The Dynamical Systems Track at ICIAM 2011 in Vancouver

By Evelyn Sander, George Mason University
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The Dynamical Systems Track at ICIAM 2011 in Vancouver

by Evelyn Sander, George Mason University

Figure 1. Vancouver Convention Center. (a) Exterior, and (b) interior regional artwork.

In July of 2011, the semi-annual ICIAM took place in Vancouver, British Columbia in Canada. Vancouver is a delightful place to hold an international conference in July, full of wonderful shops, restaurants, parks, totem poles, sushi, and more sushi. The convention center is located in downtown Vancouver, on the edge of Burrard Inlet (Figure 1), with many beautiful views and artworks in and around it -- such as a torch placed there during Vancouver's hosting of a certain non-mathematical conference in February of 2010 (see Figure 2a).


Figure 2. (a) The Vancouver Olympic torch. (b) Ducks in the waters off downtown Vancouver, courtesy of Ryan Lukeman.

As part of the mathematical program of ICIAM, the SIAM Activity Group on Dynamical Systems sponsored a track of minisymposia. Here is a brief set of highlights of the talks in this track, as well as one of the most dynamical of the plenary talks. There were plenty of other good dynamical systems talks at ICIAM, but to report on all of them from firsthand knowledge would require a much longer article written by an enoromous group of authors.

Collective Dynamics and Swarming

Organizer: Chad Topaz

This session described a number of recent results in mathematical biology on the subject of collective behaviors within a group of animals. The research for the first talk took place right next to the conference site. Ryan Lukeman took time lapse photos of the ducks in the Burrard Inlet. He used this to study the collective behavior of ducks as they dive in formation for fish. (See Figures 2b and 3.) Matching a swarming model to his duck diving data, Lukeman has shown that the social interaction between ducks contains an interaction preference for the ducks who are in the forward looking cone of vision. Following this, Razvan Fetecau gave a talk on some of the theoretical aspects of biological aggregation models. Next, Pawel Romanczuk used swarming models to study the pursuit and escape of a local vortex of crickets and locusts. Unlike the case of ducks, vision to the back is most critical in this case. Erik Fogelson moved the discussion from biological to robotic swarms, using numerical continuation methods to understand the types of collective behaviors possible in swarms of robotic individuals interacting via sensors.


Figure 3. How to get your ducks in a row (a) Ducks exhibit collective behavior in the waters off downtown Vancouver. (b-d) Ryan Lukeman studies their social interactions via swarming models. By converting head direction into a vector field, he has been able to show the dominant social interactions for the collective swimming and diving behavior. Courtesy of Ryan Lukeman.

The Neuromechanics of Insect Locomotion: How Cockroaches Run Fast and Stably Without Much Thought

A plenary talk by Phil Holmes

Phil Holmes gave a delightful plenary talk on a long term project of himself and his collaborators on cockroach locomotion. Unlike quadripedal animals, insect locomotion is quite consistent, in that at least while running, they do not use a variety of gaits. Thus it is reasonable to attempt a full model for insect locomotion. The group's results span the range from modeling to experiment, and from mechanics of the leg motion to the feedback mechanisms within the insect's brain. Holmes started with passive models for the purely mechanistic theory, which he justified as reasonable since "you can walk without a head - briefly." He then described active models for neuromechanical interactions directing the pattern of motion. The model must account for the experimental finding that there is a very short time scale of error correction when the animal gets thrown off course. This was found by attaching a small cannon to the back of a cockroach which fired ball bearings as it walked. (See movie in Figure 4a.) This compared favorably to the stable sets found in a model cockroach. (See movie in Figure 4b.)


Figure 4. Cockroach locomotion. (a) A cockroach with a firing cannon corrects its motion in a few strides. (b) A model simulation for error correction. The real roach appears in: D. Jindrich & Full, J Exp. Biol. 205, 2002, and the model roach in: R. Kukillaya, J. Proctor & P. Holmes, CHAOS 19, 2009. Movies courtesy of these authors.

