Cyclops States in Repulsive Kuramoto Networks: The Role of Higher-Order Coupling

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 rotators with higher-order Fourier modes in the 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' 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. Beyond the Kuramoto oscillators, we show that this effect is robustly present in networks of canonical theta neurons with adaptive coupling. At a more general level, our results suggest clues for finding dominant rhythms in repulsive physical and biological networks.

Automated summary

We investigate the expression of multiple cooperative rhythms in Kuramoto-Sakaguchi networks with higher-order Fourier modes. We find that three-cluster splay states with two distinct coherent clusters and a solitary oscillator are the prevailing rhythms in networks with an odd number of units. These states, called cyclops states, become global attractors across the full range of repulsion when the second or third harmonics are added to the coupling function. Our findings also extend to networks of theta neurons with adaptive coupling. Overall, our results provide clues for finding dominant rhythms in repulsive physical and biological networks.