Algebraic graph theory and the Kuramoto model

Abstract

This project is embedded in a major proposal that includes applications to graph theory to coupled dynamical systems. It provides an introduction to the algebraic theory of graphs applied to dynamical systems, a topic that is rapidly expanding in results and applications during the last twenty years. The studies are directed to the recognition of graphs as a tool in the investigation of networks of coupled dynamical systems, which we call networks. A graph is a representation of symmetrical or asymmetrical relationships between discrete objects and, in this sense, a network is defined as a graph in which vertices and edges have attributes. The perspectives of this project are to learn about the spectral theory from linear algebra applied to the study of graphs and also to learn the tools for the realization of a vector field of a network as an admissible vector field to the graph associated with this network. One case study will be considered: the well-known Kuramoto model, a classical and important example of coupling with applications in Physics and Biology, where synchrony of coupled cells is investigated. The continuation as an extended high-level second undergraduate project is promising along this line and has potential for the publication of original results.(AU)