| Instructor: Bill Briggs | Office: CU Building 611 |
| Phone: 556-4809 | Email: wbriggs@math.cudenver.edu |
| Textbook: Chaos, by Alligood, Sauer, and Yorke | Office hours: TTh 10:00-11:00, 12:30-2:00 in Science 130/132 or by arrangement |
| Web page: http://www-math.cudenver.edu/~wbriggs/chaoss99/top.html | |
Course Overview
The goal of this course is to survey the relatively new, rapidly
expanding,
and endlessly exciting field of nonlinear dynamics and chaos. Having
said
that, we might as well admit that such a goal is far too ambitious for a
single semester. The subject has already become too vast to fit into a
one-semester
package. So let's say we will do our best to survey the field.
One reason the subject is so broad is that it encompasses the entire
spectrum
from the theoretical to the applied. We will try to give a fair coverage
of both the pure and the applied aspects of the subject. There will
certainly
be some theorem proving to be done, but that will be balanced by many
applications
from fields such as physics, engineering, biology and ecology.
There are many good books that might have been used as a text. The book
we will use primarily is written by experts in the field; it also
has a good selection of topics, and it is written with many nice
teaching
and learning features. However, it is imperative that throughout the
semester
you use and explore other sources: textbooks, research monographs, web
sites,
and journal papers. This will become particularly important when it
comes
to projects. Please waste no time in visiting libraries and book stores
(including amazon.com) to find other sources that work for you. The text
book has an ample bibliography. A reading list will also be kept on the
course web page.
Topics
The following selection of topics is tentative and meant to provide a
general
framework for the course. As a topics course, the selection of topics
should
be determined in part by student interests. I would hope that perhaps
half
of what you learn is directly from the text and the class periods; the
other
half should be from projects, explorations, and your own reading.
1. Discrete Systems (Map and Difference Equations)
a. One-Dimensional Maps (Chapter 1)
b. Two-Dimensional Maps (Chapter 2)
c. Chaos in One-Dimensional Maps (Chapter 3)
d. Chaos in Two-Dimensional Maps (Chapter 5)
e. Chaotic (Strange) Attractors (Chapter 6)
2. Continuous Systems (Differential Equations)
a. Nonlinear ODEs (Chapter 7)
b. ODEs in the Plane (Chapter 8)
c. Chaos in ODEs (Chapter 9)
d. The Geometry of Manifolds (Chapter 10)
Computation
The subject of nonlinear dynamics necessarily requires both analytical
skills
and computation methods. You should have access to a computer on which
you
can write simple programs and obtain graphical output. The computer is
just
a tool for doing laboratory explorations. The language or software that
you use does not matter: you can write programs in C, PASCAL or even
BASIC;
you can use mathematical environments such as MATLAB or Mathematica; or
you can use ODE software. For the second half of the course, you should
write or have access to a numerical ODE solver.
Grading
The final grade in the course will be determined by
There will be approximately 6 problem sets consisting of problems at the end of the chapter, computer experiments in the chapter, and T-exercises in the chapter. Graduate students will have additional problems.
Undergraduates must do at least one and graduate students must do at least two Challenges at the end of each chapter.
It is not too soon to begin thinking about a topic for a term project. It must deal with a specific topic in nonlinear dynamics. It could be analytical or numerical in nature; it could be theoretical or applied; or it could a combination of all of the above. Otherwise, there are no other conditions. The actual paper should be on the order of 15-20 pages for undergraduates and 25-30 pages for graduate students. It should be carefully written and well-organized, preferably in LaTeX, and should contain figures, tables, numerical output as needed. The paper must include references (five minimum) and show evidence of library or web research. The paper itself needn't contain original work. Once again, start thinking, talking with me, and planning today!
A detailed outline with an annotated bibliography is due no later than March 15. The project is due no later than the last day of classes.
Drops and Incomplete
You have until the tenth week of classes to drop the course with only
the
instructor's (but not a Dean's) signature. The incomplete policy of the
Department and College is strictly enforced: incompletes are given only
in situations in which a student who has been in good standing all
semester,
is prevented from completing a course assignment (for example, the final
exam) by circumstances beyond his/her control (for example,
hospitalization,
death in the family).