• Instructor: Bruce Peckham, Assoc. Professor, Dept. of Mathematics and Statistics
• Office: 104 Solon Campus Center, 726-6188, bpeckham@d.umn.edu
• Office Hours: M 10:30-11:30, T 1:30-2, W 11-11:50, Th 1:30-3, F 10:30-11:30 or by appointment
  
  
• Meeting times: MWF 2-2:50 in Cina 102
  
  
• Course Prerequisites Math 8980 Graduate Seminar (usually taken by graduate students in their first semester). For Graduate students only.
  
  

Course description

1. Developing critical listening skills
2. Developing presentation skills, including facility with presentation software (such as Power Point)
3. Developing "rapid process" communication skills: listen to others' presentations, process information, ask intelligent questions
4. Developing independent critical thinking skills: decide for yourself what is interesting and important, and take the initiative to following up on such items
5. Devloping the ability to apply material learned in previous courses to other settings and applications
6. Developing appreciation for mathematics at many different levels

The course will be run as a series of seminars. Some topics may be presented in a single seminar, while others may be continued for several weeks. Each student will present at least four seminars. Student seminars will cover:

1. Plans for the student's masters' project (or thesis), background on the problem, and progress report on work done to date. (20 min; in the first 3 weeks)
2. Report on one scientific paper, preferably published in the last 5 year, preferably related to the student's masters' project. (50 min; scheduled sometime in the middle of the semester)
3. Progress report on project work to date. (50 min; sometime after the midpoint of the semester)
4. A practice run of the student's final project presentation, making full use of technology.

1. Real one-dimensional dynamical systems and Chaos
2. Complex dynamical systems: Julia sets and Mandelbrot sets
3. Legesgue measure
4. Lebesgue integration
5. Modelling with differential equations
6. Two-dimensional real dynamical systems: periodic points, stable and unstable manifolds, homoclinic tangles (chaos)
7. Bifurcation theory, including "continuation" algorithms
8. Fractal Geometry
9. Use of Software: download, install, run

1. Life without chaos is death An introduction to dynamical systems in one dimension.
2. The "Real" Mandelbrot set versus the "real" Mandelbrot set An introduction to complex dynamics.
3. The Chaos of multiplication by An introduction to both real and complex dynamical systems.
4. Modelling Peatland development with Differential Equations

1. Attend all seminars
2. Participate in seminar discussions
3. Make 4 presentations: three on project/thesis topic, one on a related scientific paper (see above)
4. Do readings and/or HW sets related to past or future presentations
5. Extend material from any seminar presentation (with instructor approval). Report will be written. Oral presentation may be given as well.
6. Make satisfactory progress on masters' project/thesis in order to make presentations in class.

   Grading (Tentative)

   Students will be evaluated on a combination of written homework assignments, computer experiments, projects, class preparation and participation, and oral presentations.

   More specifically:

   Class attendence:	 						10%
   Class presentations							40%
   Class participation							15%
   HW/Outside clas prep.			   				20%
   Outside class extension: student choice					15%
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   Total:								       100%
   

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   References:

   • CHAOS: Making a New Science by James Gleick, Penguin Books 1987.
   • ITERATION, by Choate, Devaney and Foster
   • Nonlinear Dynamics and Chaos, by Steven H. Strogatz
   • Chaos, Fractals, and Dynamics: Computer Experiments in Mathematics, by R. L. Devaney, Addison-Wesley, 1990. Introduction to discrete dynamical systems, aimed at high school level (but containing much more).
   • A First Course in Chaotic Dynamical Systems: Theory and Experiment by R. L. Devaney, Addison-Wesley, 1992. Text used for the old quarter course: Discrete Dynamical Systems (Math Q5695). Excellent introductory text for discrete dynamical systems, including complex dynamics (Mandelbrot set, Julia Sets).
   • An Introduction to Chaotic Dynamical Systems, second edition, by R. L. Devaney, Addison-Wesley, 1989. (Discrete dynamical systems. Intro graduate level text.)
   • A Toolkit of Dynamics Activities. A series of four interactive textbooks for Dynamical Systems published by Key Curriculum Press
     • Iteration, by Choate J, Devaney R L and Foster A, 1999
     • Fractals, by Choate J, Devaney R L and Foster A, 1999
     • Chaos, by Choate J and Devaney R L, 2000
     • The Mandelbrot and Julia Sets, by Devaney R L, 2000