Equilibria in Kuramoto Oscillator Networks: An Algebraic Approach

.Kuramoto networks constitute a paradigmatic model for the investigation of collective behavior in networked systems. Despite many advances in recent years, many open questions remain on the solutions for systems composed of coupled Kuramoto oscillators on complex networks. In this article, we describe an algebraic method to find equilibria in this kind of system without approximation. To do this, we use a recently introduced algebraic approach to the Kuramoto dynamics, which results in an explicitly solvable complex-valued equation that captures the dynamics of the original Kuramoto model. Using this new approach, we obtain equilibria for both the nonlinear original Kuramoto and complex-valued systems considering the case of homogeneous natural frequency. We then completely classify all equilibria in the case of complete graphs originally studied by Kuramoto. Finally, we study equilibria in networks of coupled oscillators with phase-lag in generalized circulant networks, multilayer networks, and also random networks.