On the number of stable solutions in the Kuramoto model

We consider a system of n coupled oscillators described by the Kuramoto model with the dynamics given by (theta) over dot = omega + Kf(theta). In this system, an equilibrium solution theta* is considered stable when omega + Kf(theta*) = 0, and the Jacobian matrix Df(theta*) has a simple eigenvalue of zero, indicating the presence of a direction in which the oscillators can adjust their phases. Additionally, the remaining eigenvalues of Df(theta*) are negative, indicating stability in orthogonal directions. A crucial constraint imposed on the equilibrium solution is that vertical bar Gamma(theta*)vertical bar <= pi, where vertical bar Gamma(theta*)vertical bar represents the length of the shortest arc on the unit circle that contains the equilibrium solution theta*. We provide a proof that there exists a unique solution satisfying the aforementioned stability criteria. This analysis enhances our understanding of the stability and uniqueness of these solutions, offering valuable insights into the dynamics of coupled oscillators in this system. (c) 2023 Author(s).