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MA3J3 Bifurcations, Catastrophes and Symmetry

Lecturer: Dr. David Wood

Term(s): Term 2

Status for Mathematics students: List A

Commitment: 30 Lectures

Assessment: 100% exam

Formal registration prerequisites: None

Assumed knowledge: This module will assume knowledge from core mathematics modules, in particular the whole of MA133 Differential Equations (although MA113 Differential Equations A will be sufficient provided the following is also covered), and bits of both MA249 Algebra II and MA259 Multivariable Calculus. In more detail:

MA249 Algebra II: Groups, in particular permutation groups and groups and groups of non-singular matrices (Dihedral groups especially). Quotient Groups, isomorphism theorems and orbit-stabilizer.

MA259 Multivariable Calculus: Differentiable functions, Inverse Function Theorem and Implicit Function Theorem.

Useful background: Having taken MA254 Theory of ODEs (or taking in parallel) will be beneficial, but not essential.

Synergies: Although containing a fair amount of "pure" maths, this is largely applying those theories and so this sits well beside modules such as MA256 Introduction to Mathematical Biology, MA390 Topics in Mathematical Biology, MA4E7 Population Dynamics, MA3J4 Mathematical Modelling with PDEs and MA4M1 Epidemiology by Example. It will also add extra context for modules such as MA3H5 Manifolds, MA3E1 Groups and Representations and MA4E0 Lie Groups, although here we only intersect with the basics of those.

Leads to: This module doesn't formally lead to any further modules, but sits alongside, and gives useful context to, many of the modules in "Synergies" above.

Content: 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.

Indicative content:

1. Typical one-parameter bifurcations: transcritical, saddle-node, pitchfork bifurcations, Bogdanov-Takens, Hopf bifurcations leading to periodic solutions. Structural stability.
2. Motivating examples from catastrophe and equivariant bifurcation theories, for example Zeeman Catastrophe 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, caustics. 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).

Aims: Understand how steady states can be dramatically affected by smoothly changing one or more parameters, how these ideas can be applied to real world applications and appreciate this work in the historical context of the department.

Objectives:

Books:

There is no one text book for this module, but the following may be useful references:

• 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.

Additional Resources