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

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.