Gathering Thoughts Part II: Dynamical Systems: Chaos Theory and Fractals

Originally Posted 12-20-2011

Let’s continue the broad overview of what is to be done in the next few weeks by taking the next course on my list.

Dynamical Systems: Chaos Theory and Fractals

This is to be an undergraduate introduction to the basics of dynamical system ideas. The Course Catalog copy reads as follows:

Dynamical Systems: Chaos Theory and Fractals -- 3 hrs. 
(MATH 3410/MATH 5410) 
Historical background, including examples of dynamical systems;
orbits, fixed points, and periodic points; one-dimensional and
two-dimensional chaos; fractals: Julia sets, the Mandelbrot
set, and fractal dimension; computer programs and dynamical
systems. Prerequisite(s): 800:061 (MATH 1421); 800:076 (MATH
2500); junior standing.

Those pre-requisites are Calculus II and Linear Algebra. Right now, there are 15 undergraduates and 3 graduate students enrolled. (That is why the course has two numbers. The grad students take the 5000 version.)

All of the introductions to dynamical systems that I know well are written at the graduate level. In fact, that is when I got my introduction to the subject, during my second year of graduate studies. So, my favorite references are certainly out as inaccessible to my students.

I have found two undergraduate level introductions that seem to hit the mark. The first is Devaney’s A First Course in Chaotic Dynamical Systems, and the second is Chaos: A Mathematical Introduction by Banks, Dragan and Jones. From a cursory inspection, the two books are very similar in outlook and basic design.

Now, I plan to run the course as an IBL experience. Which means that I will be writing a sequence of problems for the students to solve and present to each other. I intend to use a structure suggested by some people I met at an IBL workshop at UT Austin: students work on the problems before class meetings, and are given the first 35 minutes of class to work together and streamline their work into one solid and well-presented solution. Then the last 15 minutes is dedicated to presenting arguments to the whole class.

I will be writing my own sequence of problems, and I think I shall follow Devaney’s lead. It is easier to steal and adapt than to shear the sheep, comb the wool, … and finally weave my own blanket.

In some way, I will incorporate using a computer. I want the students to get a flavor of the subject, and that includes using a machine to compute orbits and draw pictures. I plan to use Sage).

It seems important to me to stick to low dimensional phase spaces (d = 1 or 2) for visualization purposes. We will also stick to discrete time systems (iteration of mappings) and leave the continuous time systems (differential equations) alone. I want to study some famous and important systems like the tent map, the family of quadratic maps, the continued fraction map, the full shift on two symbols, and the Newton-Raphson method. The major goal of the course is to show that even the simplest non-linear deterministic dynamical systems can exhibit what is called chaos: sensitive dependence on initial conditions, a dense orbit, and a dense set of periodic orbits all at the same time.

If all goes well, we might just get to seeing the connection between the Julia sets for functions f(z) = z^2 + c and the Mandelbrot set. I bet we won’t have time, but that is such cool thing. It is always good to end with a mind-blowing topic.

I am concerned about the way the course description includes fractals in such a cavalier way. I mean, yes some fractals will come up. But I don’t envision being able to devote enough class time to the concept of a fractal that would justify the word in the course title. I think it is there because it helps draw students to an elective. I’ll have to deal with this somehow.

So, my main objectives for this class for the next two weeks are as follows:

  • Write a few weeks worth of problems which introduce students to the basic terminology of discrete dynamical systems and our major examples for the course. (Maybe 6 problems per meeting, 3 meetings a week, four weeks of class \implies 70 small problems. This is certainly an overestimate. I’ll adjust it.)

  • Design an “introduction to Sage” workshop to be run the first week of class.

  • Make a “Sage: dynamical systems” cheat sheet to help students with syntax.

  • Build a course web page.

I feel a bit funny about having to leave out continuous time systems, given the history of the subject. So I might prepare some stuff on a couple of striking examples for when the time is right. Of course, there are other things from the history of the subject, too. I hope to be able to show them Lorenz’s equations and the “butterfly’’ strange attractor in it, Smale’s Horseshoe system, the Baker transformation, the locations of the poles in the Earth’s magnetic field over time, the three body problem, and several others. I am certain that this is not going to all get done this first time I teach the course.