Bifurcation Theory
24-28 October 2005
DynamIC: Dynamical Systems
at Imperial College
Organiser: Jeroen Lamb
This session of the London Dynamical Systems Group Graduate School on bifurcation theory is designed to provide a basic introduction to the subject. It is meant to be accessible to PhD students of all levels and thus has no non-standard pre-requisites. In line with the general philosophy of the Graduate School it's aim is primarily to present an introduction to the main ideas and results so that post-graduate students in all areas of dynamics have an opportunity to become familiar with such notions and become aware of the central definition and open problems. It is not therefore directed specifically at students already working in topics related to those discussed in the lectures. Quite the opposite ! However, lectures may present topics in a way which are likely to be new even to students working in related areas.
Precise schedule of the lectures will be posted here closer to the time.
Homoclinic Bifurcation
Ale Jan Homburg (University of Amsterdam)
Local Bifurcation
Jeroen Lamb
(Imperial College London)
This course provides an introducation into local bifurcation theory
for equilibria and periodic solutions of flows of vector fields
and diffeomorphisms. Topics that will be covered include:
- Genericity and transversality.
- Centre manifold reduction.
- Linear systems with parameters: normal forms and unfoldings.
- Nonlinear normal form theory for equilibria and periodic solutions.
- Liapunov-Schmidt reduction for steady-state and Hopf bifurcation.
Some written notes will be provided.
Additional literature:
D.K. Arrowsmith and C.M. Place, An Introduction to Dynamical Systems, 1990.
Numerical bifurcation analysis with AUTO (course material)
Jan Sieber and
Thomas Wagenknecht
(University of Bristol)
The course gives an introduction to solving nonlinear equations and
boundary-value problems using numerical continuation. It serves both as
an illustration for the theoretical concepts, introduced in the other
lectures, and as an introduction to the usage of the popular
continuation package AUTO.
Using AUTO we will go through worked examples that illustrate how
equilibria, periodic orbits, and their local bifurcations can be
computed. In addition, we demonstrate the usage of the HomCont Toolbox
in AUTO, which can be used to compute homoclinic bifurcations. Again,
examples will help to understand the concepts introduced in the
theoretical lectures.
The participants will have the opportunity to gain hands-on experience
by exploring the AUTO user interface and various examples in computer
lab sessions.
Additional literature:
Y Kuznetsov: Elements of Applied Bifurcation Theory, Springer Verlag,
1995.
Downloadable from
http://cmvl.cs.concordia.ca/auto/:
E J Doedel et al: AUTO97, Continuation and bifurcation software for
ordinary differential equations, 1998. (AUTO manual)
E J Doedel: Lecture notes on "Numerical Analysis of Bifurcation
Problems", 1997.
Further software:
http://www.dynamicalsystems.org/sw/sw/
Schedule LDSG Graduate School Bifurcation Theory