Math 3543 (Dynamics, Chaos, and Fractals), Spring 2025
| Course Information | |||||
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| Instructor | Class Times | Office Hours | |||
| Evan Dummit edummit at northeastern dot edu 571 Lake Hall |
MWR 1:35pm-2:40pm 275 Richards Hall |
M 12:05pm-1:05pm W 3:00pm-4:00pm Lake 571 |
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| For detailed information about the course, please consult the 3543 Course Syllabus. (Note: any information given in class or on this webpage supersedes the written syllabus.) | |||||
| We will use Piazza for any course-related discussion: here is the 3543 Piazza page, which can also be accessed through the course Canvas page. Access is restricted to students in Math 3543. | |||||
| Teaching Assistant | Problem Session | ||||
| Ali Cataltepe | Tuesdays 11:45am-1:15pm, 126 Ryder Hall | ||||
| Jordan Martino | Fridays 1:15pm-2:45pm, 147 Ryder Hall | ||||
| All homework assignments will be posted on this webpage (see below). Homework assignments will be collected via Gradescope on Canvas. Please submit scans of your homework pages by 11:59pm Eastern on the due date. Late assignments may be penalized at the grader's discretion. |
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| The instructor will write lecture notes for the course (see below) in lieu of an official textbook as the semester progresses. | |||||
| Homework Assignments |
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| Note that some homework problems may require computational software, such as Mathematica, which is available free for all Northeastern students. The cloud version of Mathematica is available through the 3453 Canvas page; see this page for instructions on how to install Mathematica locally. |
Homework #1, due Wed Jan 15th.
Homework #2, due Wed Jan 22nd. (Mathematica code). |
| Homework assignments are to be submitted to Gradescope via the course's Canvas page. To submit an assignment, navigate to "Assignments" and select the appropriate homework assignment. Then attach scans of each page of your assignment (or a pdf) and click Submit. Please submit the pages in order and verify that all pages are included and uploaded correctly. You may resubmit as many times as you like. Assignments are due at 11:59pm eastern time. Late submissions, messy submissions, or otherwise unreadable submissions will be penalized at the grader's discretion. Ensure you mark all problem pages when submitting to Gradescope; failure to do so may result in point penalties. |
| Handouts / Lecture Notes | |||
|---|---|---|---|
| Handout | Topics | ||
| Chapter 1: Introduction to Real Dynamics (29pp, v3.50, posted 1/4) | 1.1 ~ Dynamics on the Real Line (Examples and Motivation, Orbits and Fixed Points, Periodic Points and Cycles, Examples) 1.2 ~ Qualitative and Quantitative Behavior of Orbits (Graphical Analysis, Attracting and Repelling Fixed Points and Cycles, Weak Attraction and Repulsion, Basins of Attraction) 1.3 ~ Newton's Method |
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| Exam Information | ||||
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| Exams will be held in class during the course lecture period (or during the final exam period, for the third exam). Each exam covers one-third of the course material. | ||||
| Exam | Date, Time, Location | Topics | Review Material | |
| Midterm 1 | Mon, Feb 10th In lecture |
Homeworks 1-4 Notes §1.1-2.2 |
Review material to be posted | |
| Midterm 2 | Mon, Mar 24th In lecture |
Homeworks 5-8 Notes §2.3-4.1 |
Review material to be posted | |
| Midterm 3 | Date and time TBA (Final exam week) |
Homeworks 9-11 Notes §4.2-5.3 |
Review material to be posted | |
| For exams, you are permitted one "cheat sheet", consisting of a double-sided 8.5in-by-11in piece of paper. Calculators are permitted, but cell phones, other electronic devices, books, and additional notes of any kind will NOT be permitted in exams. | ||||
| Tips For Success In This Course | |||
|---|---|---|---|
| Attend Lecture | Missing lecture is a bad idea! If for any reason you cannot make it to a class, you should review the notes and watch the lecture recording. You are responsible for all material covered in lecture. | ||
| Read the Lecture Notes | The lecture notes are a comprehensive source of material for the course. The notes are intended as review material, although many students like to read them as preparation before attending the lecture on the corresponding topics. Please note that the electronic notes are not identical to the material covered in class: this is by design, so as to provide you a slightly different perspective on the material. | ||
