Undergraduate Research Project

Department of Mathematics
University of Arizona

Dynamical systems, chaos and control

The faculty involved in this project were Nick Ercolani, Lucas Hsu, Robert Indik, Joceline Lega, and Jerry Moloney. Kenton White, a physics graduate student working with Dr. Moloney, also helped guide the students in the computer laboratory. As a result of his efforts in this project, Kenton White was recognized as the Graduate Student Mentor of the Year by the Graduate and Professional Student Council at the University of Arizona.

The participants were three senior Mathematics majors, one junior Physics major, one sophomore Physics major and one sophomore Mathematics major. The sophomores became interested in this project through contacts with Mathematics faculty who had piqued the students' interest in mathematics in lower division Mathematics courses. It was possible to financially support all of the student participants. The research activities took place in the Arizona Center for Mathematical Sciences (ACMS) where involved faculty have offices and where conference rooms and computational facilities for the project are available.

Students' Report

The report from Adam Arluke, Chris Bergevin, Todd Cadwallader, Hyun-jeong Han and Robert Thompson is available by clicking on the picture below.

Project Organization

The organization of this project was based on a weekly 2 hour working session and an open ended weekly computer lab session. During the Fall semester students were divided into teams, exploring different versions of the basic laser model, each of which displayed a different dynamical character. Initially, in the working sessions, faculty introduced the dynamical systems models and explained how the model corresponded to the physical laser. The faculty introduced basic ideas and methods from dynamical systems and bifurcation theory. In the computer lab, primarily under the guidance of Kenton White, the students learned how to use a number of diagnostic software tools, such as DSTOOL and AUTO, to confirm and extend their analysis. As the semester progressed and the students developed their understanding of the projects, the working session transformed to a forum in which the students presented their results and engaged in an interactive discussion amongst themselves and faculty about problems and discoveries. Also they began to independently explore and use software tools to formulate and confirm conjectures. As a final project for the semester they wrote up up their results in LaTex.

Each student was assigned to one of the faculty participants who serves as the student's mentor. This is the person whom the student consults regularly and who assesses his progress.

The students came to this project with a wide range of backgrounds but by the end of the Fall semester almost all were able to identify fixed points or symmetry induced periodic orbits and assess their stability as well as to carry out a bifurcation analysis of these points and orbits as a parameter is varied. The students were encouraged to proceed at their own speed and also to learn from one another. Students really became quite independent in proposing ideas and conjectures to faculty and seeking out new tools. This is particularly the case in the computational environment. In this regard it is especially beneficial that the working sessions and computer lab took place in the ACMS environment where students could see faculty and graduate students actively pursuing research problems related to the ones in which they are participating.