Emergence of higher-order interactions in systems of coupled Kuramoto oscillators with time delay

TL;DR

This paper demonstrates that time-delayed pairwise coupling in Kuramoto oscillators naturally leads to higher-order interactions, which can be analyzed more easily and reproduce key dynamical features of the original delayed system.

Contribution

It introduces a method to derive an effective higher-order interaction model from time-delayed pairwise coupling in Kuramoto oscillators, simplifying analysis and understanding of delay effects.

Findings

Higher-order interactions emerge from time delays in coupled oscillators.

The reduced model reproduces bistability and synchronization transitions.

Stability diagrams match those of the original delayed system.

Abstract

Understanding the mechanisms that govern collective synchronization is a paramount task in nonlinear dynamics. While higher-order (many-body) interactions have recently emerged as a powerful framework for capturing collective behaviors, real-world examples regarding dynamics remain scarce. Here, we show that higher-order interactions naturally emerge from time-delayed pairwise coupling in Kuramoto oscillators. By expanding the delay term up to second order in the coupling strength, we derive an effective Kuramoto model featuring both two-body and three-body interactions, but without delay, hence, easier to be analyzed. Numerical simulations show that this reduced model can reproduce the bistability and synchronization transitions of the original time-delayed system. Furthermore, applying the Ott-Antonsen ansatz, we obtain a stability diagram for incoherent and synchronized states that closely matches the results of the original model. Our findings reveal that time delays can be effectively recast in the form of higher-order interactions, offering a new perspective on how delayed interactions shape the dynamics.