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)

This course is an introduction to global bifurcation theory of connecting orbits, and in particular to bifurcations of homoclinic loops. The following topics are included:
- Invariant manifolds and foliations
- Blue sky catastrophe
- Shilnikov homoclinic loops
- Shilnikov variables and other techniques to derive bifurcation equations
- Global bifurcation phenomena: bifurcations of horseshoes, intermittency

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

All lectures are in the Huxley Building, 180 Queens Gate, London SW7 2AZ

monday 24 October

09:00-11:00 Rm 139: Local Bifurcation (Lamb)
12:00-14:00 Rm 130: Global Bifurcation (Homburg)
15:00-17:00 Rm 139: AUTO (Sieber/Wagenknecht)

tuesday 25 October

09:00-11:00 Rm 139: Global Bifurcation (Homburg)
12:00-14:00 Rm 130: Local Bifurcation (Lamb)
15:00-17:00 Rm 215: AUTO (Sieber/Wagenknecht)

wednesday 26 October

10:00-12:00 Rm 139: Local Bifurcation (Lamb)
13:00-15:00 Rm 130: Global Bifurcation (Homburg)
15:30-17:30 Rm 215: AUTO (Sieber/Wagenknecht)

thursday 27 October

09:00-11:00 Rm 139: Local Bifurcation (Lamb)
12:00-14:00 Rm 130: Global Bifurcation (Homburg)
15:00-17:00 Rm 215: AUTO (Sieber/Wagenknecht)

friday 28 October

09:00-11:00 Rm 215: AUTO (Sieber/Wagenknecht)
11:30-13:30 Rm 139: Local Bifurcation (Lamb)
14:30-16:30 Rm 145: Global Bifurcation (Homburg)