Handling editor: Chad Topaz

Dynamical Systems at the 2008 SIAM Annual Meeting

Jeff Moehlis, Program Director of SIAG/DS and Department of Mechanical Engineering, University of California, Santa Barbara

The 2008 SIAM Annual Meeting was held from July 7-11 at the Town and Country Resort Hotel and Convention Center in San Diego, California. Here are some reflections on this conference that I have written to share with members of SIAG/DS and the dynamical systems community at large.

Figure 1. Marty Golubitsky with Margot Gerritsen, Chris Budd, Andy Wathen, and Russell Davies at the reception following Golubitsky's Past President's Address on the WhyDoMath multimedia website [1]. Photo courtesy of Carol Mehne.

Applications

The scientific program highlighted several application areas in which there are opportunities for increased contributions from the dynamical systems community. One such area is weather and climate modeling, which was featured in Invited Presentations by Cecile Penland from the National Oceanic and Atmospheric Administration and John Boyd from the University of Michigan. Climate modeling and dynamical systems research have an interesting historical relationship. For instance, the celebrated Lorenz equations, arguably the prototypical chaotic system, came from Edward Lorenz's attempt to model atmospheric dynamics. The study of these equations has contributed to a framework for understanding sensitive dependence of initial conditions for weather prediction and many other systems. As another example, stochastic resonance was introduced by Roberto Benzi and coworkers as a mechanism for the roughly periodic occurrence of Ice Ages.

In Cecile Penland's talk [2], she discussed El Nino, the temperature fluctuations in the surface water of the Eastern Pacific Ocean that are associated with extreme weather in the United States. Penland argued that the onset of El Nino can be predicted reasonably well by an empirical model consisting of a stochastically forced non-normal linear system that can display transient growth. While it might seem surprising that such a relatively simple linear model works so well, Penland points out that "El Nino is a combination of so many highly chaotic nonlinear systems varying on a widely diverse range of timescales that stochastic limit theorems (Khas'minskii 1966; Papanicolaou and Kohler 1974) apply and they are indistinguishable from stochastic forcing. That is, the system is so nonlinear that only the slowly-varying, linear part of the system can be predicted on timescales larger than about a week." Penland goes on to suggest that "What we really need are broader classes of limit theorems that show under what conditions combinations of chaotic systems converge to which classes of stochastic systems. Does synchronized chaos lead to correlated additive noise at all, must it be a multiplicative noise in some limiting dynamics, or what? Can the 'chaotic hypothesis' of Gallavotti and Cohen be given a firm mathematical basis? Do we really need an attractor to be Anasov, or Axiom A, to support Markov partitions? Is it possible to perturb externally a system lying on a non-Anasov attractor so that Markov partitions can be supported on the perturbed system? If so, how? If not, will an approximation do? Can any of these structures converge to Lèvy processes? Under what conditions?"

In John Boyd's talk [3], he explained that ensemble forecasting "is an attempt to compute some dynamical systems measures from real forecasting models, such as the Lyapunov exponent. The ensemble is seeded with singular vectors as perturbations which represent the directions of fastest growth of perturbations in the billion-dimensional phase space of the model." He also highlighted that an important, but underappreciated, difficulty with weather forecasting is initializing computer models to be consistent with the slow manifold corresponding to the weather's evolution. This is an issue of "compatibility conditions," which are relations between the governing partial differential equations, boundary conditions, and initial conditions that need to be satisfied to get well-behaved numerical results. After his talk, he mentioned (personal communication) that the dynamical systems community might be able to develop more rigorous techniques for identifying when, where, and for how long atmospheric blocking (a stagnation of weather patterns) will occur.

Of course, researchers in the dynamical systems community are already working on these and related problems in climate modeling, and are making crucial contributions. Nonetheless, it may be time for a stronger effort from our community to address these important problems, especially considering growing concerns about global warming. On this note, one of the highlighted themes of the upcoming 2009 SIAM Conference on Applications of Dynamical Systems (at Snowbird) [4] will be climate modeling, and Henk Dijkstra from Utrecht University [5] will be giving an invited presentation on this subject.

