John Hogan,
Alan Champneys,
Mario di Bernardo,
Bernd Krauskopf,
Eddie Wilson,
Hinke Osinga,
Martin Homer
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Book of presentationsThe spirit of the meeting has been captured in the book Nonlinear Dynamics and Chaos: where do we go from here? edited by John Hogan, Alan Champneys, Bernd Krauskopf, Mario di Bernardo, Eddie Wilson, Hinke Osinga and Martin Homer, published by IoP Publishing, and available from amazon.co.uk, and all good bookshops! The book was reviewed in the May 2003 issue of UK Nonlinear News, and in the October 2003 issue of DSweb magazine. |
A programme for the meeting, and abstracts for all speakers' invited talks and contributed posters is available.
Lorenz's work in 1963 provided the first modern evidence that systems of nonlinear equations could behave in a manner that cannot be predicted. Of course, he was not the first to notice this sort of behaviour but the earlier mathematical work of Poincaré (1899), Birkhoff (1927) and others had not been taken seriously by a wide enough group of people. In addition, the use of computers by Lorenz was crucial to his discoveries, but ironically led to his work being less seriously considered by contemporary mathematicians than it otherwise should have been. It was only after the work of May (1976) on the logistic equation that significant numbers of researchers became interested in this field. Here was complicated behaviour in an equation of utter simplicity that could be reproduced on a pocket calculator.
Since then enormous mathematical strides have been taken and every field where mathematics has even a tangential involvement has been swamped by computations of nonlinear differential equations and maps. However, most of the existing theory and its relation to applications lies in systems of low dimension. This is significant because the theoretical underpinning is often geometric in nature and such systems lend themselves to representation in three dimensions. Most systems of practical importance are of much higher dimension. Chemical systems (governed by the law of mass action) can involve one hundred or more coupled nonlinear ordinary differential equations, realistic neural systems are several orders of magnitude larger and the partial differential equations involved in fluid mechanics are infinite dimensional (and even their numerical representation by finite element or finite difference codes has dimensions running into many thousands). In these areas theory advances understanding either by system reduction methods where some lower dimensional behaviour is found within the system (Carr 1981) or by symmetry assumptions (e.g. Armbruster & Chossat 1999). Other approaches usually lead to intense numerical computation. Nevertheless most systems do not possess the required symmetries, may have a dimensional manifold that is too big for simple representation, and numerical computation with a large number of free parameters seldom leads to understanding of the system under investigation.
It is this general situation we feel must be addressed, but it is not clear what the future problems should be, still less how to solve them. Our idea is to bring together theoreticians and experimentalists and to stimulate discussion with the next generation of nonlinear scientists to find an agenda for research directions of the future. Three themes have been selected, which seem particularly relevant to the future development of nonlinear mathematics
The local organisers are all members of the Applied Nonlinear Mathematics Group at Bristol, which is in the Department of Engineering Mathematics. It has risen rapidly since its formation in 1992 and has become a significant worldwide focus for applications of nonlinear mathematics (see, e.g. Hogan's review in SIAM Reviews 41 (1999) 375-382). The group was recently part of a successful bid for 15 million pounds of UK government funding to equip the Faculty of Engineering with the Bristol Laboratory for Advanced Dynamics Engineering - BLADE - to study all aspects of nonlinear dynamics in engineering.
The meeting took place in Burwalls, a dedicated University of Bristol conference centre set in its own grounds next to the famous Clifton Suspension Bridge. Its largest room holds 70 people giving us our maximum possible attendance, whilst at the same time still providing the intimate atmosphere that we feel the meeting required.
The 14 main speakers have been chosen for their role in the development of the applications of nonlinear mathematics. The remaining participants are intended to be among the next generation of researchers who will make a contribution to nonlinear science. With that in mind we prioritised applications from those who are current PhD students or who have obtained a PhD since 1990.
The structure of the meeting was talks of 45 minutes duration with lots of time for discussion; there were no parallel sessions. Each main speaker was invited to give a talk that reflects on what their field has learnt from the explosion of interest in nonlinear dynamics and what needs to be learnt. The other participants presented posters on their work. Special provisions were made in the programme to stimulate the interaction between participants and speakers, with lots of time set aside for discussion. The key idea was to reduce barriers and get people talking.