This course is an introduction to chaotic dynamical systems through theory and computer experimentation. We begin by examining discrete dynamical systems, bifurcations, and transitions to chaos. We will build up analytic tools, including fractal geometry and a little complex analysis to end the course with dynamics in the complex plane, Julia sets and the Mandelbrot set. Prerequisites: Mathematics 126.

Temporary Link to Fractals

A link to Amazon.com's website for the textbook.

We'll be using ``A first course in chaotic dynamical systems", by Professor Robert Devaney, at Boston University.

Course Syllabus

- Week 1:
- Sep 2: Chapters 1 and 2 Homework (Turn all in)
- Sep 4: Chapter 3 (Turn in problems with an asterisk next Tuesday): 2, 3, 6, 7(b*, d*), 8, 10*
- Sep 5 (Meet in Computer Lab): We'll work on Chapter 3, Section 3.6
(The Computer May Lie), Exercise 11 , 12 (in Matlab). Also,
Chapter 4, Exercises 2(a,c) (in Matlab), problem 7 (by hand)
- Setting up your Desktop
- Intro to Matlab
- The Doubling Function, D(x) (See the last page of the intro handout)
- Script to use for the Computer Experiment, ch3.m in the last page of the intro handout.
- Matlab file: cobweb.m (Right-click and Save Link As...
- Matlab file: iterates.m, used by cobweb.m
- Matlab file: driver1.m

- Week 2:
- Sep 9: Be sure to look over the proofs in Chapter 5. In Chapter 5, do problems 1(a,c,f), 2(a,d,e).
- Sep 11: Turn the solutions in for these (as discussed in class):
- Homework problems
- Sample write up for the last problem
- Matlab file (Right-click this link and "Save Link As..." to save it to the local computer. Open Matlab and in the command window, type: ch5Experiment (or whatever you named the file when you saved it). Be sure that the file is in the same directory as Matlab.
- Orbits1.m (Demo in Class)

- Sep 12: No extra homework (Work on the stuff from last time)
- Week 3
- Sep 16:
- Sep 18 (No extra HW today):
- Sep 19: Finish Chapter 7, start Chapter 8 (Add to HW: p. 80, 1, 2, 3, 8)
- M-file for Experiment 6.4, p. 63 (ch6experiment.m)
- M-file to look at orbits of points when c is less than or equal to -2 (sets up Chapter 7, demoBifurcation2.m)
- M-file for plots of A_1, A_2, A_3 (QuadC1.m)
- M-file to animate the intervals A_1, A_2, etc. (QuadAnim.m)
- Java Applet for the Bifurcation Diagrams (Look for the "Bifurcation" button on the left panel)

- Week 4
- Sep 23: Finish Chapter 8, "Homework Set 3" Due.
- Sep 25: Review for Exam I: Exam is over Chapters 1-7 (the bulk is
from 3-7).

Solutions to Homework Set 3. In penance for the algebra mess this morning, I have typeset the long division in all its glory!

- Sep 26: Exam I. The take-home portion will be due at the beginning of the next class session.

Change in the format

Course Topics and class members

- Week 5
- Sep 30: Chapter 8

Orbit diagram (Bifurcation diagram) for the Quadratic map (From Boston U) - Oct 2: (Aaron presented today)
- Finish up Chapter 8. Here is the M-file from class.
- Start Chapter 9: Here is the first set of exercises. You should try them out to see if you understood today's class session You should try them out to see if you understood today's class session.

- Oct 3: (TC presented today) Finish 9.3, start 9.4

Summary and HW questions for next time

- Sep 30: Chapter 8
- Week 6
- Oct 7: Amanda presented items from 9.4
- Oct 9: Jackson introduced the concept of Chaos.
- Oct 10: Henok continued in Chapter 10.

- Week 7
- Oct 14: No Classes
- Oct 16: Continuing in Chapter 10.

Corrections to Exam 1 Due today.

Homework Set assigned today - Oct 17: Lyapunov Exponents

- Week 8
- Oct 21: Homework problems from Ch 10 finished. (Hillary did part)
- Oct 23: Section 11.1: David

**Homework:**From Chapter 11: 1, 3, 4, 7. Plus:

If f is piecewise linear and f(1)=2, f(2)=3 and f(3)=1, go through the argument from class to find the intervals I_0, I_1, A_1, A_2, A_3 and A_4 (find the intervals explicitly). A graph will help.

- Oct 24: Section 11.2: Matthew

**Homework:**Read over 11.3/11.4 for Tuesday.

- Week 9
- Oct 28: Page 140-Section 11.3: Hundley
- Oct 30: Finish 11.3: Hundley
- Oct 31: Finish Chapter 11 (Hillary)

Homework for Chapter 11 (and some from earlier)

- Week 10
- Nov 4: We finished the subshifts from Chapter 11 and Cody presented the material from 12.1.
- Nov 6: Chapter 12.2: Eric presented material from 12.2

SOLUTIONS to the Homework from Chapter 11

- Nov 7: Chapter 13

- Week 11
- Nov 11: Chapter 13 (Newton's Method)
- Nov 13 Review
- Nov 14: Exam 2 (Chapters 8-12, maybe some on 13)

- Week 12
- Nov 18: Information on Presentations
- Nov 20: Work on presentations
- Nov 21: Work on presentations

**Thanksgiving Break**- Week 13
- Dec 2: Hundley (Complex Numbers), Hillary (Intro to Fractals)
- Dec 4: Aaron and Cody: Julia Sets
- Dec 5: David and TC: The Mandelbrot Set

- Week 14
- Dec 9: Matthew (Complex Newton), Jackson (Fractals)
- Dec 11: Henok and Amanda: The Lorenz Attractor
- Dec 12: Open- Work on your papers- Due next Mon 5PM.