Synchronization and its slow decay in noisy oscillators with simplicial interactions

TL;DR

This paper investigates how noise affects synchronization in higher-order oscillator networks with simplicial interactions, revealing conditions for persistent or eroded synchrony and deriving a dynamical equation for the order parameter.

Contribution

It provides a comprehensive analysis of noise effects on synchronization in higher-order networks, including a weakly nonlinear analysis and a dynamical equation for the order parameter.

Findings

Synchronization can be eroded by noise when a dominant two-simplex interaction exists.

Persistent synchronization occurs when one-simplex or certain two-simplex interactions are strong.

The lifetime of synchronized states increases exponentially with two-simplex interaction strength.

Abstract

Previous studies on oscillator populations with two-simplex interaction have reported novel phenomena such as discontinuous desynchronization transitions and multistability of synchronized states. However, the noise effect is not well understood. Here, we study a higher-order network of noisy oscillators with generic interactions consisting of one-simplex and two types of two-simplex interactions. We observe that when a type of two-simplex interaction is dominant, synchrony is eroded and eventually disappears even for infinitesimally weak noise. Nevertheless, synchronized states may persist for extended periods, with the lifetime increasing approximately exponentially with the strength of the two-simplex interaction. When one-simplex or another type of two-simplex interaction is sufficiently strong, noise erosion is prevented, and synchronized states become persistent. A weakly nonlinear analysis reveals that as one-simplex coupling increases, the synchronized state appears supercritically or subscritically, depending on the interaction strength. Furthermore, assuming weak noise and using Kramers' rate theory, we derive a closed dynamical equation for the Kuramoto order parameter, from which the time scale of the erosion process is derived. Our study elucidates the synchronization and desynchronization of oscillator assemblies in higher-order networks and is expected to provide insights into such systems' design and control principles.