higher-order connection topology is preserved in the phase reduced model, the interaction topology generally changes, due to the fact that the hypergraph generally turns into a simplicial complex in the reduced Kuramoto-type model. Furthermore, we show that, when the oscillators have certain symmetries, even couplings are irrelevant for the dynamics at the first order. The power and ductility of the phase reduction approach are illustrated by applying it to a population of Stuart-Landau oscillators with an all-to-all configuration and with a ring-like hypergraph topology; in both cases, the analysis of the phase model can provide deeper insights and analytical results.