Asymptotic Analysis & Applied Nonlinear Dynamical Systems

Bifurcation theory studies how properties of dynamical systems can change, sometimes very abruptly at 'tipping points', as parameters are varied. For simple kinds of equilibria, such as steady states of ordinary differential equations, one can use linear stability analysis to understand how the stability of these equilibria 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 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. Using techniques in asymptotic analysis and perturbation theory, we can go beyond linear stability to study how bifurcation points influence dynamics in nearby regions of the phase space, and more generally act as 'organising centres' around which one can understand broad ranges of complex nonlinear behaviour.

Hence, this project will be an exploration of analytical aspects of bifurcation and perturbation theory, including but not limited to weakly nonlinear analysis and normal form theory, with an eye towards applications in biology, physics, and other systems of interest. You will start by exploring classical normal form and perturbation methods for ODEs in small numbers of dimensions, before considering more complicated models (such as delay or partial differential equations). From there you 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 could include:

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

• 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 asymptotics and bifurcation analysis, a project may only involve one of these two directions. In addition to pen and paper calculations, some programming (MATLAB or Julia preferred) and numerical analysis will be mandatory, however, so you 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.

Prerequisites

Familiarity with basic dynamical systems modelling (phase plane analysis, systems of ODEs) is necessary. Dynamical Systems III is ideal, but Mathematical Biology III and an eagerness to learn more theoretical tools would suffice. Familiarity with or interest in scientific programming (MATLAB, Julia) and numerical solutions of differential equations will also be required. 

Resources

Nonlinear Dynamics and Chaos by Steven Strogatz covers a large variety of basic bifurcation theory and numerous examples, but should mostly be used as a refresher on (hopefully) familiar concepts.

Perturbation Methods by John Hinch is one of the standard references for asymptotic techniques.

Elements of Applied Bifurcation Theory by Yuri Kuznetsov, and Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields by John Guckenheimer amd Philip Holmes are the standard texts for advanced aspects of bifurcation theory.