# Prize Spotlight: Martin Wechselberger

The SIAM Activity Group on Dynamical Systems (SIAG/DS) awards the J. D. Crawford Prize every two years to an individual for recent outstanding work on a topic in nonlinear science, as evidenced by a publication in English in a peer-reviewed journal within the four calendar years preceding the award year.

Wechselberger was recognized for breakthrough insights, highlighting the importance of canards and mixed-mode oscillations, that have application to neuroscience and open up new fields of research for dynamical systems with multiple time-scales.

Wechselberger is a professor in the School of Mathematics and Statistics at the University of Sydney, Australia. He earned his MSc and his PhD in the Department of Mathematics at the Vienna University of Technology, where he went on to hold a postdoctoral research fellowship. After a three-year postdoctoral research fellowship in the Mathematical Biosciences Institute at Ohio State University, he joined the faculty of the School of Mathematics and Statistics at the University of Sydney in 2005. His research focuses on nonlinear dynamics; in particular, on developing mathematical tools to understand the genesis of complex patterns and rhythms observed in physiology.

**Q:** *Why are you excited about winning the prize?*

**A:** I am extremely honored to receive this prize, which recognizes recent outstanding work within nonlinear science. I see this award as a broader recognition of the research on “canard theory and its applications,” which has become very popular over the years with many applications in the sciences. The success of canard theory is also reflected in the significant number of talks and minisymposia related to this exciting subject at this year's SIAM Conference on Applications of Dynamical Systems (DS17).

**Q:** *Could you tell us a bit about the research that won you the prize?*

**A:** My research is concerned with “canards” - peculiar creatures in slow-fast dynamical systems that have the ability to delay the relaxation mechanism from the slow to the fast dynamics. The term “canard” was introduced by a group of French mathematicians who studied 2D relaxation oscillators. There the classical canard phenomenon explains the very fast transition upon variation of a parameter from a small amplitude limit cycle via “canard cycles” to a large amplitude relaxation cycle. This very fast transition, called “canard explosion,” happens within an exponentially small range of a control parameter. Thus this phenomenon is very hard to detect (well, back when computer power was way less than that of today's smart phones!). So, it seemed to be more like a “canard,” a dubious newspaper report. Furthermore, the shape of a canard cycle in the 2D phase space resembles that of a duck, “canard” in French.

I focused originally on 3D problems where I developed an analytical framework in the field of geometric singular perturbation theory that proves the existence and bifurcation of canards. Combined with a suitable global return mechanism, this canard theory provides one now widely accepted explanation for complex oscillatory patterns known as mixed-mode oscillations. Such patterns correspond to switching between small amplitude oscillations and large relaxation oscillations and have been observed in a plethora of neural and physiological models. I also resolved the restriction of canard theory to lower dimensional multiple scales problems, i.e., I was able to extend canard theory to problems in arbitrary dimensions which have more than two slow variables.

**Q:** *What does your research mean to the public?*

**A:** The power of canard theory lies in its generality. It can be applied to a diverse range of research areas from mathematical physiology, fluid dynamics, magneto-hydrodynamics, and even climate modelling. For example, Peter Cox's Climate System Dynamics research group at the University of Exeter applied canard theory to explain the “compost-bomb instability” - a potentially catastrophic explosive release of peatland soil carbon into the atmosphere as the greenhouse gas carbon dioxide, which could significantly accelerate anthropogenic global warming [S. Wieczorek et al (2011), Proc. R. Soc. A 467, 1243-1269].

**Q:** *What does being a SIAM member mean to you?*

**A:**I really enjoy being a member of the SIAM Activity Group on Dynamical Systems (SIAG/DS), which provides ample opportunity for fostering collaboration and building an international research network. I have held several positions within SIAG/DS such as Portal-Editor-in-Chief of DSWeb, the electronic portal of SIAG/DS, which I highly recommend. More recently, I am a co-organizer (with Evelyn Sander of George Mason University) of the 2017 SIAM Conference on Applications of Dynamical Systems (DS17). The SIAG/DS conference is the leading biennial international dynamical systems conference. And, in 2017, we set a new record with almost 1000 participants. If you work in dynamical systems then get involved with SIAG/DS!