Symmetries as Building Blocks for Synchronization Clusters: Theory and Experiment

Networks of coupled oscillators are sometimes observed to produce patterns of synchronized clusters where all the oscillators in each cluster have identical trajectories, but not the same as oscillators in other clusters. We show the intimate connection between network symmetry and cluster synchronization using computational group theory to reveal the symmetry clusters (SC) and determine their stability. Other synchronization clusters such as equitable partitions clusters (EC) or Laplacian coupling clusters (LC) also exist. We show that EC's and LC's can be constructed by the merging of appropriate SC's. We show that these types of dynamical behaviors can exist experimentally in a set of fiber ring lasers from which we form complex networks. We also demonstrate that the stability theory for such clusters applies to the experiment.