Thinking Fast and Slow: Tipping Points and Critical Transitions

This project will be an introduction to areas of dynamical systems theory where timescales matter. 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 nonlinear optics, 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 systems 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. 

This project will in particular explore aspects of these theories going beyond the study of when simple equilibria bifurcate, to include time-dependent solutions, as well as measures for predicting the onset of bifurcations. Possible topics of study include:

• The study of critical transitions and early-warning signals for systems which are slowly changing;

• The theory of slow/fast dynamical systems, where one can approximate the system's behaviour on different timescales and relate these with asymptotic methods;

• Applications of these ideas arising in a variety of real-world systems.

Students will pick a topic of their choice within the broad scope of dynamical systems where time matters, either based on interest in a particular kind of theory, class of mathematical model, or physical or biological phenomena they are interested in exploring.

Please do email me at krause@maths.ox.ac.uk if you have any questions, and I'd be happy to chat in more detail about the project.

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 be helpful, though alternatively an interest and ability in more rigorous analytical work would be suitable. 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 independent exploration of a topic will be helpful.

Resources

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

Several papers, such as this one, do a good job overviewing different kinds of tipping points and related phenomena.

Multiple Time Scale Analysis is a much more advanced book, covering a large range of material from the mathematical point of view