This is a one semester course in dynamical systems theory devoted mostly to the study of iteration of mappings of dimension one and two. Most of the basic ideas of dynamical systems theory can be introduced in this setting. Topics to be covered include fixed points, periodic points, stability, bifurcations, conjugacies, chaos, symbolic dynamics, fractal dimension, iterated function systems, Julia sets, Mandelbrot set. Many ideas from topology and analysis will be introduced on the way.
A First Course in Chaotic Dynamical Systems:Theory and Experiment, by Robert Devaney. We will cover most of the book. The lectures will sometimes go beyond the material in the text. A good supplementary reference which covers many of the topics in the course in greater depth is Dynamical Systems, by Shlomo Sternberg.
There will be grades for homework and three midterm exams, weighted as follows:
| Homework | 25 % |
| Midterm Exams (25 % each) | 75 % |
| Midterm I | Wednesday, October 4 |
| Midterm II | Wednesday, November 8 |
| Midterm III | Wednesday, December 13 |
Homework assignments will be posted on the course website. The first half of the class on most Monday's will be devoted to problem sessions. We will break up into small groups to discuss homework problems from the previous week and do some presentations at the board. I will help out if necessary. In addition, you will have to write up solutions to some of the problems to be graded. To see the assignments as they become available, click on the link above.
Computer experiments can give a lot of insight about dynamical systems. I will present some computer demonstrations in class using the software Mathematica. A Mathematica notebook with some of these demonstrations will be available for download on the course website. You are encouraged to use Mathematica or some other program to do some explorations of your own. You can use Mathematica at the CSE computer labs. To get a CSE computer account go to CSE Accounts. Once you have a CSE lab account, you can also download a copy for your personal computer at: Get Mathematica
Experimental Mandelbrot/ Julia set program which runs in web browsers. Try it at: Mandalia
Here is a Mathematica notebook which will allow you to produce plots like those presented in class. You can easily modify the commands to make similar plots for other dynamical systems. More functions will be added as the semester goes on.
Here is very tentative week by week outline of the course. This may change as the semester goes on.
| Week | Topic | Reading |
| 9/6 | Iteration, examples, different type of orbits | Ch. 1-3 |
| 9/111-9/13 | Graphical analysis, attracting and repelling points | Ch. 4-5 |
| 9/18-9/20 | Newton's method, saddle-node bifurcation | Ch. 13, 6.1,6.2 |
| 9/27-9/29 | Period-doubling bifurcation, orbit diagram | 6.3, Ch. 8 |
| 10/2-10/4 | Review, Midterm I | |
| 10/9-10/11 | Schwartzian derivative, invariant Cantor sets | Ch.12, Ch. 7 |
| 10/16-10/18 | Symbolic dynamics, Conjugacies | Ch 9 |
| 10/23-10/25 | Chaos | Ch. 10 |
| 10/29-11/1 | Sarkovskii's theorem, subshifts | Ch. 11 |
| 11/6-11/8 | Review, Midterm II | |
| 11/13-11/15 | Fractals, dimension theory | Ch. 14 |
| 11/20-11/22 | More fractals, Complex functions | Ch. 15 |
| 11/29-12/1 | Julia sets | Ch. 16 |
| 12/4-12/6 | Mandelbrot set | Ch. 17 |
| 12/11-12/13 | Review, Midterm III |