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General

Description

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.

Semester

Spring 2013 (Monday, January 21 to Friday, March 29)

Timetable

• Wed 09:05 - 09:55

Prerequisites

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.

Syllabus

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.

Lecturer

 Email P.Ashwin@ex.ac.uk Phone (01392) 725225 Interests Nonlinear dynamics, applications
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.

Students

 Reem Alomair (Manchester) Dana Alsaleh (Manchester) Burhan Bezekci (Exeter) Katy Gallagher (Liverpool) Clare Hobbs (Exeter) Xinhe Liu (Loughborough) Ummu Atiqah Mohd Roslan (Exeter) Paul Ritchie (Exeter) Fatih Say (Nottingham) Wessel Woldman (Exeter)

Bibliography

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

Note:

Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.)

Assessment

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) Instructions: 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.

Files

Files marked L are intended to be displayed on the main screen during lectures.

Week(s) File magic046_examples_2013rev.pdf magic046_examples_2013rev_sols.pdf magic046_exam_2013a.pdf magic046_exam_sols_2013b.pdf magic046_slides_2013.pdf 0 magic046_examples2_2013.pdf