Content: Equivariant Bifurcation Theory is the study of systems of equations which have an inherent symmetry within them, it is a perfect fusion of pure and applied mathematics, leading to some quite powerful and aesthetically pleasing results. The module will concentrate primarily on systems of ordinary differential equations, and both steady-state and periodic solutions that are formed through bifurcations as a parameter is varied, but the theory can also be applied to discrete systems and systems of partial differential equations.

Essential background required includes solving systems of first order differential equations from Differential Equations, and knowledge of symmetry groups from Algebra II: in particular we concentrate on those that have physical symmetry interpretations such as permutation groups ($S_n$), symmetries of regular n-gons ($D_n$) and circle group symmetries ($O(2)$ and $SO(2)$). Additional knowledge from the second year Theory of ODEs or third year Qualitative Theory of ODEs will help but not having taken neither should not prove a major obstacle.

The module should appeal to both students who wish to study applications of mathematics, and those who enjoy the beauty of mathematics for its own sake.

In more detail the topics to be covered will include:

0. Overview: Basic bifurcation theory (standard one parameter bifurcations) and symmetry groups (essentially revision).

1. Steady-State Bifurcation: Symmetries of ODEs, Liapunov Schmidt reduction, Equivariant Branching Lemma, applications (inc. coupled cell networks and speciation).

2. Linear Stability: Symmetry of the Jacobian, Hilbert Bases and Equivariant Mappings, examples of Dn Steady State Bifurcations and Sn Invariant Theory.

3. Time Periocicity and Spatio-Temporal Symmetry: Animal gaits, characterization of possible spatio-temporal symmetries, rings of cells, coupled cell networks.

4. Hopf Bifurcation with Symmetry: linear analysis, the Equivariant Hopf Theorem, Poincaré-Birkhoff Normal Form, Hopf Bifurcation in Coupled Cell Networks (esp Dn), mode interactions.

Further topics from (depending on time and interest): Forced symmetry breaking, Euclidean Equivariant systems (example of liquid crystals), bifurcation from group orbits (Taylor Couette), heteroclinic cycles, symmetric chaos, Reaction-Diffusion equations, hidden symmetries, networks of cells (groupoid formalism).

Aims:

Objectives:

Books: Printed lecture notes will be made available as the module progresses, but other good textbooks for reference:

The Symmetry Perspective, Golubitsky and Stewart, 2002
Singularities and Groups in Bifurcation Theory Vol 2, Golubitsky/Stewart/Schaeffer 1988
Pattern Formation, Hoyle 2006.

For more general background of relevant ODE theory:

Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Guckenheimer/Holmes 1983