After leaving Austin I went to Caltech for a postdoc with Philip Saffman.
This introduced me to more traditional applied mathematics, as well as
introducing me to many members of our community (Tasso Kaper, Phil Holmes, and
others). I then moved to the Princeton Program in Applied and Computational
Mathematics to work with Yannis Kevrekidis and Steve Orszag.
Not long after, while on an NSF postdoc, I developed the first version of
EZ-Spiral, a code for simulating waves in excitable media that I still
maintain. My original motivation for releasing the code was that there were
competing (cellular automaton) models for fast simulations of waves in
excitable media. The only way I could think of to really compare simulations
was to run them side by side on the same hardware. So I packaged up my code
and sent it out to colleagues. It turned out to be very successful and
helpful to the community. I continue to have a strong interest in software for
research and teaching.
In December 1993 my wife, Laurette Tuckerman, and I moved permanently to
Europe, I at Warwick and she with the CNRS in Paris. We have been there ever
since.
And yes, I still commmute.
My main research interest for the past several years has been the study of
patterns in shear flows. These patterns are not the laminar ones that arise
through hydrodynamic instabilities, but rather patterns within highly
fluctuating flow fields in which turbulent and laminar flow self-organizes on
long length scales.
Initial Picture: Illustration of pattern formation within a turbulent flow. Simulation of plane Couette flow shows an oblique stripped pattern formed from turbulent and laminar flow midway between moving plates. (Work with Laurette Tuckerman).
This remarkable phenomenon was first reported in the 1960's, but only
relatively recently was the near-universal occurrence of such states within
parallel shear flows recognized.
From a fluid dynamics perspective, the problem is important because it is
intimately tied to the onset of turbulence. From an applied dynamics point of
view, the problem is fascinating because it raises many questions about how to
describe systems that clearly show patterns visually, but which involve highly
fluctuating fields. Understanding these patterns brings together many ideas
in bifurcation theory, pattern formation, and symmetry breaking, and also
techniques from statistical mechanics and non-equilibrium phase transitions.
My most recent work in this area has been on pipe flow, where I have been able
to identify the essential physical features and to produce qualitatively
reasonable models. What has most surprised me is to see just how much pipe
flow resembles excitable media. In fact, a recording of a standard flow
feature known as a puff could easily be mistaken for the recording of an
action potential. I recently released version 0.2 of EZ-Pipe.
A turbulent puff in pipe flow. Localized turbulence in a background of laminar flow propagates downstream. These states have a strong connection to action potentials in excitable media. (Collaboration with David Moxey at Warwick and Kerstin Avila, Marc Avila, Alberto de Lozar, and Björn Hof at Göttingen)
I have been asked to comment on what I think are the "important problems in
applied dynamical systems". This is a hard and perhaps dangerous question to
try to answer, and nominating particular areas as hot or especially important
strikes me as counter-productive.
Our field spans the range from proofs of basic theorems in dynamical systems
theory to large-scale numerical simulations and experiments. Borrowing from
Jim Yorke when handing out red socks -- all of these are important.
One of the main challenges is to keep ideas flowing both up and down from
basic theory through to application areas. This challenge will only increase
as more research in applied dynamical systems involves large collaborative
research projects. Using the area of cardiology as an example, there is no
fundamental reason why eventually there will not be physiologically accurate
computer simulations of whole hearts. Before then, and perhaps in the not too
distant future, computer simulations of more limited regions of the heart will
provide valuable information to practicing cardiologists. Dynamics is very
much at the core of this field. It has played, and will continue to play, a
central role in the understanding of heart arrhythmias. I suspect that as
waves in complex, moving tissue are explored, there will be more fundamental
issues for applied dynamics. This is merely one example that happens to be
familiar to me, but I think it is representative of where many future problems
in dynamical systems will come from as highly-complex real-world applications
are attacked.