A large number of important structures and phenomena in nature, society, and technology can be modeled by networks of interacting dynamical systems. Examples include power and communication networks in technology, neuronal and genetic networks in biology, as well as social and economic networks, to name a few. Understanding behavior of these systems requires advanced mathematical techniques for studying dynamics in complex networks. This project will use mathematical modeling, analysis, and numerical simulations to elucidate the link between the structure and dynamics in complex networks. In particular, the investigator aims to find new ways for describing network organization in dynamical models and study critical phenomena, such as the onset of synchronization and emergence and bifurcations of spatial patterns in networks of coupled oscillators. The results of this research and the tools that will be developed in its course, will enhance our ability to understand, predict, and control the behavior of real world networks.
The mean field approximation is one of the most effective analytical tools available for studying large ensembles of interacting dynamical systems. Originally developed for problems in statistical physics, this method has been extremely successful for studying collective dynamics in coupled oscillator models of various physical, chemical, biological, and technological systems. The analysis of synchronization in the Kuramoto model of coupled phase oscillators and the bifurcation analysis of chimera states rely on the mean field equation formally derived in the limit, as the number of oscillators goes to infinity. Despite its spectacular success in applications, the mathematical basis of the mean field approximation of coupled dynamical systems on networks is not well understood.  In this research, the aim is to derive and rigorously justify the mean field description of dynamics of coupled systems on convergent families of graphs. This advances the mean field theory for coupled systems in two ways: first, by extending the main results of this theory to a large class of random networks, including those with small-world and scale free connectivity, and, second, by relaxing the key regularity assumptions, blocking the application of this theory to real world networks. The new theoretical results are used to study the onset of synchronization in systems of coupled phase oscillators with spatially structured interactions, stable dynamical regimes in the Kuramoto model on power law graphs, and pattern formation in neural fields in random media.
