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

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.

Figures (3)

• Figure 1: Synchronization transitions in the case of 2 oscillators. The black solid and dotted lines represent the stable and unstable branches, respectively.
• Figure 2: Synchronization transitions in the case of (a) fixed time delay $\tau=2.8$ and varying coupling strength $\epsilon$,and (b) fixed coupling strength $\epsilon=0.3$ and varying time delay $\tau$. The black circles show the numerical results of Eq. (\ref{['eq:Kuramoto_time_delay']}), the green solid circles show that of Eq. (\ref{['eq:dynamics_second_identical']}), and the red $\times$ symbols show the results of the Sakaguchi-Kuramoto model, in the case of identical oscillators with $\omega_0=\pi/2$ and $N=300$. Note that bistable regions of full synchrony and incoherence exist for Eq. \ref{['eq:Kuramoto_time_delay']} and \ref{['eq:dynamics_second_identical']}, while no bistable regions exist in the Sakaguchi-Kuramoto model. The blue solid and dotted lines represent the stable and unstable branches of the order parameter dynamics by the Ott-Antonsen ansatz for Eq. \ref{['eq:dynamics_second_identical']} in Eq. \ref{['eq:dynamics_of_R']}.
• Figure 3: Stability diagram of the incoherent state and the fully synchronized state obtained from (a) the order parameter dynamics in Eq. (\ref{['eq:dynamics_of_R']}) and (b) the time-delay Kuramoto model in Eq. (\ref{['eq:Kuramoto_time_delay']}) derived in Yeung_1999. The red regions represent the domains where the incoherent state is linearly stable, and the blue regions represent the domains where the fully synchronized state is linearly stable. The red solid lines and the blue dashed lines indicate the boundaries of these regions, and the black dotted line indicates $\tau=1/\epsilon$. The purple stripe regions show the bistable regions.