DSWeb Tutorials Competition '07 By Evelyn Sander George Mason University

This year DSWeb ran a student competition for tutorials on dynamical systems. The first such contest took place in 2005 (as reported in the July, 2005 issue of this magazine). Prizes were awarded to Christian Kuehn of Cornell University and John Wordsworth of Exeter University. Both winning entries are published in the DSWeb Tutorials section. Kuehn and Wordsworth received a cash prize and a travel award to attend the SIAM Conference on Applied Dynamical Systems in Snowbird, Utah. Dwight Barkley, Charlie Doering, Bard Ermentrout, Mary Silber, and Mary Lou Zeeman served as judges.

Evelyn Sander gives prizes to John Wordsworth (middle) and Christian Kuehn (right) at the DSWeb Student Competition awards ceremony at the SIAM Conference on Applied Dynamical Systems in Snowbird, Utah, May, 2007; photo by Hinke Osinga.

Christian Kuehn's tutorial An Introduction to Rotation Theory is an excellent supplement for a basic dynamical systems course for mathematics students. The text includes exercises with hints and answers, as well as animated images to make this work as a self-paced tutorial for a beginning student. The topics start out quite simply with an introduction to circle maps and rotation numbers. However, the discussion progresses quickly to quite advanced topics, such as the relationship between rotation numbers and ergodic theory, omega-limit sets, and wandering intervals. There is also a discussion of Denjoy theory for irrational rotation numbers, followed by an introduction to Arnol'd tongues for families of diffeomorphisms. Amazingly, such sophisticated topics are presented without making the tutorial unreadable for a beginning student with an undergraduate analysis background. The tutorial contains many references, including links to other online material.

An animated image from Christian Kuehn's tutorial illustrates the rotation number for circle maps.

John Wordsworth's An Introduction to Coupled Oscillators: Exploring the Kuramoto Model is a good introduction to coupled oscillators for novice students in dynamical systems. It does not assume any analysis background, which would make it well suited to any science students. It gives a derivation of the Kuramoto model, a discussion of the phase response curve, the nature of the Hopf bifurcations occuring for weakly coupled oscillators, and a description of synchronization that occurs at a critical parameter of coupling. There are many graphical images to clarify explanations, but the most compelling piece is an excellent interactive Kuramoto simulator for experimentation, with suggestions of interesting experiments to try. The tutorial ends with further suggestions for modifications, as well as a set of references.

John Wordsworth's tutorial includes an interative simulation of a Kuromoto oscillator.

This contest was limited to graduate students. However, DSWeb encourages contributions to the Tutorials Section from all members of the dynamical systems community. A link on DSWeb spreads the word about your interactive tutorial. It is also a good way to share lecture notes or talks with the rest of the community.