Tiered synchronization to traveling-wave state through a metastable state

A first-order phase transition often occurs through a metastable state in diverse systems. However, in synchronization transitions (STs), existence of such a metastable state has been rarely focused up to now, even though its effect will be intriguing. Here we consider a Kuramoto model with two competing types of oscillators which exhibits a hybrid ST. We find that the system stays at the metastable state for a long time to reach a synchronized traveling-wave state. In the metastable state, the order parameter shows large temporal and sample-to-sample fluctuations as in k-core percolation and the number of clusters having similar velocities discretely increases dynamically as in the consciousness recovery process in the human brain.