Fisher information of the Kuramoto model: A geometric reading on synchronization

In this paper, we make a geometric investigation of the synchronization described by the Kuramoto model. The model consists of two-coupled oscillators with distinct frequencies, phase X, coupling strength K, and control parameter M. Here, we use information theory to derive the Riemannian metric and the curvature scalar as a new attempt to obtain information from the phenomenon of synchronization. The components of the metric are represented by second moments of stochastic variables. The scalar curvature R is a function of the second and third moments. It is found that the emergence of synchronization is associated with the divergence of curvature scalar. Nearby the phase transition from incoherence to synchronization, the following scaling law holds R∼M−MC−2. Critical exponents and scaling relations are assigned through standard scaling assumptions. The method presented here is general extendable to physical systems in nonlinear sciences, including those who possess normal forms and critical points.