DSWeb Tutorials Competition By Evelyn Sander George Mason University

This spring, DSWeb was pleased to sponsor the first graduate student competition for dynamical systems tutorials. Robert Devaney, John Guckenheimer, Jerrold Marsden, Björn Sandstede, and Steven Strogatz served as judges. Winners and runner-ups received cash prizes and now appear in the Tutorials section of DSWeb. The contest was a great success. It gave students an opportunity and incentive to hone their expository skills and be judged by leading expositors in dynamical systems. The winners were recognized at an awards ceremony at the SIAM Dynamical Systems Meeting in Snowbird, Utah. In addition, the winning entries are a fun, interesting, and informative set of tutorials which take advantage of the web as an interactive educational tool. Each would be an excellent supplement to a dynamical systems or differential equations course.

 

Tutorials Section Chief Editor Evelyn Sander and contest winners Sam Reid, David Winterbottom, and Shawn Shadden at the awards ceremony at the SIAM Dynamical Systems Meeting in Snowbird, Utah. Photo by Bernd Krauskopf.

 

The first prize went to Sam Reid of Department of Computer Science University of Colorado at Boulder for his entry Elementary Cellular Automata as Dynamical Systems. The tutorial teaches dynamical concepts using cellular automata. It is carefully constructed - a model of how to teach using the web. Concepts are introduced slowly and reinforced using teaching tools such as thoughtful quizzes, detailed help buttons, and illustrative use of color. Any undergraduate mathematics student would find it a joy to read. Reid explained "I am currently a student of computer science, but cellular automata have been a long-time hobby since my days as a physics major. I really enjoyed working on this tutorial. I am very thankful to DSWeb for hosting this contest and acknowledging my work."

David Winterbottom of the School of Mathematical Sciences, University of Nottingham, UK took second place with The complex Ginzburg-Landau equation. The tutorial is a comprehensive introduction to this famous family of partial differential equations. The many different behaviors seen - such as stationary patterns, oscillations, waves, spirals, and turbulence - are described and illustrated using pictures and movies. Historical background and applications are also given. Says Winterbottom, "My advisor pointed this contest out to me since I like to write for the web and use java. It turned out to be a lot of work, but it was also a lot of fun."

 

Sam Reid's winning entry uses interactive Java as a educational tool for explaining the dynamics of cellular automata. (Click on each figure to link directly to the contest entry.)		The second place tutorial by David Winterbottom uses figures and movies to give a broad explanation of the Complex Ginzburg-Landau equations. This figure shows a spatiotemporal plot of defect-mediated turbulence.

 

Third prize winner Shawn Shadden, Control and Dynamical Systems, Caltech, described a dynamical tool for analysis of experimental data in Lagrangian Coherent Structures: Analysis of time-dependent dynamical systems using finite-time Lyapunov exponents. The tutorial gives an in depth look at the use of finite-time Lyapunov exponents in the study of systems with arbitrary time dependence. Shadden explains how to use finite-time Lyapunov exponents in the context of fluid flow to find time-dependent separatrices known as Lagrangian coherent structures. Using both images and animations, the theoretical techniques are illustrated through analysis of oceanographic and aerodynamic data.

Runner-up Brian Bockelman, Department of Mathematics University of Nebraska-Lincoln, wrote the tutorial Dynamic Equations on the Sierpinski Gasket. This comprehensive introduction to harmonic analysis on the Sierpinski gasket presents an unusual but illuminating point of view for differentiation. His writing style is refreshingly light, making the material quite accessible to any mathematical reader. However, the tutorial is by no means elementary - Bockelman has developed a sophisticated software package for Sierpinski gasket spline and finite element methods and illustrates the use of Green's functions, calculation of bi- and tri-harmonic functions and eigenvalues and eigenfunctions. Thus readers can use the package for further experimentation.

Honorable mentions went to Sebastian Marotta, Boston University, for Mathematica notebooks for Iterated Function Systems, and Gouhei Tanaka, Department of Complexity Science and Engineering University of Tokyo, Japan, for Crisis-induced Intermittency in Coupled Chaotic Maps.

 

Shawn Shadden's third place entry is an in-depth tutorial on finite time Lyapunov exponents to compute dynamics of real data. This figure is a separatrix for ocean currents in Monterey Bay, CA.		Runner-up Brian Bockelman's Tutorial on harmonic functions on the Sierpinski Gasket, such as the one shown here. It is a witty self-contained introduction to the subject with sophisticated interactive computational tools for experimentation.

 

Both the awards ceremony in Snowbird and this article end with the following invitation: This contest was limited to graduate students. However, DSWeb encourages Tutorials contributions from all members of the dynamical systems community. For example, a link from DSWeb spreads the word about your interactive tutorial. It is also a good way to share typed lecture notes with the rest of the community.