Course Description:  Dynamical systems are generated by iterative, or repeated, mathematical processes.  As such, they have many applications in physics, biology, meteorology, and even finance.  For example, suppose you write a mathematical function that predicts the population growth of bacteria in a petri dish.  If you input the size of the population, the function outputs the number you expect to have one hour later.  Repeating this process allows you to predict the number of bacteria in the dish over the course of several days or weeks.  The list you generate of successive values is called a dynamical system. In some instances, we can predict exactly how a dynamical system will behave.  In others, called chaotic dynamical systems, the behavior is suprisingly unpredictable.  Finally, fractals are the beautiful, complex, and strangely organic mathematical objects that arise through the study of dynamical systems.

Prerequisite: Calculus III or instructor's permission.

Text: A course packet is available for purchase at the University Press.  You should read each section before the appropriate lecture.  The packet consists of selections from

• Robert L. Devaney, A First Course in Chaotic Dynamical Systems, Westview Press, 1992.
• Marc Frantz, Lessons in Mathematics and Art, Indiana University, 2000.
• H. Peitgen, H. Jurgens, and D. Saupe, Chaos and Fractals: New Frontiers of Science, Springer-Verlag, 1992.

Homework: Homework assignments will be given weekly and posted at http://www.sju.edu/~rhall/Chaos/homework.html.   You should start working on the homework problems for a section as soon as that section is covered in class.  Although you are encouraged to consult with other students and seek help from me, homework should ultimately be written in your own words. Answers unsupported by work will not receive credit.  Homework should be neatly handwritten (or typed) on one side of the page only and stapled.

Tests: There will be three tests, given on Thursday, September 30th, Thursday, November 11th, and during finals week (December 15th-21st). Makeup tests will only be given to students who contact me within 48 hours of missing a test.  Students with a valid, verifiable reason for missing a test may take a makeup without penalty;  those who have missed a test without a valid, verifiable reason may take a makeup with a 30% penalty.

Grades: Grades will be weighted as follows:

65%  three exams
30%  homework (lowest two grades dropped) and projects
  5%  class participation

The grading scale is 94-100% A, 90-93% A-, 87-89% B+, 84-86% B, 80-83% B-, 77-79% C+, 74-76% C, 70-73% C-, 67-69% D+, 60-66% D, and below 60% F. Grades may be curved at the end of the semester.

Academic Honesty: Dishonesty includes cheating on a test, falsifying data, misrepresenting the work of others as your own (plagiarism), and helping another student cheat or plagiarize. Academic dishonesty will result in a grade of zero on that particular assignment; serious or repeated infractions of the Academic Honesty policy will result in failure of the course. For complete information about the University's policy on Academic Honesty, consult the Student Handbook 2004-2005.

Attendance: Class attendance is mandatory.  Although I do not have a rigid cut policy, anyone who has missed lots of classes and is doing poorly in the course should not expect much sympathy from me.  If you do miss a class, it is your responsibility to make up the material and make sure your homework is turned in on time.

Schedule:
 

Weeks 1-4	Orbits, graphical analysis, fixed and periodic points, bifurcations
September 30th	First test
Weeks 6-9	The quadratic family, chaos, introduction to fractals, iterated function systems
November 18th	Second test
Weeks 11-14	The Julia set, the Mandelbrot set
December 15th-21st	Third test

Rachel W. Hall / Department of Math and Computer Science / Saint Joseph's University / rhall@sju.edu