Rotating clusters in phase-lagged Kuramoto oscillators with higher-order interactions

TL;DR

This paper explores how phase-lag influences cluster synchronization in higher-order Kuramoto oscillators modeled with simplicial complexes, revealing control mechanisms for cluster frequencies and critical transition points.

Contribution

It extends the Kuramoto model to include phase-lag in simplicial interactions and derives analytical expressions for cluster and global synchronization, advancing understanding of complex oscillator networks.

Findings

Phase-lag shifts critical points of synchronization transition.

Derived reduced equations for cluster and global order parameters.

Phase-lag acts as a control parameter for cluster frequencies.

Abstract

The effect of phase-lag parameter in pairwise interactions has been a topic of great interest for long. However, real-world systems often have interactions that are beyond pairwise and can be modeled using simplicial complexes. We investigate the effect of the inclusion of phase-lag in coupled Kuramoto oscillators with simplicial interactions and find that it shifts the critical points at which first-order transition from cluster synchronized state to incoherent state occurs. In the thermodynamic limit, using the Ott-Antonsen approach we derive a reduced equation for order parameter measuring cluster synchronization. Further, we progress through the self-consistency method to achieve a closed form of the order parameter measuring global synchronization which was lacking in Ott-Antonsen approach. Moreover, considering polar coordinates framework we obtain rotation frequency of the clusters which comes out to be a function of the phase-lag parameter further indicating that phase-lag can be used as a control parameter to achieve a desired cluster frequency.