Synchronous Clustering in Multilayered Networks

The topic of cluster synchronization of oscillators in networks has an active history and has recently experienced renewed interest in the nonlinear dynamics community because of the use of such graph theory tools as symmetries and equitable partitions to predict what cluster structures are possible, calculate their dynamic stability, and characterize their bifurcations as they desynchronize. Another recent network structure that has caught the interest of the dynamics community is multilayered networks. We show how to extend the graph theory tools to multilayered networks. Certain multilayered networks have a structure that allows their dynamics to be simplified and fit into the existing structure of analysis of synchronous clusters. We also present a recent experiment on a simple 4-oscillator multilayer system. We show what can be discerned from experimental data including bifurcation plots and desynchronization patterns. This is the first experiment on cluster synchronization that we know of that uses totally analog oscillators with no computer-aided control of the oscillators. We also will discuss how some of the concepts can be generalized.