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.
The course should be accessible to applied mathematicians working with bifurcations in nonlinear systems, either from an analytic or a numerical viewpoint, and the necessary group theory will be introduced.
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
- Fundamental ideas from group theory.
The course follows on from the core course MAGIC056 Dynamical Systems I, and should complement Dynamical Systems II.
The first part of the course aims to give an idea of the classification of bifurcations by codimension for systems with symmetries. The last part gives further examples of dynamical phenomena that appear in systems with symmetries, and examples of where these appear.
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 D4 Hopf bifurcation, mode interaction and bifurcation to robust heteroclinic cycles.