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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)


  • Wed 09:05 - 09:55


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.


Peter Ashwin
Phone (01392) 725225
Interests Nonlinear dynamics, applications
Photo of Peter Ashwin
Profile: My research is into various aspects of nonlinear dynamical systems and its applications, including bifurcations with symmetry, coupled dynamical systems spatio-temporal dynamics and low dimensional maps.


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Reem Alomair
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Dana Alsaleh
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Burhan Bezekci
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Katy Gallagher
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Clare Hobbs
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Xinhe Liu
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Paul Ritchie
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Fatih Say
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Wessel Woldman


The symmetry perspective: from equilibrium to chaos in phase space and ...Golubitsky and Stewart
Singularities and groups in bifurcation theoryGolubitsky, Schaeffer and Stewart
Methods in equivariant bifurcations and dynamical systemsChossat and Lauterbach
Pattern formation: an introduction to methodsHoyle
Equivariant bifurcation theory - ScholarpediaJ. Moehlis and E. Knobloch
Equivariant dynamical systems - ScholarpediaJ. Moehlis and E. Knobloch


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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

Files:Exam paper
Deadline: Friday 26 April 2013 (1610.6 days ago)

The take-home exam for this will be available from here at mid-day on 16th April 2013. It will consist of three questions of equal weight (50 marks each). The mark will be based on the best two attempted questions. In principle, someone who has taken the course, attended the lectures and done the examples should be able to pass the exam with about two hours of work.

Recorded Lectures

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