A study of the Kuramoto model for synchronization phenomena based on degenerate Kolmogorov-Fokker-Planck equations

Abstract: We study a nonlinear partial differential equation that arises when introducing inertial effects in the Kuramoto model. Based on the known theory of degenerate Kolmogorov operators, we prove existence, uniqueness and a priori estimates of the solution to the relevant Cauchy problem. Moreover, a stable numerical operator, which is consistent with the degenerate Kolmogorov operator, is introduced in order to produce numerical solutions. Finally, numerical experiments show how the synchronization phenomena depend on the parameters of the Kuramoto model with inertia.