| Solve Homework Problems | Much of the learning in this course will take place as you solve the homework problems. Like many other activities, problem-solving and proof-writing are things that are learned by doing them, not by hearing someone else tell you about them or reading about them in a book. As such, the homework assignments are an integral part of the course, and are fundamental to learning the material. It is highly recommended that you look over the homework assignments as soon as they are available, and work on them well in advance of the deadline: many problems will take substantial time and effort to solve, and you should expect to spend as much time as you need to finish the assignments. | ||
| Attend Problem Sessions | There are weekly problem sessions run by the course TAs. The goal of the problem session is to provide you a location where you can work collaboratively with other students on assignments, and also get TA assistance. Students are highly encouraged to attend the problem sessions and work on the homework assignments there. | ||
| Attend Office Hours | Office hours are specifically reserved for you to receive individual, one-on-one help from the instructor. Office hours will be the most effective when you have already put in effort to learn the material on your own (including trying to solve the homework problems), and when you come in with a list of specific questions or topics you are struggling with. | ||
| Course Schedule | |||
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| The schedule is subject to change! All sections refer to the course lecture notes. | |||
| Week | Schedule | ||
| Week of Jan 6 (class starts 1/6) |
§1.1.1: Motivation for Dynamical Systems §1.1.2: Orbits and Fixed Points §1.1.3: Periodic Points and Cycles No homework this week. |
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| Week of Jan 13 | §1.1.4: The Doubling Function, the Logistic Maps, and Computational Di fficulties §1.2.1: Orbit Analysis Using Graphs §1.2.2: Attracting and Repelling Fixed Points §1.2.3: Attracting and Repelling Cycles Homework #1 due Wednesday, Jan 15th. |
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| Week of Jan 20 (no class 1/20) |
§1.2.4: Weakly Attracting and Weakly Repelling Points (and Cycles) §1.2.5: Basins of Attraction Homework #2 due Wednesday, Jan 22nd. |
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| Week of Jan 27 |
§1.3: Newton's Method §2.1.1: The Quadratic Family qc(x) = x2+c §2.1.2: General Properties of Bifurcations Homework #3 due Wednesday, Jan 28th. |
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| Week of Feb 3 |
§2.1.3: Saddle-Node Bifurcations §2.1.4: Period-Doubling Bifurcations §2.2.1: The Schwarzian Derivative Homework #4 due Wednesday, Feb 4th. |
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| Week of Feb 10 | MIDTERM 1 in class on Monday, Feb 10th §2.2.2: Numerical Computation of Attracting Cycles §2.3: Orbit Diagrams §3.1.1: Motivation for Symbolic Dynamics No homework this week. |
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| Week of Feb 17 (no class 2/17) |
§3.1.2: Nested Intervals, Itineraries, and Cantor Sets §3.1.3: Metric Spaces and the Sequence Space §3.1.4: Equivalence of Dynamical Systems Homework #5 due Wednesday, Feb 19th. |
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| Week of Feb 24 | §3.1.5: Equivalence of the Quadratic Maps and the Shift Map §3.2.1: Motivation for Chaos §3.2.2: Devaney's Definition of Chaos Homework #6 due Wednesday, Feb 26th. |
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| Spring Break (no classes) from Mar 3 to Mar 7 | |||
| Week of Mar 10 | §3.3.1: The Period-3 Theorem §3.3.2: The Sarkovskii Ordering and Sarkovskii's Theorem §4.1: Classical Fractal Constructions Homework #7 due Wednesday, Mar 12th. |
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| Week of Mar 17 | §4.2.1: Topological Dimension §4.2.2: Minkowski Dimension §4.3.1: Iterated Function Systems and Self-Similarity Homework #8 due Wednesday, Mar 19th. |
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| Week of Mar 24 | MIDTERM 2 in class on Monday, Mar 24th. §4.3.2: Minkowski Dimensions of Self-Similar Sets §4.3.3: The "Chaos Game" No homework this week. |
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| Week of Mar 31 | §5.1: Dynamical Properties of Complex-Valued Functions §5.2: Julia Sets For Polynomials §5.3.1: General Julia Sets Homework #9 due Wednesday, Apr 2nd. |
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| Week of Apr 7 | §5.3.2: Julia Sets and Cantor Sets §5.3.3: Critical Orbits and the Fundamental Dichotomy §5.3.4: The Mandelbrot Set Homework #10 due Wednesday, Apr 9th. |
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| Week of Apr 14 (classes end 4/15) |
§5.3.5: Analysis of the Mandelbrot Set Homework #11 due Wednesday, Apr 16th. |
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| MIDTERM 3 is scheduled for date TBA at time TBA in location TBA | |||