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.

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.