Higher-order interactions induce anomalous transitions to synchrony

TL;DR

This work analyzes a minimal phase model with two-body and three-body couplings for globally coupled identical oscillators, showing that higher-order interactions can trigger anomalous transitions to synchrony and multistability beyond the Kuramoto framework. By deriving and analyzing the phase-reduced dynamics, the authors demonstrate that three-body coupling induces coexistence of incoherence, full synchronization, and two-cluster states, as well as slow switching via heteroclinic cycles for certain parameter regimes; analytical stability conditions and phase diagrams corroborate the numerical findings. Importantly, these phenomena persist under small heterogeneity in natural frequencies, suggesting broad applicability to real systems. The study advances understanding of how higher-order interactions shape collective synchronization with potential implications across physics, biology, and engineering.

Abstract

We analyze the simplest model of identical coupled phase oscillators subject to two-body and three-body interactions with permutation symmetry. This model is derived from an ensemble of weakly coupled nonlinear oscillators by phase reduction. Our study indicates that higher-order interactions induce anomalous transitions to synchrony. Unlike the conventional Kuramoto model, higher-order interactions lead to anomalous phenomena such as multistability of full synchronization, incoherent, and two-cluster states, and transitions to synchrony through slow switching and clustering. Phase diagrams of the dynamical regimes are constructed theoretically and verified by direct numerical simulations. We also show that similar transition scenarios are observed even if a small heterogeneity in the oscillators' frequency is included.

Figures (5)

• Figure 1: Synchronization transitions with pure two-body coupling (a) and with three-body (higher-order) coupling (b,c). Kuramoto order parameter vs. phase lag $\alpha$ for $N=1000$, $\beta=0$, and $K=0$ (a), $K=0.45$ (b), and $K=-0.45$ (c). The yellow (diamonds), blue (stars), red (circles), and green (triangles) indicate the incoherent state, full synchronization, two-cluster state, and slow switching, respectively. A typical snapshot of each state is shown in the inset to ease understanding of the dynamics.
• Figure 2: Phase diagrams of Eq. \ref{['eq.phasemodel']} for $\beta=0$ (a) and $\beta=1$ (b). In the blue, yellow, red, and green regions, full synchronization, incoherent state, two-cluster state, and slow switching are stable, respectively. The hatching indicates that more than one state is stable using the same color code.
• Figure B1: Kuramoto order parameter vs. phase lag $\alpha$ for $N=1000$, $\beta=0$, and $K=1.2$. The yellow (diamonds), blue (stars) and red (circles) indicate the incoherent state, full synchronization and two-cluster state, respectively.
• Figure B2: Time series of Kuramoto order parameter $R$ of the system Eq. \ref{['eq.phasemodel']} in the main text for $K=-0.45$ , $\alpha=-1.5$, and $\beta=0$ initialized close to the incoherent state. We observe the system get close to an unstable QPS where the order parameter rotates uniformly (hence the time series shows a plateau). The system displays slow switching after it departs from the unstable QPS.
• Figure E1: Phase diagrams of Eq. \ref{['eq.phasemodel']} for $\beta=0$ obtained by direct numerical simulations (a) and linear stability analysis (b). The light blue, yellow, red, or white color indicates the incoherent state, full synchronization, two-cluster state, or slow switching is stable, respectively, while violet, green, or dark blue indicates bistability of full synchronization and the two-cluster state, bistability of incoherence and full synchronization, and multistability of incoherence, full synchronization, and the two-cluster state, respectively.