Applied Topological Dynamics

Organizer: Evelyn Sander

This session described some of the recent applications of topological methods in dynamical systems. Jim Yorke described results using an epidemiological approach to study the HIV-AIDS outbreak. One particular question was: Which is more critical in the spread of the disease: the initial infectious period or the longer infectious period that occurs later? Jay Mireles James described a high order method of rigorous numerical computation for invariant manifolds and connecting orbits. Eric Kostelich reported on the dynamical concept of unstable dimension variability. He showed that such considerations have consequences in applications such as weather prediction. I ended the minisymposium with a description on recent results applying numerical continuation methods to materials science applications in order to understand the development certain phase separation patterns occuring in nucleation.

Dynamics of the Earth's Climate

Organizers: Hans Kaper, Mary Silber, Mary Lou Zeeman

This minisymposium features mathematics climate dynamics research, including some of members of the Mathematics and Climate Research Network (http://www.mathclimate.org/). Esther Widiasih described results on an extension of a mode of Budkyo for the temperature distribution of the earth by taking into account the ice albedo feedback. The extension allows for a dynamical change of the ice line. Jan Sieber spoke on a model for understanding bifurcations in a noisy slowly evolving system. This method has been applied to ice core data. Ka-Kit Tung talked about a model for the earth's temperature response to radiative forcing. Adam Monahan discussed a model for surface wind fluctuations, which could be used for example to estimate the wind power resource.

Stochastic Dynamics: Equilibrium and Non-equilibrium

Organizer: Grant Lythe

This session covered some modern trends in stochastic dynamical systems. Katja Lindenberg described the large qualitative differences on particles driven through a periodic potential when considering subdiffusion and superdiffusion rather than normal diffusion. Stefan Boettcher discussed an analysis of expermental data on cluster formation in colloids, showing that the data exhibits diffusion, but only on a logarithmic time scale. scale. Eli Ben-Naim discussed first passage properties of random walks in high dimensions. Grant Lythe ended the session with a talk on stochastic models for T-cells in the immune system, for which the system is not in equilibrium.

Dynamical Systems Approaches to Model Reduction

Organizers: Hans Kaper and Tasso Kaper

The subject of model reduction is clearly a critical one, represented by a total of three multi-part minisymposia at the ICIAM. In this two-part minisymposium, the focus is on how to use dynamical systems to study problems which involve widely disparate length and time scales. Clarence Rowley started off the session with results on the Complex Ginburg-Landau Equation, showing that for this equation, a nonlinear model reduction method as derived from the method of balanced truncation performs approximately an order of magnitude better than the more standard method of proper orthogonal decomposition. Marc Roussel then described a new way to compute using the ILDM method -- a popular method of model reduction based on reduction via invariant manifolds. Following this, Hans Kaper gave a nice overview of how to view many of the common model reduction methods in a unified framework, thus simplifying the proofs of their validity. In the afternoon, Christian Kuehn stood in for John Guckenheimer, presenting examples of numerical methods using geometric singular perturbation theory in the study of canards arising in FitzHugh-Nagumo and Morris-Lecar equations. Josh Mengers presented a numerical study of a reaction-diffusion equation using the SIM method. Samuel Paolucci talked about a space-time adaptive method. The session ended with a talk by Giovanni Samaey discussed a model for bacterial chemotaxis, in which bacteria search for food by always following the chemotactic gradient. In fact to study this at the smaller scale, one must take into account that each individual bacterium turns a lot before it gets to the food. This can thus either be studied using a reduction to a simple PDE, or via a stochastic simulation of a large number of particles.

Network Dynamics, Information, and Biology

Organizer: Eric Shea-Brown

The session aimed to trace the path from dynamics to biological computation in networks. First, Marty Golubitsky presented an new theorem relating a fundamental aspect of network dynamics -- "rigidly" persistent periodic orbits -- to the underlying network architecture. Taking a step towards information coding, Rachel Kuske described a number of recent results on stochastic oscillator networks, including where noise can have nonintuitive effects like decreasing the variability of the system output. Ilya Shmulevich then covered the tradeoff between stability and adaptability in computing networks, and mathematical notions of criticality at the intersection. Finally, Surya Ganguli revealed optimal network architectures for the encoding of information over long timescales, including a fascinating link to nonnormal connection matrices. Note: I was unable to attend this minisymposia, so thank you to Eric for this description of the minisymposium.

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