Instructor: Tim Chumley
Office: Clapp 423
Phone: 413-538-2525
e-mail: tchumley
Office Hours: Mondays & Wednesdays & Fridays 4:00-5:30, Thursdays 11:00-12:00; additional availability by appointment
Textbook: A First Course in Chaotic Dynamical Systems by Robert L. Devaney, ISBN: 9780429280665;
available as a free e-text
Announcements
Announcements will be posted on the Course Announcements Moodle forum throughout the semester, but essentially all other materials will be posted on this page.
Syllabus
Check the syllabus for all the important class policies (grades, attendance, etc.).
Homework
There will be weekly homework assignments throughout the semester to be turned in. Please read these guidelines for writing in our class.
- General information. A selection of problems will be assigned to be written up individually and turned in each week.
- These problems will be due Thursdays at 5 pm.
- You may work with others but the writing should be done on your own.
- Gradescope. Homework will be turned in through Gradescope.
- You should be enrolled automatically. Please let me know if you have any issues logging in.
- Gradescope has made a short tutorial on submitting homework.
- Collaboration. I want you to work together on the homework! The process of explaining your ideas to one another really helps in learning math. However, you must write up the assignments on your own and avoid copying others’ work directly. Please only write what you understand so that I know where to help, and please avoid using AI chat bots, online forums, solutions manuals, or similar resources. A huge part of learning in this class comes from figuring out how to get unstuck by spending time thinking on your own or talking to me and classmates; these other resources can end up being counter-productive in the long term.
- Rewrites. Homework is for practice, and you are not expected to write perfect answers from the start!
- You will be allowed to submit revisions of most problems for full credit each week.
- Your revisions will be due on Thursdays at 5 pm. This means each week you’ll have two things to turn in on Fridays: an initial submission and a redo submission.
- Please resubmit (only the problems you’re revising) on Gradescope by using the resubmit function. I’ll be able to see your submission history in order to see what was initially correct or incorrect.
| Assignment | Due |
|---|---|
| Homework 0 | Jan 29 |
| Homework 1 | Feb 5 |
| Homework 2 | Feb 12 |
| Homework 3 | Feb 19 |
| Homework 4 | Feb 26 |
| Homework 5 | Mar 5 |
| Homework 6 | Mar 26 |
| Homework 7 | Apr 2 |
| Homework 8 | Apr 9 |
| Homework 9 | Apr 16 |
Quizzes
There will be quizzes most weeks that will be given on Wednesdays at the start of class. The purpose of these is to check in to see that you’re comfortable with fundamental material and homework problems. Topics will be announced in advance.
| Quiz | Date |
|---|---|
| Quiz 1 | Feb 4 |
| Quiz 2 | Feb 11 |
| Quiz 3 | Feb 18 |
| Quiz 4 | Feb 25 |
| Quiz 5 | Mar 4 |
| Quiz 6 | Apr 1 |
| Quiz 7 | Apr 8 |
| Quiz 8 | Apr 15 |
Exams
There will be two midterm exams. The dates for the exams are subject to change slightly.
| Exam | Date | Format | Material |
|---|---|---|---|
| Exam 1 | Mar 11 | in-class | Chapters 2-6 |
| Exam 2 | Apr 22 | in-class | TBA |
Project
In lieu of a final exam, we’ll devote the last few class meetings to a short project involving some selection of topics we wouldn’t otherwise have time to cover. The project will involve a presentation or report. Details will be discussed in the middle of the semester.
Assignments
Some assignments will be posted here in the lead-up to the project.
| Assignment | Due |
|---|---|
| Project 0 | Apr 3 |
Course plan
Our plan is to cover parts of the first 10 chapters of the textbook, as well as parts of later chapters, time permitting. Below is a rough week by week outline of the semester which will be updated regularly. Please check back regularly for precise details on what is covered, as well as postings of class materials like lecture notes.
Chapters 1-3
Wednesday
- Topic: Chapters 1-3: Introduction. We introduce the concept of a dynamical system and some terminology (iteration, orbits, fixed points).
- Class materials: Lecture notes, worksheet, iterator.m script, time_series.m script
- After class: Work on and submit Homework 0. Please make sure to install MATLAB and come ask for help if you have trouble.
Chapters 3-4
Monday
- Topic: Chapter 3: Iteration and Orbits. We discuss fixed points, periodic points, and eventually periodic points. We also introduce and study the doubling map.
- Class materials: Lecture notes, worksheet, doubling_map.m script
- After class: Read sections 4.1 and 4.2.
Wednesday
- Topic: Chapter 4: Graphical analysis. We introduce cobweb diagrams in order to visualize orbits of a dynamical system. We also discuss complete orbit analysis and phase portraits.
- Class materials: Lecture notes, worksheet, Desmos cobweb plotter
- After class: Think about the role slope plays in today’s worksheet examples and its relationship to attracting and repelling fixed points.
Chapter 5
Monday
- Topic: Chapter 5: Fixed points. We continue our discussion from last time, talking about complete orbit analysis and phase portraits. We also talk about a theorem that guarantees the existence of fixed points.
- Class materials: Lecture notes, worksheet
- After class: Finish today’s worksheet and read Section 5.4, including the proof of the Attracting Fixed Point Theorem.
Wednesday
- Topic: Chapter 5: Attraction and repulsion. We begin to formalize the observations we have made on attraction and repulsion using calculus. Our aim is to summarize the possible behaviors of orbits based on the value of \(|F'(x_0)|\) when \(x_0\) is a fixed point.
- Class materials: Lecture notes
- After class: Read Section 5.5.
Chapters 5, 6
Monday
- Topic: Chapter 5: Periodic points. We wrap up our discussion from last time and discuss attracting, repelling, and neutral cycles.
