This is a one-semester 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, chaos, fractals, Julia sets and the Mandelbrot set, stable manifold theorem, Smale’s horseshoe map, homoclinic chaos, strange attractors and Poincaré maps. Many ideas from topology and analysis will be introduced along the way.
Discrete Chaos (2nd edition), by Saber Elaydi. We will try to cover most of chapters 1, 2, 3, 6, 7 and parts of chapters 4, 5. The lectures will differ somewhat from the text however and some extra topics may be covered.
There will be grades for homework, two midterm exams and a final exam. All exams and quizzes are open book/notes, calculators allowed. For a general university policy statement about grades, academic honesty and workload, go to: University Policy Statement.
| Homework | 20 % |
| Midterm Exams (25 % each) | 50 % |
| Final Exam | 30 % |
| Midterm I | Monday, October 12 |
| Midterm II | Wednesday, November 18 |
| Final | Friday, December 18, 10:30-12:30 (CORRECTION TO EXAM TIME) |
Homework will be due about every two or three weeks. Students are encouraged to work together on the assignments but everyone should write up their own version of the solutions. Not all of the assigned problems will 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 tentative week by week outline of the course. More later ...
| Week | Topic | Reading |
| 9/9 | Iteration, fixed points, cobweb plots | 1.1-1.6 |
| 9/14-9/18 | Periodic points. Stability. Basis of Attraction. | 1.7-1.8,2.2-2.3 |
| 9/21-9/23 | Bifurcations, period doubling | 1.9, 2.4-2.5,2.7-2.9 |
| 9/28-9/30 | Singer;s theorem, chaos (Li-Yorke, decimal shift) | 2.6, 3.1-3.5 |
| 10/5-10/7 | More chaos | 3.1-3.5 |
| 10/12-10/14 | MT I ,Conjugacy | 3.6-3.8 |
| 10/19-10/21 | Cantor sets, symbolic dynamics | 3.6-3.8 |
| 10/26-10/28 | Topology of Cantor sets, 2D maps, periodic points | Handout on 2D maps |
| 11/2-11/4 | Linear maps, linearization, multipliers | Handout on 2D maps, 4.1-4.7 |
| 11/9-11/11 | Stable manifolds, Bifurcations | Handout on 2D maps, 4.11 |
| 11/16-11/18 | More 2D maps, Midterm II | Handout on 2D maps |
| 11/23-11/25 | Fractals, Iterated Function Systems, Attractors | 6.1,6.4,6.5 |
| 11/30-12/2 | Fractal Dimension, Collage Theorem | 6.6,6.3 |
| 12/7-12/9 | Complex Dynamics | Ch. 7 |
| 12/14-12/16 | Complex Dynamics | Ch. 7 |