Exponential synchronization of Kuramoto oscillators using spatially local coupling

We study the generalized Kuramoto model of coupled phase oscillators with a finite size, and discuss the asymptotic complete phase–frequency synchronization. The generalized Kuramoto model has inherent difficulties in mathematical approaches that this model is governed by nonlinear equations and the Kuramoto oscillator is arbitrarily connected with the others. To overcome these mathematical barriers, many researchers have adopted a linearization of homogeneous solutions, and applied a perturbation method. However, we introduce a new method which just requires some conditions on the smallest and largest positive eigenvalues of the graph Laplacian, and directly compute the bounds of homogeneous solutions. Using this method, we present analytic results for the generalized Kuramoto model. More specifically, we give a few sufficient conditions for initial configurations leading to the exponential decay toward the completely synchronized states. Our sufficient conditions and decay rate depend on the coupling strength, the initial phase and natural frequency configurations, and the graph Laplacian, but the conditions are independent of the system size. Moreover, we estimate the time evolution of deviations for the phase and frequency, and show that the smallest and largest positive eigenvalues for the graph Laplacian affect the stability region and convergence rate for the synchronized states. Finally, we compare our analytic results with numerical simulations using a few examples.