MA 771: Discrete Dynamical Systems, Spring 2007

M-W 2-3:30 PM

The aim of this course is to introduce mathematicians to current research in discrete dynamics. We will begin with an analysis of the dynamics of one-dimensional maps of both the interval and the circle. Topics to be covered include chaos, elementary bifurcations, Sarkovskii's Theorem, Schwarzian derivative, symbolic dynamics, chaotic behavior, and homoclinic phenomena.

Midway through the course we will discuss higher dimensional dynamics, including special examples like the horseshoe and strange attractors. We will discuss the stable manifold theorem in its most elementary setting and also such higher dimensional bifurcations such as the Hopf bifurcation.

The latter part of the course will be devoted to special topics in discrete dynamics such as complex dynamics or the Lorenz attractor.

Prerequisites for the course include a solid grounding in real analysis such as MA 511. The undergrad course MA 471 is an excellent prerequisite for this course. Much of the material here is the same; it is presented in a much more rigorous fashion in MA 771. Other dynamics courses such as MA 573 are helpful, but not necessary prerequisites.

This course should be quite different from other advanced mathematics courses you have taken. The goal of the course is not only to introduce the ideas of discrete and chaotic dynamical systems, but also to get you actively involved in the ongoing research in this field. Accordingly, during the first two weeks of the course, you will be assigned a "pet function" which will travel with you throughout the course. (Or you can choose your own pet, if you can find a specific function that relates to your area of interest.) Your job is to relate EVERYTHING we do in the course to this function as we move through the curriculum. When it is a baby, your pet will be a one-dimensional function. But as it grows up, your pet will become a higher dimensional little thing, most likely a complex function. Each homework will be to relate what is currently going on in class to your little pet. There will be no assigned problems: your job is use what you have learned to delve more deeply into what is happening to your pet. Who knows? Your little pet could turn into a research paper or REU project by the end! Be sure to feed it with plenty of good ideas... Also, please name your pet; but any name that rhymes with or sounds like any faculty member's name will probably be sent to the pound by mid-semester!

One of the most important things in science and mathematics is to be able to present your ideas in a coherent fashion both in writing and in speaking. Therefore, during the semester, you will be required to make two short presentations to the class on something that is both relevant to the material covered in the class and that is particularly interesting about your pet. You will also make a 50 minute presentation at the end of the semester on a topic of interest dealing with your pet.

The textbook for the course is that () book
called **
An Introduction to Chaotic
Dynamical Systems, Second Edition ** by what's-his-name,
published by Perseus Press (Westview).

This course will also serve as a prerequisite for a seminar in complex dynamics to be offered (hopefully) in Fall 2007. Bring your pet along!

- c (x - x
^{3}/3) * - c sin(x) **
- c cos(x) **
- cx(1 - x)(1 -2x)
^{2}** - c tan(x) ***
- c x e
^{x}*** **c x sin(x) *****

* = appropriate for undergrads

** = OK for beginning graduate students

*** = I don't know what the hell is going on here, so this might be good for a research project

** Assignments:**

1. ** Due Weds., Feb. 7**
Describe any bifurcations that occur as the parameter varies in your
pet family. Try to prove what you
can; discuss numerical evidence for other bifurcations that you
observe; and highlight in your paper
properties that seem to be different from those discussed in class.
Don't forget to consider all values of **c**, not just positive
values.

2. ** Due Mon., Feb. 26** For your pet, find **c**-values for
which you can prove the existence of
chaotic behavior on a "Cantor-like" set via symbolic dynamics. I am
especially interested in chaotic regimes for which you need a
"different" sequence space in order to prove chaos. This means you
need to specify the different space, its metric, the chaotic properties of
the analogous shift map, etc. Provide all details where the proofs
differ from that for the logistic function.

3. ** Due Mon., March 26** Apply Newton's Method to your pet. Are
there any bifurcations? Chaotic behavior? Other interesting stuff?
Maybe look in the complex plane for a preview of what's to come?

4. ** Due Weds., April 18** Now complexify your pet. Feel free to
change your pet slightly if you wish to make it more "delicious."
What can you now say? By the way, nobody (except you) knows anything
about this topic.

5. ** Monday, May 7 (or TBA) ** "Dynamics Daze" Time TBA.
Every participating student
will make a 50 minute presentation on this date. Topics could involve
your favorite things about your pet, or you could opt to read and
report on a research paper on a different topic. If you decide to do
the latter, please stop by my office to choose a paper.