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

TL;DR

This work develops a general phase-reduction theory for nonlinear oscillators coupled through higher-order interactions on hypergraphs. By applying averaging to a phase description, it derives a reduced phase model where the original adjacency is preserved at first order, but the interaction topology can become a simplicial complex as lower-order terms emerge; under certain oscillator symmetries, even couplings can vanish at first order, leaving only odd terms. The theory is instantiated with Stuart-Landau oscillators in all-to-all and ring-like topologies, yielding tractable phase equations that reproduce key dynamical states (synchrony, clusters, twisted states) and reveal topology changes. The results provide a principled bridge between complex higher-order interactions and concise phase models, enabling analytical insights and practical analysis for real-world systems with many-body couplings.

Abstract

Phase reduction is a powerful technique in the study of nonlinear oscillatory systems. Under certain assumptions, it allows us to describe each multidimensional oscillator by a single phase variable, giving rise to simple phase models such as the Kuramoto model. Classically, the method has been applied in the case where the interactions are only pairwise (two-body). However, increasing evidence shows that interactions in real-world systems are not pairwise but higher-order, i.e., many-body. Although synchronization in higher-order systems has received much attention, analytical results are scarce because of the highly nonlinear nature of the framework. In this paper, we fill the gap by presenting a general theory of phase reduction for the case of higher-order interactions. We show that the higher-order topology is preserved in the phase reduced model at the first order and that only odd couplings have an effect on the dynamics when certain symmetries are present. Additionally, we show the power and ductility of the phase reduction approach 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, only the analysis of the phase model can provide insights and analytical results.

Figures (4)

• Figure 1: Schematic representation of the definition of phase on the limit cycle (a) and the asymptotic phase and isochrons (b).
• Figure 2: Numerically obtained phase diagram for the ensemble of Stuart-Landau oscillators all-to-all coupled through Eq. \ref{['eq:alltoall']} with $\epsilon=0.1$ and $K_1=1$ (a) and for the reduced phase model Eq. \ref{['eq.phalltoall']} with $K_1=1$ (b). In the blue and yellow regions, only two-cluster and fully synchronized states are stable, respectively, while in the red region, we find multi-stability between those states. In panel (b) for the phase model, we depict the theoretical prediction for the stability of synchronization and two-cluster in blue and red, respectively. Panel (c) shows the snapshots of the dynamics for two-cluster and synchronization, respectively.
• Figure 3: (a) Schematic representation of the twisted state with two rotations on a ring-like hypergraph: node are circles, links are black lines and hyperedges are gray bubbles. The arrows and colors in the link represent a twisted state in which the phase rotates twice around the ring. (b) Time series of the $x$ component of each SL oscillator obtained by numerically simulating the original model. (c) Time series of $\cos{\theta}$ obtained by numerically simulating the reduced phase model. In (b) and (c), the parameters are $\epsilon=0.1$, $K_1=0.5$, and $K_2=0.5$. The color indicated the value of $x_j$ and $\cos\theta_j(t)$ according to the color bar.
• Figure 4: Effect of phase reduction on the topology (a) for three different interactions, (b) completely antisymmetric, (c) completely symmetric, (d) symmetric in one of the variables.