Analytical solutions of equilibrium points of the standard Kuramoto model: 3 and 4 oscillators

Abstract

This paper concerns the problem of how to obtain analytically all equilibrium points of the standard Kuramoto model (with all-to-all uniform coupling) having any natural frequencies and coupling gain. As an initial effort to solve this challenging problem, for the Kuramoto model with 3 (respectively 4) oscillators, to obtain analytically all equilibrium points, we show that we need to solve a polynomial equation of the sine of the phase of an oscillator with the highest order of 6 (respectively 14). For 3 oscillators, this polynomial equation with numerical examples shows that the maximal number of distinct equilibrium points for any natural frequencies and coupling gain is 6. For 4 oscillators, this paper shows theoretically that the maximal number of distinct equilibrium points is not greater than 14, and presents two numerical examples to show the existence of 10 distinct equilibrium points. From the numerical investigation carried out in this study, it is a conjecture that the maximal number of distinct equilibrium points of 4 oscillators for all natural frequencies and coupling gain is 10. This paper also presents numerical examples to investigate the synchronization of the oscillators and convergence to stable equilibrium points.