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, stable manifold theorem, Smale's horseshoe map, fractal dimension, iterated function systems. Many ideas from topology and analysis will be introduced on the way.
Introduction to Dynamical Systems, Continuous and Discrete (2nd edition), by R. Clark Robinson. We will cover much, but not all, of the second half of the book: chapters 8, 9, 10 and parts of chapters 11, 12, 13, 14. The lectures will differ somewhat from the text however.
There will be grades for homework and three midterm exams, weighted as follows:
|Midterm Exams (25 % each)||75 %|
|Midterm I||Monday, October 5|
|Midterm II||Wednesday, November 9|
|Midterm III||Wednesday, December 14|
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
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.
|9/7||Iteration, fixed points, periodic points||8.1, 9.1|
|9/12-9/14||Graphical iterateion. Stability. Basis of Attraction.||9.2,9.3|
|9/19-9/21||Singer's theorem, Bifurcations||9.4,9.5|
|9/28-9/30||More bifurcations, conjugacies||9.5,9.6|
|10/3-10/5||Review, Midterm I|
|10/10-10/12||Transition graphs, topological transitivity||10.1,10.2|
|10/17-10/19||Shift map, itineraries, sensitive dependence||10.3,10.4|
|10/24-10/26||Invariant Cantor sets||10.5|
|11/7-11/9||Review, Midterm II|
|11/14-11/16||Higher dimensional maps, Periodic points||12.1,12.2|
|11/21-11/23||Stable manifolds, toral automorphisms||12.3, 12.4|
|11/30-12/2||Horseshoe map, Fractal dimension||13.1,14.1|
|12/5-12/7||Iterated function systems||14.3|
|12/12-12/14||Review, Midterm III|