Cyclops states in repulsive Kuramoto networks: the role of higher-order coupling

TL;DR

This paper investigates the dynamics of repulsive Kuramoto networks with higher-order coupling, revealing that specific three-cluster splay states, called cyclops states, dominate the system's behavior especially with added harmonics.

Contribution

It introduces the concept of cyclops states in repulsive Kuramoto networks and demonstrates their dominance through analysis and numerical simulations, highlighting the impact of higher-order coupling.

Findings

Cyclops states are prevalent in odd-numbered networks with weak repulsion.

Adding higher harmonics makes cyclops states global attractors.

Cyclops states dominate phase space in weakly repulsive networks.

Abstract

Repulsive oscillator networks can exhibit multiple cooperative rhythms, including chimera and cluster splay states. Yet, understanding which rhythm prevails remains challenging. Here, we address this fundamental question in the context of Kuramoto-Sakaguchi networks of identical rotators with higher-order coupling. Through analysis and numerics, we show that three-cluster splay states with two distinct coherent clusters and a solitary oscillator are the prevalent rhythms in networks with an odd number of units. We denote such tripod patterns cyclops states with the solitary oscillator reminiscent of the Cyclops's eye. As their mythological counterparts, the cyclops states are giants that dominate the system's phase space in weakly repulsive networks with first-order coupling. Astonishingly, the addition of the second or third harmonics to the Kuramoto coupling function makes the cyclops states global attractors practically across the full range of coupling's repulsion. At a more general level, our results suggest clues for finding dominant rhythms in repulsive physical and biological networks.