Handling editor: Hinke Osinga
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.
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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.
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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.
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Christian Pötzsche
(München); photograph by Thorsten
Hüls. |
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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.
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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.
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Frank Schilder (Surrey);
photograph by Hinke Osinga. |
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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.
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Virginie De Witte (Ghent);
photograph by Hinke Osinga. |
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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.
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Bernd Krauskopf (Bristol); photograph by Thorsten Hüls. |
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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.
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Herman
the German; photograph by Andre
Vanderbauwhede. |