Introduction to Chaotic Dynamical Systems
Math 204 ST, Spr 2024
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.
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
- Course Calendar
Daily Class Work and Homework
- Week 1:
- Slides for the first day
- Wednesday, Jan 17: Read chapters 1 and 2 (they are short), and
answer the questions here.
- Friday, Jan 18: Partial computer lab today.
- Read over Chapter 3 and work through the computer lab below before working on the homework.
- Homework for Chapter 3 ((*) due next Wed): 2, 3, 6, 7(b*, d*), 8, 10*, 11, 12.
- Homework from the computer lab: Section 3.6 (The Computer May Lie). You only need to do the experiment for the doubling function (*). Note that exercise 14 may help.
- Links from Friday's lab:
- Week 2
- Monday cancelled due to ice. Read chapters 4 and 5 for Wednesday. Due date for homework extended to Thursday.
Homework to turn in: Day 1 HW: From the reading, problems 4, 7. From the sequences and limits: 3(a,b).
Ch 3: 7(b,d) and 10. Section 3.6 (Computer may lie), but only for the doubling function (use Matlab/Octave to help).
- Wed: Notes on chapter 4:
HW for Chapter 4: 1(a,c,f*,g*), 2(a*,b), 3(a,c,f*,g*), 7.
- Fri: Chapter 4, intro to Ch 5.
HW for Chapter 5: Experiment (a-d), 1(a*,c*,f,j), 2(a,d*), 3*,4(a*,b)
- Week 3 (Jan 29-Feb 2)
- Mon: Finish Ch 5. HW for Wednesday will just be HW from Ch 4, 5.
- Wed: Ch 6, Lab (HW from Ch 6: The experiment, 1(a,b,c,e)*, 3*, 4-16)
- Fri: Finished Ch 6
- Week 4 (Feb 5-9)
- Mon: Ch 7: HW from Ch 7: 2*, 3, 5*, 8, 9-15, 22(a,b,c), 23*, 24*, 25*.
- Wed: Topics from Ch 7, Ch 8
HW from Ch 8: #4-8 (#7 our software doesn't have that option, but try the others- these problems are all "graph and explain").
- Fri: Ch 8- Ch 9 (part 1)
Nice bifurcation plotting tool! (Same
link as before)
- Week 5
- Monday- Working through Chapter 9.
Handout for the homework (this has some review material
about 1-1, onto, preimage and homeomorphism). Turn in #3, 8.
- Wed: Finish up Chapter 9
HW To Turn in:
Ch 7: 2*, 3, 5*, 8, 9-15, 22(a,b,c), 23*, 24*, 25*
Ch 8: 4-8, except 7. (Turn all in- Use the bifurcation plotting tool to help you - linked online from Week 3, Wed).
- Fri: Chaos.
- Week 6
- Mon: No classes
- Wed: Finish up Chapter 10
- Fri: Review for Exam on Monday (see review materials below)
- Week 7
- Mon: Exam 1
- Wed: Chapter 11 (Period 3 implies chaos). Hw: 1-4
- Fri: Chapter 11-12 (Sharkovsky's theorem, The Schwarzian Derivative)
Also: Exam 1 handed back today. Turn in your solutions on or before Tuesday next week.
Overall grade will be a weighted average of the two scores.
- Week 8
- Week 9 (Mar 25-29)
- Mon
- We'll be writing up a paper on a topic in chaos theory- please think about a possible topic.
- We reviewed an IFS, and saw that the Sierpinski Triangle is related to our old friend, symbol
space. We further reviewed "chaos", and looked at particular orbits in symbol space (using the
shift map) versus points on the gasket (using function iteration). In particuar, be able to locate
fixed points and period two points using linear algebra.
- The orbit of a new point: The random orbit. The orbit of this point is very interesting- it gets arbitrarily close to every point of our set.
- Wed: Dimension (Finishing Ch 14).
Homework for Friday: LaTeX tutorial (about 30 minutes)
Link for LaTeX tutorial: How to use LaTeX in Overleaf. You'll need to set up a free account.
Homework for Monday: Go through Experiment 14.9, and write up your results using LaTeX. Be sure to get an early start, so you can ask questions on Friday.
- Fri: Finish Ch 14.
- Week 10 (Apr 1-5)
- Mon: Finish Ch 14, work on Ch 15.
Homework due: Write up of Experiment 14.9
- Wed: Finish Ch 15. Homework for Ch 15: 1(a,d), 2(a,d,e), 3(a,d,e)*, 9*, 10.
- Fri: More on complex functions, How to plot Julia sets, Computer Lab
- Week 11 (Apr 8-12)
- Mon: Working through Ch 16- Some lab time today (from Fri)
Homework due Friday: Ch 15, Ch 16, Lab 1-3 (See the Canvas assignment page).
- Wed: Continue through Ch 16- Some lab time at the end of class
- Fri: Finished Ch 16, started Ch 17
- Week 12 (Apr 15-19)
- Mon: Chapter 17, continued.
- Wed: Ch 17, continued. Mandelbrot links below. Take-home exam (Exam 2) distributed.
- Fri: No class- Work on your take-home exam (due Monday at 11:59PM)
- Week 13 (Apr 22-26)
- Mon: Finish Ch 17 Notes for Monday
- Wed: Meet in the Lab- Work on 17.4, 17.5
Matlab script: orbits.m. A short Matlab script to plot the orbit of a point c (you can use either Matlab or Octave-online).
- Fri: Continue in the lab (17.7, 17.8); Intro to LaTeX for those that need it.
For the Lab: I would like for you to type up your lab responses in LaTeX (17.4, 17.5 for one set,
17.7, 17.8 for the other set). You may work in groups of up to three- please be sure everyone in
the group is getting practice typesetting in LaTeX (for the final project).
- Week 14 (Apr 29-May 3)
- Mon: Meet in the Lab, finish up the Mandelbrot labs. Extensions to the Mandelbrot set.
Tell Prof. Hundley what your final project will be on.
- Wed: Finish up all labs. Both are due tonight.
Rubric for the final project.
- Fri: Discussion of final project. Time to work on final project. Due: Mon, May 13th (no extensions!).
- Week 15 (May 6)
- Mon: No class today- work on your project (due one week from today!)
Exam Materials
- Exam 1
- Exam 2: Take home exam (email me for a copy if you need it).
Other Links (plotters, applets, etc.)
Links for Julia sets
Links for Mandelbrot sets
