# applied math seminar

### Event Description:

Title: Cluster Synchronization in Networks with Symmetries

We first review the master stability function (MSF) approach to synchronization of networks of coupled identical oscillators. The original formulation proposed by Pecora and Carroll applies to the case that all the oscillators in the network converge on the same time-evolution (complete synchronization) and that the couplings between the oscillators are all of the same type. Here we relax both these assumptions. We first assume an arbitrary coupling topology, for which we observe the emergence of cluster synchronization. We show that the emergence of clusters can be explained in terms of the symmetries of the underlying network structure. We observe both in simulations and in an experiment an interesting phenomenon in which some clusters lose synchrony while leaving others synchronized. We apply tools from computational group theory to reveal the hidden network symmetries and predict the patterns of synchronization that can arise.

We also consider a situation in which different coupling mechanisms coexist, and each coupling mechanism can be associated with a specific network layer, which corresponds to a hypernetwork or multilayer network. We use an approach based on simultaneous block diagonalization of matrices to reduce the stability problem in a lower dimensional form. This work is in collaboration with Lou Pecora (NRL), Aaron Hagerstrom (NIST), Abu Bakar Siddique (UNM), Joe Hart (UMD), Tom Murphy (UMD), and Raj Roy (UMD).

REFERENCES

L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, R. Roy, "Cluster Synchronization and Isolated Desynchronization in Complex Networks with Symmetries", Nature Communications, 5, 4079 (2014).

F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, T. E. Murphy, R. Roy, "Complete characterization of stability of cluster synchronization in complex dynamical networks", Science Advances 2, e1501737 (2016).

SHORT BIO

Francesco Sorrentino received a PhD in Control Engineering from the University of Naples Federico II (Italy). He was first a postdoc and then visiting assistant professor in the Nonlinear Dynamics and Chaos Group at the University of Maryland at College Park. In 2008 he became assistant professor at the University of Napoli Parthenope. In 2012 he joined the Department of Mechanical Engineering at the University of New Mexico. His research primarily focuses on cutting-edge topics in Nonlinear Dynamics and Chaos Theory. His work includes studies on dynamics and control of complex networks, adaptive sensor networks, adaptation in complex systems, and identification of nonlinear systems. Other subjects of interest are the dynamics of large networks of coupled neurons and evolutionary game theory. His research is funded by the National Science Foundation, the Office of Naval Research, and the Defense Threat Reduction Agency.

### Event Contact

**Contact Name: **Pavel Lushnikov