Phaselocked Solutions Of The Finite Size Kuramoto Coupled Oscillator Model

Abstract

We investigate properties of phaselocked solutions of the finite size Kuramoto coupled oscillator model. There are minimal assumptions on the frequencies $\omega_{1},\ldots,\omega_{N}$ in the model. The coupling constant is denoted by $\alpha,$ and a key parameter $\gamma$ is introduced which is proportional to ${\frac {1}{\alpha}}.$ Theorem 1.1(III) gives new predictions for the location of $\gamma^{*},$ the “phaselocking threshold.” When $N$ is large these predictions show that the size of the $\gamma$ interval $[\gamma_{*},\gamma^{*})$ where two phaselocked solutions coexist (one stable and the other unstable) is uniformly bounded away from zero. To demonstrate the utility of this result we give an application which is motivated by recent reports of an increasing number of cyber-attacks on electric power grids. In this application, which is related to a large scale electric grid model, we identify a three-step mechanism which shows how a subtle change in frequencies $\omega_{1},\ldots,\omega_{N}$ causes phaselocked synchronization collapse as the system undergoes a transition from stable synchronization to decoherence. In an electric grid such synchronization collapse may potentially cause electricity blackouts.