Math 3543 (Dynamics, Chaos, and Fractals), Summer-II 2023
| Course Information | |||||
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| Instructor | Class Times | Office Hours | |||
| Evan Dummit edummit at northeastern dot edu |
MTWR 1:30pm-3:10pm, Richards 235 | T 12:05pm-1:05pm (Zoom only) R 3:30pm-5:30pm or by appointment Lake 571 or online via Zoom |
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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.) | |||||
| Teaching Assistant | Problem Session / Office Hours | ||||
| Ryan Kannanaikal kannanaikal.r at northeastern dot edu |
F, 1pm-3:30pm Richards 235 |
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| Math 3543 uses a Piazza page for course discussion. Links to all of the live lectures, office hours, problem sessions, and lecture recordings are hosted there. | |||||
| 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 submissions received up to 30 hours after the due date will be accepted with a small point penalty. | |||||
| Midterm 2 will be held in the course classroom (235 Richards) from 1pm-3:30pm on Tuesday August 22nd . | |||||
| 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. |
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Homework #1, due Fri Jul 7th. (solutions)
Homework #2, due Fri Jul 14th. (solutions), (Mathematica code). Homework #3, due Fri Jul 21st. (solutions), (Mathematica code). Homework #4, due Fri Jul 28th. (solutions), (Mathematica code). Homework #5, due Sat Aug 5th. (solutions), (Mathematica code). Homework #6, due Sat Aug 12th. (solutions), (Mathematica code). Homework #7, due Sat Aug 19th. (solutions), (Mathematica code). (Link to Mandel, a free utility designed for visualizing the Mandelbrot set and Julia sets.) |
| Homework assignments are to be submitted via Gradescope. To submit an assignment, either follow the direct Gradescope link above, or navigate in Canvas to "Assignments" and select the appropriate homework assignment. Then attach scans of each page of your assignment (or a pdf) and click Submit. Please verify that all pages are included and uploaded correctly, and also select each page on which each problem appears. 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 | |||||
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| The instructor will write lecture notes for the course in place of an official textbook as the semester progresses. Most of the course material will follow the presentation in Devaney's "A First Course in Chaotic Dynamical Systems", with approximate equivalences as follows: Notes ch1: Devaney chapters 1, 2, 3, 4, 5, 13 Notes ch2: Devaney chapters 6, 7, 8, 12 Notes ch3: Devaney chapters 9, 10, 11 Notes ch4: Devaney chapter 14 Notes ch5: Devaney chapters 15, 16, 17 |
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| Handout | Topics | ||||
| Chapter 1: Introduction to Real Dynamics (29pp, v2.00, posted 6/30) | 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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| Chapter 2: Dynamics of One-Parameter Families (21pp, v2.00, posted 7/13) | 2.1 ~ Bifurcations in One-Parameter Families (The Quadratic Family, General Properties, Saddle-Node Bifurcations, Period-Doubling Bifurcations) 2.2 ~ Critical Orbits and Attracting Cycles (Schwarzian Derivatives, Numerical Computations) 2.3 ~ Orbit Diagrams |
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| Chapter 3: Chaotic Dynamics (26pp, v2.00, posted 7/19) | 3.1 ~ Symbolic Dynamics (Nested Intervals and Itineraries, Metric Spaces, Sequence Spaces, Equivalence of Dynamical Systems) 3.2 ~ Chaotic Dynamical Systems (Motivation for Chaos, Devaney's Definition of Chaos, Examples) 3.3 ~ Sarkovskii's Theorem and Applications |
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| Chapter 4: Fractals (21pp, v2.00, posted 8/3) | 4.1 ~ Classical Fractal Constructions (Generalized Cantor Sets, the Koch Curve and Snowflake, the Sierpinski Triangle and Carpet) 4.2 ~ Topological Dimension and Minkowski Dimension 4.3 ~ Iterated Function Systems and Invariant Sets (Iterated Function Systems, Self-Similarity, the "Chaos Game") |
