In many mathematical models of applications, symmetries are present; either from approximations of homogeneity in a system, or as a modelling assumption to give models that are simpler and therefore amenable to analysis.
The presence of symmetries in a system may however have symmetry broken solutions, and these are created at bifurcations when one varies a system parameter. The main aim of this course is to give an introduction to symmetric or equivariant bifurcations of vector fields, using a number of examples and techniques from group theory and singularity theory.
We will present a selection of topics in bifurcation with symmetry including the equivariant branching lemma, equivariant Hopf lemma and robust heteroclinic cycles for ordinary differential equations.
I will use a selection of material from various sources including
M Golubitsky and I Stewart, The Symmetry Perspective, Birkhauser (2000).
I will assume students are familiar with
* Solution of ordinary differential equations (ODEs) by analytical methods
* Fundamentals of qualitative theory of ODEs
* Fundamentals of bifurcations for parametrized ODEs
* Fundemental ideas from group theory
1. ODEs and bifurcations; introduction.
2. Saddle-node, transcritical, pitchfork and Hopf bifurcations.
3. Normal forms and reduction.
4. Center manifold and Liapunov-Schmidt methods.
5. Symmetries and equivariant singularities.
6. Classification of bifurcations by codimension.
7-10. Examples from the literature including D_4 Hopf bifurcation, mode interaction and bifurcation to robust heteroclinic cycles.