Introductory description

This module investigates how solutions to systems of ODEs (in particular) change as parameters are smoothly varied resulting in smooth changes to steady states (bifurcations), sudden changes (catastrophes) and how inherent symmetry in the system can also be exploited. The module will be application driven with suitable reference to the historical significance of the material in relation to the Mathematics Institute (chiefly through the work of Christopher Zeeman and later Ian Stewart). It will be most suitable for third year BSc. students with an interest in modelling and applications of mathematics to the real world relying only on core modules from previous years as prerequisites and concentrating more on the application of theories rather than rigorous proof.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

1. Typical one-parameter bifurcations: transcritical, saddle-node, pitchfork bifurcations,
   Bogdanov-Takens, Hopf bifurcations leading to periodic solutions. Structural stability.

2. Motivating examples from catastophe and equivariant bifurcation theories, for example
   Zeeman Catastophe Machine, ship dynamics, deformations of an elastic cube, D_4-invariant
   functional.

3. Germs, equivalence of germs, unfoldings. The cusp catastrophe, examples including Spruce-
   Budworm, speciation, stock market. Thom’s 7 Elementary Catastrophes (largely through
   exposition rather than proof). Some discussion on the historical controversies.

4. Steady-State Bifurcations in symmetric systems, equivariance, Equivariant Branching Lemma,
   linear stability and applications including coupled cell networks and speciation.

5. Time Periocicity and Spatio-Temporal Symmetry: Animal gaits, characterization of possible
   spatio-temporal symmetries, rings of cells, coupled cell networks, H/K Theorem, Equivariant Hopf
   Theorem.

Further topics from (if time and interest):

Euclidean Equivariant systems (example of liquid crystals), bifurcation from group orbits (Taylor
Couette), heteroclinic cycles, symmetric chaos, Reaction-Diffusion equations, networks of cells
(groupoid formalism).

Learning outcomes

By the end of the module, students should be able to:

• Understand how observable (stable) solutions to systems of ODEs can be dramatically changed through a smooth change of parameter(s) through bifurcations and catastrophes.
• apply techniques from the whole module to examples, and understand the implications of the results, not limited to the idea that a dramatic change to solutions need not arise from a dramatic change to the equations (environment).
• Have a good grasp of the underlying theory, without necessarily being in a position to rigorously prove the main results. Appreciate how abstract material from previous core modules fit into it and can be applied.
• Appreciate the historical nature of the topics covered in the context of the Mathematics Institute.

Indicative reading list

• Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,
  Guckenheimer/Holmes 1983
• Catastrophe Theory and its Applications, Poston and Stewart, 1978
• The Symmetry Perspective, Golubitsky and Stewart, 2002
• Singularities and Groups in Bifurcation Theory Vol 2, Golubitsky/Stewart/Schaeffer 1988
• Pattern Formation, an introduction to methods, Hoyle 2006.

Subject specific skills

See learning outcomes.

Transferable skills

Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.