Graduate students are most often concerned with the jobs they might end up with after graduation. Opportunities vary from discipline to discipline, but in the area of dynamical systems choices are mainly oriented towards careers in academia. University jobs are attractive. They give scientists a lot of freedom in choosing their research projects – from modeling Bose-Einstein condensates [1] to studying the dynamics of wooden toys and Lego blocks [2]. However, for a scientific discipline to have a long term future, it must have an applied side. It must deliver results that have a short term impact on the lives of ordinary people. A demand for a particular scientific skill set on the job market is usually a good indicator of how well that scientific discipline fares in the applied world. In that respect, dynamical systems theory could be doing much better.

I work for the United Technologies Research Center, one of the few industrial research institutions that employ dynamical systems experts to advance knowledge in dynamical systems theory and not to simply provide “service” for other fields. Let me share some of my experiences. I hope to initiate a discussion on how to steer dynamical systems research so it can find more applications in industry.

Dynamical systems theory is a relatively new discipline. It has not had enough time to make sufficient impact to gain wide name recognition in industry. There have been some spectacular success stories (e.g. [3,4]). These, however, have not been sufficient to push dynamical systems into the mainstream of industrial research.

In industrial research, dynamical systems expertise alone is not enough – it has to be accompanied with a proficiency in the engineering disciplines where it is applied. I still remember my first meeting after I started working in industry. My colleagues explained the problems they were working on and wanted my opinion. My immediate response was to ask for the governing equations. Everybody in the room chuckled – it was a rookie mistake. In industry, dynamical systems experts are pretty much on their own. They have to familiarize themselves with the engineering problem of the day well enough to be able to produce the governing equations and to recognize when solutions to these equations make no physical sense. What makes this task more difficult is that engineers use computer aided design (CAD) tools to design their systems and typically do not deal with mathematical equations directly.

Furthermore, engineers are often reluctant to invest their time and effort in learning dynamical systems theory. After all, they mastered methodologies that served the engineering community well for decades. The burden of proof is on us in the dynamical systems community. We have to show that dynamical systems theory can bring significant improvements in engineering design. To do so, we have to gain expertise in areas where we want to apply our skills. This is a tough spot to be in, but it comes with the status of newcomer.

Making excursions into areas of engineering may be beneficial for dynamical systems researchers in academia, as well. Even simple devices such as electrical machines [5] and ejector pumps [6] exhibit intricate dynamics and may provide a rich source of graduate student projects. Studying these problems is a little more difficult than, say, studying van der Pol oscillator networks or the Lorenz system, but it is far more rewarding. The problems arise naturally in industrial research and results can be easily tested. If promising, they can even be implemented in real products. Students working on such projects may find more career opportunities opening up.

Anyone willing to pursue industrial research has to be ready to face some serious “language barriers”. They may be as simple as using the symbol “p” in electrical machinery literature to denote time derivative instead of commonly accepted “d/dt”. Others run deeper. Some areas of engineering are even abandoning the use of traditional mathematical language altogether. There, dynamical systems are modeled as Simulink [7] diagrams, and mathematical equations are used as auxiliary tools to help chart the resulting diagrams. This approach is introduced early at the undergraduate level in engineering education, and it has already become the primary means of communication in a number of engineering disciplines (see e.g. [8]). At first glance this may seem sacrilegious, but one should remember that graphical representations of mathematical models have been used successfully in the past. Perhaps the most notable are the descriptions of path integrals using Feynman diagrams.

I am not suggesting we should do all our research using Simulink. After all this is an opaque, proprietary tool that may not fit everybody’s budget or technical needs. But, we may need a comprehensive, hierarchical graphical way of representing dynamical systems. This is particularly important in complex systems analysis. Systems that I encounter in my work consist of thousands of differential algebraic equations. Without graphical tools that allow one to see the coarse structure of the system and then to scan down and reveal its detailed structure, dealing with such systems would be a futile exercise. The dynamical systems community can help define standards and methods for graphical representation and analysis of complex dynamical systems. The coupled cell approach introduced by Golubitsky and Stewart [9] is a step in that direction.

These are just a few of many opportunities for dynamical systems theorists to leave their mark on current technological development. It is the dynamical systems community that has to reach out and demonstrate what it can contribute.

References:
[1] Porter, Mason A. and Cvitanovic, Predrag [2004]. Modulated Amplitude Waves in Bose-Einstein Condensates. Physical Review E, Vol. 69, No. 4: 047201.
[2] Leine, R.I., Van Campen, D.H and Glocker, Ch., "Nonlinear dynamics and modelling of various wooden toys with impact and friction", Journal of Vibration and Control, Vol. 9, pp. 25-78, 2003.
[3] Prashant G. Mehta, Gregory Hagen and Andrzej Banaszuk, “Symmetry and Symmetry-Breaking for a Wave Equation with Feedback” SIAM J. Applied Dynamical Systems Vol. 6, No. 3, pp. 549–575, 2007.
[4] B Eisenhower, G Hagen, A Banaszuk, I Mezić, “Passive Control of Limit Cycle Oscillations in a Thermoacoustic System Using Asymmetry”, J. Appl. Mech. Volume 75, Issue 1, 011021, 2008.
[5] P.C. Krause, O. Wasynczuk and S.D. Sudhoff, “Analysis of Electric Machinery and Drive Systems”, IEEE Press 2002.
[6] R. Yapici and H. K. Ersoy, Energy Conversion and Management 46 pp. 3117-3135 (2005).
[7] Simulink © 1994-2010 The MathWorks, Inc. http://www.mathworks.com/products/simulink/
[8] A. Veltman, D.W.J. Puelle and R.W. Doncker, “Fundamentals of Electrical Drives”, Springer 2007.
[9] M. Golubitsky and I. Stewart. "Nonlinear dynamics of networks: the groupoid formalism," Bull. Amer. Math. Soc. Vol. 43, No. 3, pp. 305-364, 2006.