The global climate affects us all. Another phenomenon that touches our everyday lives was also mentioned frequently at the Annual Meeting: the Internet. The scientific program included talks on subjects ranging from search algorithms to the analysis of statistical properties of nodes and links in the network that is the Internet. The Internet-related talks focused heavily on network structure and not on dynamics. Nonetheless, the Internet is a dynamical system, and as stated by Jon Kleinberg from Cornell University [6], who gave an Invited Presentation on social and information networks, "the dynamical systems community has a number of powerful mathematical techniques at its disposal that could contribute to this area." This work could include a consideration of the dynamics of the spread of information through the Internet, the dynamics of group and community structure on social sites (which has been studied in the computer science community, but less from a mathematical perspective) and how the Internet is affected by topological changes such as the addition and subtraction of nodes and links.

Figure 2. Sketch of the interplanetary superhighway from Shane Ross' WhyDoMath entry on invariant manifolds on the Why Do Math website [1]. shown during Marty Golubitsky's past President's Address. Artist: Cici Koenig (image obtained from images.google.com).

Why Do Math

Marty Golubitsky gave the Past President's Address (see Figure 1) on a multimedia website [1] that he and others are developing to highlight exciting contributions that applied mathematics and computational scientists have made in science, society, and everyday life. This website features articles aimed at the undergraduate freshman/sophomore level, and strives to convey the excitement and importance of the research that our broad community does. Several articles can be viewed on the development page http://dev.whydomath.org. Two of these articles have a clear dynamical systems flavor, namely one by Shane Ross on the interplanetary superhighway, shown in Figure 2 (related to invariant manifolds) and one by Brent Doiron and Eric Shea-Brown on mathematical neuroscience. Golubitsky made a plea for help with this important effort. If you have a mathematics success story (one that has already come to fruition), are an excellent writer, have web skills, and/or have suggestions for articles for the Reading Room on the WhyDoMath site, contact [email protected].

SIAM Fellows

A major topic of discussion at the SIAM Business Meeting at the conference was whether or not SIAM should institute a Fellows program [7]. Many readers may already be familiar with the idea of "fellows" from other professional societies. Various pros and cons regarding a Fellows Program have been articulated, and should be carefully considered. Visit http://fellows.siam.org to find more information. In particular, the Fellows FAQ and The Proposal links may be of particular interest. Voting on this issue is currently underway.

Figure 3. Mathieu Desroches, a graduate student at the University Bristol, discusses numerical computations of canards and mixed-mode oscillations for a self-coupled FitzHugh-Nagumo system. Photo courtesy of Jeff Moehlis.

Final Thoughts

Other dynamical systems highlights at the meeting included (but were not limited to) Bard Ermentrout's SIAG/DS-sponsored talk called "The Olfaction Way" (a play on the title of the W.C. Fields movie "The Old Fashioned Way") as well as minisymposia on neuroscience, device applications of nonlinear dynamics, and computational techniques, shown in Figure 3. There was also a successful minisymposium on undergraduate dynamical systems research organized by Chad Topaz and Andy Bernoff. In conclusion, the dynamical systems community was well represented at the Annual Meeting, and that it is clear that we will have plenty of interesting problems to think about in the immediate future.

References

[1]		Why Do Math, http://www.whydomath.org and http://dev.whydomath.org
[2]		Cecile Penland, http://www.cdc.noaa.gov/people/cecile.penland/
[3]		John Boyd, http://aoss.engin.umich.edu/people/jpboyd
[4]		SIAM Meeting on Applied Dynamical Systems, http://www.siam.org/meetings/ds09/, Snowbird Utah, 2009.
[5]		Henk Dijkstra, http://www.phys.uu.nl/~dijkstra/
[6]		Jon Kleiberg, http://www.cs.cornell.edu/home/kleinber/
[7]		SIAM Fellows, http://fellows.siam.org
[8]		A full list of talks is available at the SIAM Annual Meeting website, http://www.siam.org/meetings/an08/