Large Deviations and Metastability in Dynamical Networks

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We live in the age of networks. Our everyday lives depend on robust and predictable performance of different technological networks around us (such as power grids and communication networks). Human physiology relies on coordinated activity of hundreds of cellular networks (for example, networks of neural cells in the brain, cells in the heart and pancreas, to name a few). The patterns of connections in real world networks can be complex and exhibit nontrivial statistical properties. Understanding how the structural organization of a network affects its dynamics is the principal challenge in the theory of interacting dynamical systems that sets it apart from the theories for classical spatially extended dynamical systems, such as partial differential equations or lattice dynamical systems. The Principal Investigator (PI) seeks to develop a systematic mathematical approach to the analysis of dynamical networks. Through the development of new mathematical techniques and analyzing representative mathematical models, the PI aims to elucidate the relation between the structure and dynamics in complex networks. The PI is committed to teaching and training students. A six-month long Research Co-op for two undergraduate students will be organized in the course of this research. The PI will continue to organize minisymposia on dynamical networks at conferences on differential equations and dynamical systems.This research advances the theory for interacting dynamical systems through the development of new theoretical results and analyzing selected models. The PI and colleagues identify new dynamical phenomena and study them using the combination of tools from graph theory, probability, and analysis. The emphasis will be on the effects of random spatial organization and noise on dynamics of large networks. The following problems are addressed: the Large Deviation Principle for interacting dynamical systems on random graphs, metastability in the continuum Kuramoto model forced by noise, and synchronization and pattern formation in the Kuramoto model with inertia. The Kuramoto model of coupled phase oscillators plays a central role in the theory of synchronization with many important applications in science and technology. In particular, the analysis of synchronization and stability of clusters developed in this research elucidates the dynamics emerging in high-voltage power grids.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.