Optimal interaction functions realizing higher-order Kuramoto dynamics with arbitrary limit-cycle oscillators

Abstract

The Kuramoto model is the simplest case of globally coupled phase oscillators with a purely sinusoidal fundamental-harmonic phase coupling function, whose dynamical properties have been extensively studied. While coupled phase oscillators are derived from weakly interacting limit-cycle oscillators via phase reduction, this procedure does not necessarily yield the Kuramoto model or its higher-order extensions exactly for general limit-cycle oscillators and interaction functions, except in the special case of interacting Stuart-Landau oscillators. In this study, we artificially design optimal pairwise and higher-order interaction functions between limit-cycle oscillators, from which higher-order Kuramoto models can be exactly derived via phase reduction for arbitrary smooth limit-cycle oscillators. We validate the results through numerical simulations of FitzHugh-Nagumo oscillators, demonstrating that the collective synchronization dynamics predicted by the reduced higher-order Kuramoto models are realized. Control of the collective phase of the FitzHugh-Nagumo oscillators based on Ott-Antonsen reduction of the higher-order Kuramoto model is also demonstrated.

Figures (8)

• Figure 1: (a) The limit-cycle orbit $\boldsymbol{\chi}(\theta) = \left[ \chi_{x}(\theta)\; \chi_{y}(\theta) \right]^{\top}$ of the vector field $\bm{F}$ of the FHN oscillator as a function of $\theta$. (b) The PSF $\bm{Z}(\theta) = \left[ Z_{x}(\theta)\; Z_{y}(\theta) \right]^{\top}$ of the vector field $\bm{F}$ of the FHN oscillator as a function of $\theta$.
• Figure 2: Comparison between the distribution $P_{\mathrm{G}}(\omega) = \mathcal{N}(\omega;\omega_{0},\sigma^{2})$ and the normalized histogram of the natural frequencies of the FHN oscillators with the number of $N = 10^{4}$, which are represented as the red curve and gray bars, respectively.
• Figure 3: The order parameter $R$ vs. the pairwise coupling strength $K_{1}$. The red and black plots represent the order parameters of the system of the FHN oscillators and pairwise Kuramoto model, respectively. The blue dotted vertical line represents the critical value $K_{1} = K_{\mathrm{c}}$. We can observe that the system of the FHN oscillators also exhibits the synchronization transition when $K_{1}$ exceeds $K_{\mathrm{c}}$.
• Figure 4: Distributions of the coupled FHN oscillators on the $xy$ plane when $K_{1} = 0.04$. (a) Incoherent initial state. (b) Collectively synchronized steady state. In each panel, the red dots represent the oscillator states and the black periodic orbit represents the limit cycle of the FHN oscillator with the common part $\bm{F}$.
• Figure 5: The order parameter $R$ vs. the phase lag parameter $\alpha$. The stable completely incoherent states, two-cluster states, and fully synchronized states are indicated by crosses, diamonds, and circles, respectively. For slow-switching phenomena, the maximum and minimum values of the order parameters after the destabilization are indicated by upward and downward triangles, respectively. The red and black marks represent the results by the system of the FHN oscillators and higher-order Kuramoto model, respectively.