Bifurcations inDynamical Systems with Applications

By Peter Kloeden, Johann Wolfgang Goethe Universität,Frankfurt, Germany
Print

Handling editor: Hinke Osinga

Bifurcations in Dynamical Systems with Applications

May 19 - May 21, 2008

Department of Mathematics, Bielefeld University
Logo of Bifurcations in Dynamical Systems with Applications
 

by Peter Kloeden, Johann Wolfgang Goethe Universität, Frankfurt

Bielefeld is a wonderful place to have a mathematics conference. Firstly, the University of Bielefeld has a very strong, well-funded mathematics department and their current Sonderforschungsbereich (a DFG-funded special research center) contributed generously to the costs of running this workshop. Another good reason is that, being located on the first high ground inland from the North Sea, it rains very often in Bielefeld, some say for at least 400 days a year, which eliminates many distractions. Indeed the university buildings all abut onto a 250-metre long covered mall-like hall, which even contains an indoor swimming pool for those who don't like to get wet when they swim. The third good reason for having a workshop there is that Wolf-Jürgen Beyn and Thorsten Hüls did a superb job organizing it. Indeed, due to their good connections, it did not rain at all during the meeting, not even during the excursion on the last afternoon.

Peter Kloeden and Wolf-Juergen Beyn at an evening party in Beyn's house; photograph by Andre Vanderbauwhede     Thorsten Huels during the excursion; photograph by Hinke Osinga
Peter Kloeden and Wolf-Jürgen Beyn at an evening party in Beyn's house; photograph by Andre Vanderbauwhede. Thorsten Hüls during the excursion; photograph by Hinke Osinga.

The workshop is part of a series that was started by Yuri Kuznetsov in Amsterdam in the early 90s. Over the past years woorkshops took place in Sevilla (May 2004), Ghent (June 2006) and Montreal (July 2007). And Yuri Kuzentzov announced that he would organise the next one in the Netherlands in 2010.

Participants of the workshop Bifurcations in Dynamical Systems with Applications; May 21, 2008
Participants of the workshop Bifurcations in Dynamical Systems with Applications; May 21, 2008.

Central topics in Bielefeld were the analysis of local and global bifurcations, as well as the development of numerical tools for their continuation in parameters. The aim was to review recent trends, focusing on constructive and computational aspects. Like the earlier workshops a hard core of AUTO fans formed the main contingent of participants and speakers, with Eusebius Doedel being there in person to make sure that AUTO continued to be treated with due reverence. But there were also a good number of non-AUTO topics to provide hints about theoretical developments and future challenges to numerical methods.

Making a first appearance in this workshop series were a number of talks on bifurcations and attractors in nonautonomous systems by Peter Kloeden, Christian Pötzsche and Martin Rasmussen. Nonautonomous bifurcations indeed present a real computational challenge since nobody quite knows what a nonautonomous bifurcation actually is. In fact, there is still no systematic theory of nonautonomous bifurcation, just examples and special cases, though there are now suitable concepts and formalisms of nonautonomous dynamical systems and their attractors which were presented in the talks. Since many important applications involve nonautonomous systems, we are sure to hear much more about this topic at future meetings and one can already envisage that AUTO will one day evolve into NonAUTO.

  Christian Poetzsche (Muenchen); photograph by Thorsten Huels
Christian Pötzsche (München); photograph by Thorsten Hüls.
Khashayar Pakdaman (Paris); photograph by Hinke Osinga  

There were also some very interesting talks on applications to biological systems. Indeed some of us thought we had at long last found the dream research area when Khashayar Pakdaman from Paris gave a talk on the dynamics of neuronal coding, which reported on experiments and data measurements on crayfish combined with modeling of dynamics and computations. However, we were brought back to reality when Khashayar told us that one is not allowed to eat the crayfish at the end of the day. His models involved delay differential equations with an asymptotically stable steady state solution, which meant that it is the transient solutions that explain the observed oscillatory behavior rather than the limiting steady state. Hinke Osinga spoke about bursting behavior in secretory pituitary cells, while Dirk Roose's talk was on parameter estimation in hyperbolic PDEs used to model virus population dynamics.

Khashayar Pakdaman (Paris); photograph by Hinke Osinga.

There were also some exciting developments in software for bifurcation analysis. Frank Schilder (Surrey) presented his initiative with Harry Dankowicz (Urbana-Champaign) to develop a standardized toolbox that is split into fully independent modules that enable development of bifurcation toolboxes on top of a core implementation of key tasks, such as continuation, bifurcation detection, and branch switching. He specifically focused on how the toolbox allows a generalization to the continuation of invariant tori with sophisticated mesh adaptation.

  Frank Schilder (Surrey); photograph by Hinke Osinga
Frank Schilder (Surrey); photograph by Hinke Osinga.
Virginie De Witte (Ghent); photograph by Hinke Osinga  

Often algorithms are being implemented and used even if a formal proof of convergence does not yet exist. Virginie De Witte discussed the convergence properties of the adjoint problem in the context of computing periodic orbits, where it is useful e.g. for finding the phase response curve. The AUTO fans were relieved to hear that yet again the convergence properties strongly depend on whether Gaussian points are chosen as collocation points.

The end of the Tuesday afternoon was devoted to a demo session, a tradition that gives participants an opportunity to demonstrate their software or techniques in informal small-group sessions. The relaxed atmosphere made it easy to wander around and get an impression of the latest news in computational methods, while the informal set-up allowed for direct questioning and intense scrutiny of the details.

Virginie De Witte (Ghent); photograph by Hinke Osinga.

Clare Lee (Bristol, background left) demonstrates singularities in high-dimensional Poincaré maps; photograph by Hinke Osinga   Jan Sieber (Aberdeen, foreground right) demonstrates parameter continuation directly in an experiment without using equations; photograph by Hinke Osinga
Clare Lee (Bristol, background left) demonstrates singularities in high-dimensional Poincaré maps and Jan Sieber (Aberdeen, foreground right) demonstrates parameter continuation directly in an experiment without using equations; photographs by Hinke Osinga.

The final talk by Bernd Krauskopf was perhaps at the wrong time in the program. It is certainly interesting to learn about wobbling nose wheels of planes and to see some very convincing simulations of how planes can go into a spin on the ground if they turn at too high a speed, but not just before one has to fly somewhere. Bernd reassured us that planes are a very safe means of transport, but nevertheless some of us were glad that we were going home by train.

  Bernd Krauskopf (Bristol); photograph by Thorsten Huels
Bernd Krauskopf (Bristol); photograph by Thorsten Hüls.
Herman the German; photograph by Andre Vanderbauwhede  

Before leaving, most of us first went on the excursion to the Teutoburger Wald and the statue of Herman the German. He is known as Hermann to locals and Arminius to the Romans, whom he and his followers defeated in a battle two thousand years ago in the vicinity --- meaning up to a hundred kilometers away if one believes the disputed claims of a good dozen local town mayors. Astute observers noted that this statue of Hermann with sword held high, which was built in the 1870s after the Franco-Prussian war, actually faces France, but felt it tactful not pursue the matter.

 
  Herman the German; photograph by Andre Vanderbauwhede.
Tags:

Please login or register to post comments.

Name:
Email:
Subject:
Message:
x