No synchronization for coupled oscillators on small-world networks

The synchronization of oscillators is seen in many different systems, such as fireflies and cold atom clouds. Nonetheless, there's a limit on how simple a model can be and still be capable of synchronizing. In most models, an oscillator requires at least three levels (or phase values) in order to display a synchronized solution. Therefore, much of the fundamental investigations around synchronization have been performed on such three-state models due to their relative simplicity (as opposed to continuous or models with many phase values). In this work, we show that one such model may fail to achieve synchronization even when the dynamics occur on networks with non-local coupling and even long-range interactions. Failure to synchronize happens through a long-lived meta-stable traveling wave solution and is dependent on initial conditions and fluctuations.