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. An overly ambitious list of topics to be covered include fixed points, periodic points, stability, bifurcations, fractals, rotation numbers, equidistribution, phase locking, symbolic dynamics, chaos, Julia sets and the Mandelbrot set, stable manifold theorem, Smale's horseshoe map, homoclinic chaos, strange attractors and Poincare maps. Many ideas from topology and analysis will be introduced along the way.
A First Course in Dynamics, by Hasselblat and Katok. The course will not be based exclusively around the book. The lectures will follow their own logic and will cover what you need to know for the homework and tests. The topics we will cover include much of chapters 2-7 and selected parts from other chapters in the text. Some topics covered in lecture are not treated in the book at all.
Some other references which have been put on reserve could be consulted for these extras or for a more elementary treatment of certain topics:
- Alligood, Sauer and Yorke, Chaos, An Introduction to Dynamical Systems
- Devaney, A First Course in Chaotic Dynamical Systems, Theory and Experiment
- Devaney, An Introduction to Chaotic Dynamical Systems
- Robinson, An Introduction to Dynamical Systems, Continuous and Discrete
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 Grading Policy Statement.
| Homework | 20 % |
| Midterm Exams (20 % each) | 40 % |
| Final Exam | 40 % |
| Midterm I | Monday, October 9 |
| Midterm II | Monday, November 13 |
| Final | Friday, December 15, 10:30-12:30 |
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.
Here is a Mathematica notebook which will allow you to produce plots like those in some of the handouts from class. It is structured as a self-explanatory lab. You can go to any campus computer with Mathematica, download the lab, open it using Mathematica and go through it step by step. Also, you can easily modify the commands to make similar plots for other dynamical systems. Have fun !
Here is a very tentative week by week outline of the course. The word "extra" means some topic not in the book.
| Week | Topic | Reading |
| 9/6-9/8 | Iteration. Examples, Fixed points. Cobweb plots. | 1.1-1.3, 2.1 |
| 9/11-9/15 | Periodic points. Stability. Contraction maps. | 2.1-2.6, extra |
| 9/18-9/22 | Quasiperiodic and chaotic map examples. | 4.1.1-4.1.3, extra |
| 9/25-9/29 | Linear maps, eigenvalues, contractions in R^n | Ch.3 |
| 10/2-10/6 | Nonlinear maps in R^2, periodic points, stability | extra |
| 10/9-10/13 | Midterm I. Cantor sets. | 2.7, extra |
| 10/16-10/20 | Fractals, iterated function systems | 2.7, extra |
| 10/23-10/27 | Bifurcation diagram, Singer's theorem | 11.1-11.3, extra |
| 10/30-11/3 | Circle maps, degree, uniform distribution | 4.1-4.2, extra |
| 11/6-11/10 | Rotation numbers, phase locking | 4.3-4.4 |
| 11/13-11/17 | Midterm II. Torus flows, unif. distribution | Ch.5 |
| 11/20-11/22 | Billiards, area-preserving maps, Thanksgiving | 6.1,6.3-6.4 |
| 11/27-12/1 | Homoclinic chaos, Smale horseshoe. | extra |
| 12/4-12/8 | More about chaos and symbolic dynamics | Ch. 7 |
| 12/11-12/15 | Even more about chaos, Final Exam | Ch. 7 |