Dynamical Systems on Networks

Many dynamical processes, such as disease transmission, information spread, or migrating organisms can be modelled by considering processes on graphs or networks. There is a large and growing theory of these kinds of models, across a range of modelling frameworks from discrete-time stochastic processes, to ordinary or delay differential equations. A good primer, and core text for this project, is this survey article of the field which overviews how traditional aspects of dynamical systems (such as transients, stability, chaos, etc) can be seen in models involving graphs, and interface with the topology of the underlying graph.

This project will begin with this survey article and some preliminary work on modelling coupled oscillators on simple networks. From this introduction, students will be free to pursue topics of their choice, both learning about a topic already in the literature and also extending it in their own way, either via novel mathematical analysis, numerical simulations, or by applying these tools to a particular physical phenomenon. Some examples of topics would be:

• Synchronization of Oscillators (chaotic pendula, chimeras, biological neural networks, etc)

• Ecological Dynamics on Networks

• Epidemics on Networks

While the broad topic is that of numerical bifurcation analysis, it will be valuable to look at these models from other perspectives as well. Programming and some numerical analysis will be mandatory, however, so students should be keen to spend at least some time with complex models on the computer.

Please email me at andrew.krause@durham.ac.uk if you have any questions, and I would be happy to chat in more detail about the project and possible directions it could take.