A Next-Generation Scientist's Impression: Recent Trends in Nonlinear Dynamics

May 9, 2001

Mason A. Porter

I have been attending colloquia and conferences for only a couple of years, but in my examination of the topics my fellow dynamicists---both students and established professionals---are currently studying, I have nevertheless noticed several trends. Although I was barely conscious (or not even born!) during most of the evolution of nonlinear dynamics, I am interested in the historical development of the discipline. Without attempting to be exhaustive, I discuss here three current trends I consider important. Two of the areas are already significant components of the landscape, and interest in them does not show any signs of abating. The third area, while not as ubiquitous as the other two, is one I expect to become increasingly prevalent in the next several years.

Although I lack the experience of my mentors, I offer my views in the belief that the perspective of an observer on the periphery of a forest can usefully complement that of the trees within it. I hope that I can spark some interesting discussion by presenting them. Here is what I have noticed:

(1) As in other areas of applied mathematics, there is an exponentially increasing emphasis on biological applications. The problems in this influential subdiscipline of dynamical systems are diverse---they include work in theoretical neuroscience, physiological modeling, synchronization of biological oscillators, theoretical ecology, the dynamics of disease transmission, biomechanics, biological pattern formation, and a multitude of other areas.

The trend toward increasingly quantitative studies in the biological sciences is not new (on some time-scales, at least), but its importance continues to increase rapidly. If a random group of applied mathematicians were asked about the area of science in which they focus, "biology" might now be the most popular answer. Additionally, more and more scientific disciplines have massive groups within them studying biological problems. At Cornell University, for example, it would not be much of an exaggeration to say that there are (in the above sense) more than twenty biology departments! The increasing prominence of biology in applied mathematics is further reflected by SIAM's recent formation of a life sciences activity group.

Biological applications are ubiquitous for several reasons. One of the primary ones is the huge import of several problems in biology---from finding a cure for AIDS to preventing ventricular fibrillation. Additionally, because biology has become more quantitative, mathematical scientists can now feasibly study many of its deep and interesting problems. Another reason is that many (if not most) applied mathematicians seem to base their choice of applications mainly on their interest in the subject---not on the hope that their work will receive truly practical use. Applications in biology present new challenges to mathematical scientists because they have a different flavor from those in many subjects (such as mechanics) to which nonlinear dynamics has been applied historically. In many ways, biological modeling constitutes a mathematical Holy Grail! There are a multitude of problems awaiting study by mathematicians.

From a more practical perspective, the ever-increasing importance of biology has led to increased funding for research that incorporates it. Moreover, universities are seeking professors with experience in mathematical biology (of various stripes), and the job market has profound effects on the subjects that new graduate students elect to study. Finally, these individual effects exhibit a synergistic relationship and produce a self-perpetuating cycle of funding and research.

(2) A second area of nonlinear dynamics (or---more properly---of nonlinear science) that has emerged is the paradigm of complex systems (or "complex adaptive systems"). Such systems are thought to occur regularly in the physical, biological, and social sciences. Models within the realm of complex adaptive systems arise in a multitude of disciplines, including mathematical finance (so there are practical, monetary considerations in the study of this subject as well), condensed-matter and statistical physics (which motivated many of the associated concepts and techniques), neuroscience, geology, and evolutionary biology. Complex systems encompass such diverse phenomena as the group intelligence of ants and the dynamics of large arrays of coupled oscillators.

In addition to borrowing ideas from statistical physics, the study of complex adaptive systems uses techniques from such subjects as graph theory and probability, thereby making it more important for dynamicists to study those areas of mathematics. (Graph theory, for example, is relevant because many complex systems can be expressed in terms of networks.) In addition to the "new" mathematics and science that arise in complex systems, network models (of, for example, social networks) are important because they will lead to increased collaboration between mathematicians and social scientists.

(3) A final important trend is the increasing focus on what can be termed "the mathematics of scale." Technological advancements have led to the use of increasingly small devices and device components (for example, micro- and nano-electrical-mechanical devices, or MEMs and NEMs). To study many such devices, one must model their dynamics at multiple physical scales simultaneously. Devices at the nanoscale, for example, are sometimes describable as systems of coupled classical and quantum components.

Simultaneously including the quantum and classical scales in the modeling of dynamical systems is relatively new---it is not as pervasive as, for example, biological applications of nonlinear dynamics---but I expect multiscale modeling of this type (as well as others) to become increasingly important because of the ubiquity of small scales in present and future technology. Indeed, the Mathematical Sciences Research Institute in Berkeley has scheduled a program on semiclassical analysis for the spring of 2003. (Numerous semiclassical methods and regimes are applicable to models at small scales.) Moreover, these ideas are not restricted to systems that decompose into classical and quantum subsystems. One can, for example, consider phenomena in which discrete classical components (such as rigid bodies) are coupled to continuous media. More generally, models that exhibit behavior across three or more scales can be studied.

There will likely be a lot of funding available for studying the dynamics of systems at small scales. Moreover, there are many exciting mathematical and physical problems associated with these systems. At smaller scales, for example, systems can be exactly Hamiltonian up to the precision that can be reliably measured. Thus, degenerate bifurcations are very important in these situations. Additionally, one must consider couplings of regimes that are "untraditional." For example, since the quantum regime comes into play more often, one may need to analyze quantum systems coupled to continuous media, such as fluids or solids. A second aspect of the mathematics of scale involves self-similarity across spatial and temporal scales. Methods from statistical physics (such as renormalization) can often be used to analyze systems that exhibit such behavior. Related mathematical subjects (such as ergodic theory) may thus become more important in applications as well.

Mason A. Porter is a third-year PhD student in the Center for Applied Mathematics at Cornell University. His interests lie in nonlinear dynamics and its applications to physics (and occasionally other subjects).

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