Introduction to Equivariant Bifurcation Theory (MAGIC046)
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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.
Spring 2013 (Monday, January 21 to Friday, March 29)
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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.
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There will be a take-home exam during the MAGIC assessment period 15th-26th April 2013. It should be possible to pass this with a couple of hours work if you have been following the course.
MAGIC046 Equivariant Bifurcation Theory Exam 2013
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