Using Symmetries of Complex Networks of Oscillators to Find All Synchronization Clusters – 4/24/2017
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Using Symmetries of Complex Networks of Oscillators to Find All Synchronization Clusters – 4/24/2017

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Louis M. Pecora
US Naval Research Laboratory
Washington, DC

Abstract:

Several common physical systems can be modeled as a set of oscillators of some type connected to each other in a complex network. For example, power generators in an electric grid, neurons in brain structures, and gene interactions in cells. Many networks of coupled oscillators are observed to produce patterns of synchronized clusters where all the oscillators in each cluster have exactly the same dynamical trajectories in state space, but not the same as oscillators in other clusters even though the clusters are all interconnected. It has been difficult to predict these clusters in general.  We show the intimate connection between network symmetry and cluster synchronization.  We apply computational group theory to reveal the clusters and determine their stability. Other synchronization clusters are possible in addition to the symmetry clusters (SC). These are equitable partitions (EP) of the network. We show that the EP can be constructed by the merging of appropriate SC so that the SC’s form the building blocks of all possible clusters. We show this type of behavior experimentally using videos of an electro-optic network.