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| Chapter 5: Introduction to Complex Dynamics (26pp, v2.00, posted 8/8) | 5.1 ~ Dynamical Properties of Complex-Valued Functions (Complex Arithmetic, Complex Derivatives, Attracting/Repelling/Neutral Fixed Points and Cycles, Orbits of Linear and Quadratic Maps) 5.2 ~ Julia Sets for Polynomial Maps (Julia Sets, Computing Julia Sets) 5.3 ~ Additional Properties of Julia Sets, the Mandelbrot Set |
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| Exam Information | ||||
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| Exams will be distributed in person during the course lecture period. Each exam covers half of the course material. | ||||
| Exam | Date, Time, Location | Topics | Review Material | |
| Midterm 1 (exam), (sols) |
Tue, Aug 1st In lecture |
Homeworks 1-4 Notes §1.1-3.2.1 |
Midterm 1 Topics | |
| Midterm 2 | Tue Aug 22nd 1pm-3pm, in course classroom |
Homeworks 5-7 Notes §3.2.2-5.3 |
Midterm 2 Topics | |
| 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 | |||
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| Attend Lecture | Missing lecture is a bad idea! If for any reason you cannot make it to a class, you should watch the recording of the lecture. You are responsible for all material covered in lecture. | ||
| Read the Lecture Notes (or Textbook) | The lecture notes are comprehensive sources 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 is a 2.5-hour weekly problem session run by the course TA. This session runs from 1pm-3:30pm on Fridays, ahead of the time the homework is due. 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. | ||
| Attend Office Hours | Office hours are specifically reserved for you to receive individual, one-on-one help from the instructor or TAs. 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 July 3 (class starts 7/3, no class 7/4) |
§1.1.1: Motivation for Dynamical Systems §1.1.2: Orbits and Fixed Points §1.1.3: Periodic Points and Cycles §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 Homework 1 due Saturday 7/8 at 5am Eastern. |
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| Week of July 10 | §1.2.3: Attracting and Repelling Cycles §1.2.4: Weakly Attracting and Weakly Repelling Points (and Cycles) §1.2.5: Basins of Attraction §1.3: Newton's Method §2.1.1: The Quadratic Family qc(x) = x2+c §2.1.2: General Properties of Bifurcations Homework 2 due Saturday 7/15 at 5am Eastern. |
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| Week of July 17 | §2.1.3: Saddle-Node Bifurcations §2.1.4: Period-Doubling Bifurcations §2.2.1: The Schwarzian Derivative §2.2.2: Numerical Computation of Attracting Cycles §2.3: Orbit Diagrams Homework 3 due Saturday 7/22 at 5am Eastern. |
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| Week of July 24 |
§3.1.1: Motivation for Symbolic Dynamics §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 §3.1.5: Equivalence of the Quadratic Maps and the Shift Map §3.2.1: Motivation for Chaos Review for Midterm 1. Homework 4 due Saturday 7/29 at 5am Eastern. |
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| Week of July 31 | §3.2.2: Devaney's Definition of Chaos §3.3.1: The Period-3 Theorem MIDTERM 1 in class on Tuesday, August 1st §3.3.2: The Sarkovskii Ordering and Sarkovskii's Theorem §4.1: Classical Fractal Constructions §4.2.1: Topological Dimension Homework 5 due Saturday 8/5 at 5am Eastern. |
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| Week of August 7 | §4.2.2: Minkowski Dimension §4.3.1: Iterated Function Systems and Self-Similarity §4.3.2: Minkowski Dimensions of Self-Similar Sets §4.3.3: The "Chaos Game" §5.1: Dynamical Properties of Complex-Valued Functions Homework 6 due Saturday 8/12 at 5am Eastern. |
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| Week of August 14 (class ends 8/17) |
§5.2: Julia Sets For Polynomials §5.3.1: General Julia Sets §5.3.2: Julia Sets and Cantor Sets §5.3.3: Critical Orbits and the Fundamental Dichotomy §5.3.4: The Mandelbrot Set §5.3.5: Analysis of the Mandelbrot Set Review for Midterm 2. Homework 7 due Saturday 8/19 at 5am Eastern. |
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| Week of August 21 |
MIDTERM 2, Tue Aug 22nd, 1:00pm-3:00pm, Richards 235 | ||