Syllabus for Math 5535 --- Fall 2017

Description:

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.

Text:

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.

Grades:

There will be grades for homework and three midterm exams, weighted as follows:

Homework 25 %
Midterm Exams (25 % each) 75 %

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.

Exam Dates:

 Midterm I Wednesday, October 4
 Midterm II Wednesday, November 8
 Midterm III Wednesday, December 13

Homework:

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.

Computers:

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

Mathematica Notebook:

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.

Approximate Schedule:

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