- Class materials: Lecture notes
- After class: Work this week’s homework and come ask questions about it.
Wednesday
- Topic: Chapter 6: Dynamics of the quadratic map. We begin our discussion of bifurcations by studying the quadratic map. The idea will be to study a family of maps indexed by a parameter and try to understand how changes to the parameter yield a transition in the number of fixed points or a transition in the number of 2-cycles.
- Class materials: Lecture notes, worksheet
- After class: Finish Problem 2 on today’s worksheet and prepare to discuss it at the start of our next class.
Chapter 6
Monday
- Topic: Snow day, no class.
Wednesday
- Topic: Chapter 6: Saddle-Node Bifurcations. We continue our discussion from last time and define saddle-node bifurcations.
- Class materials: Lecture notes
- After class: Read Section 6.3.
Chapters 6, 7
Monday
- Topic: Chapter 6: Period-Doubling Bifurcations. We discuss the period doubling bifurcation for the quadratic family and work on other examples.
- Class materials: Lecture notes, worksheet
- After class: Work on Homework 5.
Wednesday
- Topic: Section 7.1: The Quadratic Family. We continue our discussion on the quadratic family, focusing on the case when \(c = -2\).
- Class materials: worksheet, orbit_diagram_quadratic_map.m, orbit_diagram_logistic_map.m
- After class: Begin studying for Exam 1, which will be in class next Wednesday. Make sure to know all your definitions and how to do problems like we’ve seen on worksheets and homework.
Spring Break
Monday
- Topic: Spring break, no class.
Wednesday
- Topic: Spring break, no class.
Chapter 7
Monday
- Topic: Section 7.2: More on the quadratic family. We discuss which initial seeds give rise to orbits that remain bounded for the quadratic map. We also introduce the Cantor Middle-Thirds set by studying the same question for the tent map.
- Class materials: Lecture notes, worksheet
- After class: Read Section 7.3.
Wednesday
- Topic: Section 7.3: The Cantor Middle-Thirds set. We discuss ternary expansions and how to tell whether a real number in the unit interval is in the Cantor Middle-Thirds set.
- Class materials: Lecture notes, worksheet
- After class: Work on homework.
Chapter 9
Monday
- Topic: Chapter 9: Symbolic dynamics. We begin our discussion of a powerful tool in dynamical systems, symbolic dynamics. The idea will be to encode the orbit of a map using a binary sequence which gives an itinerary of which part of phase space is visited at each step. We’ll start with a concrete example, the doubling map.
- Class materials: Lecture notes, worksheet
- After class: Read section 9.1.
Wednesday
- Topic: TBA
- Class materials: Lecture notes
- After class: .
Chapters 9, 10
Monday
- Topic: TBA
- Class materials: Lecture notes
- After class: .
Wednesday
- Topic: TBA
- Class materials: Lecture notes
- After class: .
Chapter 10
Monday
- Topic: TBA
- Class materials: Lecture notes
- After class: .
Wednesday
- Topic: Group work time. We spend the day working in groups learning about the group project topic, discussing questions, and beginning to think about presentations.
- Class materials: Presentation rubric
- After class: Spend some time working on the ideas discussed in your group today. Check in with your group about outside-of-class meeting time. Begin preparing ideas for presentation. Think about what will be presented (you can’t say everything you think of!) and how the presented material will be divided between group mates.
Exam, Group work
Monday
- Topic: Review. We spend the day on a review worksheet and discussing any questions you have.
- Class materials: worksheet, solutions
- After class: Continue studying for Exam 1.
Wednesday
- Topic: Exam 2.
- After class: Think about your presentation topic in preparation for the work time in the next class meeting.
Group work time, presentations
Monday
- Topic: Group work time. We spend our last day on group work time before the presentations begin.
- After class: Meet with your group. Do practice runs of your presentation, particularly if you’re presenting Thursday.
Wednesday
- Topic: Presentations. We’ll spend the day on the first group presentations.
- After class: Wrap up your presentation if presenting next week.
Presentations
Monday
- Topic: Presentations. We’ll spend the day on the remaining group presentations.
- After class: Enjoy your summer! Keep in touch!
Getting help
Here are a few ways to get help:
- Office Hours: Mondays & Wednesdays & Fridays 4:00-5:30, Thursdays 11:00-12:00; additional availability by appointment
- Study groups: Other students in the class are a wonderful resource. I want our class to feel like a community of people working together. Please get in touch if you’d like me to help you find study partners, and please reach out if you’d like others to join your group. You may work together on homework, explain and check answers, but make sure you write what you know on homework in order to get good feedback.
- Message board: I’ve set up a question and answer forum for asking and answering questions about class material, including homework problems. My plan is to check the forum regularly and answer questions as they come up, but my hope is everyone in the class will pitch in and answer others’ questions as another form of participation in the class.
Resources
- We will be using MATLAB, a programming language and piece of software, to experiment with some of the theoretical ideas we learn. LITS has a short installation guide and MathWorks, the creators of MATLAB, have an onramp guide for learning the basics.
- Our textbook’s author has written a number of Java Applets to aid in our understanding. Unfortunately they seem to be out of date and it might not be possible to run them without some work but I’ve put a link here in case they can be used.
- Over the course of the semester, I’ll be posting MATLAB scripts that we will use as we learn new concepts. I’ll be posting them here:
- This Desmos cobweb plotter is useful.
- Here are some dynamical systems related videos and web pages. Please share anything interesting you come across.
- Visualizing bifurcations of the logistic map (thank you Jayleen Jiang for sharing this and a number of the other links below)
- More on the logistic map
- Cantor set and Cantor function
- The Mandelbrot set
- Newton’s method and Newton’s fractal
- Complex systems