Exponential asymptotic stability of the Kuramoto system with periodic natural frequencies and constant inertia
The authors consider the system \[ I\ddot{\theta}_i+\dot{\theta}_i=\omega_i+\frac{K}{N}\sum_{k=1}^N\sin{(\theta_k-\theta_i)} \] modelling a network of \(N\) all-to-all coupled Kuramoto phase oscillators with inertia. The frequencies \(\{\omega_i\}\) are all \(T\)-periodic in time and there is an upper bound on the absolute value of the difference between any two oscillator's frequencies, at any time. The main result proved is that, under certain assumptions, the asymptotic-in-time solution for each oscillator consists of the sum of three terms. The first is a macroscopic phase, common to all oscillators and determined by initial values of the oscillators and frequencies. The second is a ``standing wave'' independent of initial conditions for which the difference between any two phases is \(T\)-periodic. The third is a term which decays exponentially in time. A number of simulations are shown which illustrate some of the analytical results.
scientific article; zbMATH DE number 7628876
scientific article