Numerical Bifurcation Analysis of Complex Systems

The following project proposal is for third-year students taking the mathematical sciences final-year undergraduate project module (4H level) in the coming academic year (2024/2025). This project will be jointly supervised with Dr. Denis Patterson.

Bifurcation theory studies how properties of dynamical systems can change, sometimes very abruptly at 'tipping points', as parameters are varied. This kind of analysis is now commonplace in models in numerous fields of science, from mathematical biology and physics, to fluid dynamics and engineering. Examples include understanding how systems like interacting populations or the global climate change as parameters are varied, and in particular trying to predict when we might see sudden and dangerous changes such as extinctions or extreme weather events. Even with relatively simple models, there is only so much of this analysis that can be done with just pen and paper. For simple kinds of solutions, such as steady states (equilibria) of ordinary differential equations, one can use linear stability analysis to understand how the stability of these changes as parameters are varied. Various kinds of stability changes, as well as the destruction or creation of equilibria (and more complicated objects) can then be understood at critical points of a linearized system. Computational methods, however, allow for a far greater ability to analyze these bifurcations, and to understand complex nonlinear models far away from regimes which are analytically tractable.

Hence, this project will be an exploration of numerical bifurcation theory, and particularly an introduction to numerical continuation, with an eye towards applications in biology, physics, and other systems of interest. The project will start with a basic exploration of bifurcations in ODEs using BifurcationKit. From there students will pick a topic of your choice, either based on interest in a particular dynamical behaviour, class of mathematical models, or physical or biological phenomena you are interested in. Examples would be:

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

• Geometry of Turing instabilities

• Bifurcations of Limit Cycles (Blue Sky Catastrophe, Homoclinic Bifurcations, Period Doubling Route to Chaos)

• Tipping Points in Ecosystems (e.g. climate change, anthropogenic intrusion, etc)

 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 us at andrew.krause@durham.ac.uk and denis.d.patterson@durham.ac.uk if you have any questions, and we would be happy to chat in more detail about the project and possible directions it could take.

Prerequisites

Familiarity with basic dynamical systems modelling (phase plane analysis, systems of ODEs) is necessary. Familiarity with or interest in scientific programming (MATLAB, Python, Julia, C++ etc) and numerical solutions of differential equations will also be required. It is expected that students are comfortable with topics from Dynamical Systems (for example, taking Mathematical Biology III and/or Dynamical Systems III), as well as an eagerness for computational modelling, will be helpful.

Resources

Practical Bifurcation Analysis by Rüdiger Seydel will be used as the standard textbook for the theoretical background underlying the continuation methods explored. While this project is intended to be applied, some familiarity with how things work 'under the hood' is helpful.

Nonlinear Dynamics and Chaos by Steven Strogatz covers a large variety of basic bifurcation theory and numerous examples. 

The three codes linked above also have copious documentation and examples/associated papers that can be found online, and these will serve as the primary jumping off point for really getting familiar with the computational tools involved.