# Chaos and Complexity Courses for Spring 1998, UW-Madison

Math 491 --- Applied Dynamical Systems, Prof. P. Milewski
Prereq: Math 319, or Math 320, or consent of instructor.

9:30--10:45 TuTh in B105 Van Vleck, 3 credits

Text: S. H. Strogatz, Nonlinear Dynamics and Chaos

An upper level undergraduate course that introduces students to the study of nonlinear dynamical systems. The course will be taught from an applied mathematics standpoint, with emphasis given to the applications of the theory.

Although the course is taught at the 400 level, I have had graduate students from engineering, the physical sciences and mathematics that have benefited from the course.

Course description:

1. One-dimensional maps and difference equations: linear and nonlinear problems, graphical solutions, bifurcations, chaos.
2. First-order differential equations (one-dimensional flows): linear and nonlinear equations, graphical solution, bifurcations.
3. Two-dimensional flows: phase plane, stability of fixed points, periodic solutions and limit cycles; Introduction to bifurcation theory, local and global bifurcations, Tools for studying global behavior of flows: Lyapunov functions, Poincare-Bendixson Theorem, gradient flows.
4. Three-dimensional flows: Lyapunov exponents, Poincare sections, strange attractors, chaos.
5. Each of the topics will be discussed with references to applications in Mechanics, Population Dynamics, Biological Oscillators, Neurophysiological Models, Chemical Oscillators, and others. Computers will be used for mathematical experimentation and to aid in visualising solutions.

Math 812 --- Computational Neuroscience (``Topics in Applied Mathematics''), Prof. Amir Assadi
Prereq: Consent of instructor.

11-12:15 TR in B203 Van Vleck, 3 credits

Texts: Dana Ballard, An Introduction to Natural Computation (MIT Press 1997), Bill Bialek et al., Spikes (MIT Press 1997).
Recommended Texts: William Klemm, Understanding Neuroscience (Mosby's Biomedical Science Series 1996), R. Beale and T. Jackson, Neural Computing, An Introduction (Institute of Physics Publishing 1992 or the latest edition)

This course attempts to introduce a few topics in computational neuroscience, an emerging multi-disciplinary field whose progress is tied to developments in high performance computing as well as rapid progress in several fields of neuroscience and cognitive psychology. My purpose is to work with the students to explore projects in modeling that can be approached with a combination of some mathematical techniques (e.g. from differential equations, probability, geometry, ...) and basic understanding of scientific computation (e.g. basic familiarity with a programming environment such as MATLAB for visualization and numerical simulation). The final product will be a computational model that attempts to explain (approximately) an experiment in neuroscience, e.g. in vision and neurophysiology, or to propose an experiment based on theoretical conclusions that are suggested by the computational model. Needless to say, the students will learn some neuroscience, as well as how to work in a computational environment. The science part will be sometimes lectured by guest lecturers, and the mathematical lectures will have hands-on lab sessions when needed. The grade will be based on a term project that will be carried out by a team of 2-3 students.

Students from outside of mathematics are welcome. Students without advanced knowledge of mathematics are welcome, but they should have background in a complementary area, such as computer programming, biology, or engineering. Mathematics students are not required to know neuroscience, but to learn it simultanously. The main topics will be: computation in artificial and neural networks, image science (topics from vision and medical imaging) and topics in models of olfaction. Please contact me for more information.

Related Courses: There are two courses that will have considerable overlap with the neuroscience aspects and data aquisition and processing:

1. Neuroscience 675: Current concepts in higher cortical function. Instructor: Professor Lewis Haberly (lhaberly@facstaff.wisc.edu, 147 Bardeen, Tel. 262-7918.) Please contact The Neuroscience Training Program and Professor Haberly for schedule, lecture room and other information.)
2. Medical Physics 568. Magnetic Resonance Imaging. Instructor: Professor James Sorenson. MWF 2:25pm 5275 Med. Sci. Center. Please contact the Medical Physics Department for other information.

Math 821 --- Introduction to Cellular Automata and Complex Random Systems, Prof. David Griffeath
Prereq: Consent of instructor.

9:55 MWF in B223 Van Vleck, 3 credits

We will explore theoretical and experimental approaches to the study of cellular automata and other spatially distributed complex systems. The course will be interdisciplinary in spirit, stressing the use of CA dynamics, deterministic and probabilistic, as paradigms for physical phemomena from across the spectrum of applied science. Included among the models covered will be lattice gases, percolation through porous media, self-organization in excitable media, and spatial distribution of competing species.

Where possible, we will develop techniques for the rigorous mathematical analysis of such models. Tools include path representations, convex analysis, contour methods, mean-field approximation, and large deviations. But the course will also stress computer simulation and dynamic, interactive visualization. Demonstrations and tutorials will make extensive use of our web site, the Primordial Soup Kitchen, our WinCA experimentation platform, and other Java-based CA software currently being developed under the auspices of a UW Web Grant.

Little advanced (graduate-level) mathematical background will be required. However a basic familiarity with aspects of probability theory, combinatorics, differential equations, and/or computer algorithms at the advanced undergraduate level is essential.

There will be no formal textbook. Reading materials for the course, which will be distributed, are likely to include:

• Rick Durrett's ``St. Flour Lecture Notes,'' and his Annals of Probability Special Invited Paper on ``Oriented Percolation'';
• excerpts from Cellular Automaton Machines by Toffoli and Margolus, MIT Press, 1987;
• Rudy Rucker's ``Introduction to Cellular Automata'' (text bundled with his CA-Lab software);
• Several of my recent papers with J. Gravner, on Excitable CA dynamics and CA shape theory.
If interested in this seminar, please REGISTER at your earliest convenience and let me know by email that you plan to attend. We will need at least 10 registered students in order for the course to run. Based on initial feedback, it appears that there is sufficient interest.

Any questions, or requests for additional information, should be sent by email to griffeat@math.wisc.edu.

Physics 724 --- Plasma Waves and Instabilities, Prof. Stewart Prager
Prereq: NEEP/ECE/Physics 525 & Physics 721 or ECE or consent of instructor

9:30 TR in 1327 Sterling, 3 credits

Perhaps the most salient feature of plasmas is that they support a dazzling variety of waves, many of which grow spontaneously (instabilities). Such phenomena determine, in part, the heating of the solar corona, the confinement of plasmas in magnetic fusion experiments, the acceleration of cosmic rays, the stability of plasmas for materials processing, etc. We will develop the fundamentals of plasma waves --- driven and spontaneous --- and will focus on applications to laboratory and astrophysical plasmas. The applications will be guided by the interests of the students. Topics include:

• Waves in Uniform Plasmas --- the dielectric tensor
• Wave Zoology --- Alfven waves, cyclotron waves, drift waves, hybrid waves . . .
• Resonance Phenomena
• Hot Plasma Effects --- Landau damping, cyclotron damping
• Linear Microinstabilities
• MHD Instabilities --- energy principle, magnetic reconnection
• Quasilinear Saturation
• Nonlinear Waves --- solitons, wave coupling, stochasticity

(Fri Dec 12 1997)