# Math A4500: Syllabus

**Course Title:** Dynamical Systems

**Prerequisites:** C or better in Math 32404 (B or better in Math 32300 and taking 32404 at the same time is acceptable).

**Catalog Description:** Dynamical systems arise naturally from connections to the sciences and many mathematical subjects both pure and applied. Students will be able to apply techniques learned in this course to these interrelated subjects. This course provides an introduction to important classes of dynamical systems and exposure to the most important phenomena which appear in the subject.

**Semester:** Fall 2022.

**Section:** ST.

**Meeting time and place:** TuTh 6:00PM - 7:40PM in NAC 4/210.

**Instructor Information:**

**Name:**Prof. Pat Hooper**Office Hours:**3-4pm on any Tuesday our class meets in in MR 209. Or by appointment.**Office:**Marshak 209**Office Phone:**(212) 650-5149 (extension 5149 from on campus)**Email:**whooper@ccny.cuny.edu

I welcome you to contact me outside of class and office hours. I'm generally most easily reachable by email.

**Course Textbooks:**

*An Introduction To Chaotic Dynamical Systems*by Robert L. Devaney, 3rd edition.

Other sources such as journal articles and exerpts from other textbooks may be used as well.

**Topics covered:**
We will spend approximately two thirds of the course discussing One Dimensional Dynamical Systems as well as introducing Zero Dimensional Dynamical Systems. (Shift spaces have zero topological dimension but are remarkably important!) We'll spend most of the remaining one third of the class discussing Dynamical Systems of dimension two or more. I do not expect to cover Complex Dynamics. However, I will very likely also introduce some dynamical systems related to my own interests: Piecewise isometries, polygonal billiards, and systems that come from geometry.

I tentatively plan to cover at least the following topics:

- Introduction to dynamical systems (roughly following Devaney's I.1-I.2)
- Examples of Dynamical Systems (perhaps different than Devaney I.2)
- Devaney I.3: Elementary Definitions
- Devaney I.4: Hyperbolicity
- Devaney I.5: The Logistic Family of Maps
- Devaney I.6: Symbolic Dynamics
- Devaney I.7: Topological Conjugacy
- Devaney I.9: Structural Stability
- Devaney I.8: Chaos
- Devaney I.10: Sharkovsky’s Theorem
- Devaney I.11: The Schwarzian Derivative
- Devaney I.12: Bifurcations
- Devaney I.13: Shifts of Finite Type and Period Three
- Devaney I.16: Maps of the Circle
- Devaney III.27: Dynamics of Linear Maps
- Devaney III.28: The Smale Horseshoe Map
- Devaney III.29: Hyperbolic Toral Automorphisms
- Devaney III.30: Attractors

**Attendance:**
You should attend every class but
extenuating circumstances arise that can
make this difficult. If you cannot attend a
class, please let me know. If
circumstances make you miss more than
3 classes during the semester, you may
be overextended. I ask that you come see
me to discuss your options.

**Class participation:**
In order to learn mathematics, it is important to participate actively: asking questions, answering questions, and suggesting ideas. All of us in the class, you, me, your peers,
have a responsibility to create an
environment in which this is possible. It is important to note that students in this class come from a range of background and have different paparedness for the material in this course. In particular, I expect everyone to be an active participant, and supportive of other participants and to work towards maximizing collective learning.

**General expectations:** For each hour spent in the classroom, I expect you to spend at least three hours reading and understanding the book, understanding lecture notes, doing homework. Practice (doing problems) is an important part of learning. Only adequate practice will guarantee that you can complete exam problems in a timely manner.

**Expectations of written work:**Grades assigned to mathematical arguments will be partially based on presentation. In order to receive full credit, a student who reads your answer should be able to easily understand how you solved the problem. Written work is expected to be legible and arguments are expected to be well articulated.

**Grades:**Grades will be computed from the following weighted average:

- Homework, Classwork, and Quizzes (20%)
- Exam grade (60%)
- Final Presentation (20%)

The exam grade will be a weighted average of the midterm exams taken. I expect to offer three midterms.

Your grade percentage will be tabulated out of 100% as indicated by the percentages above. A letter grade will be assigned to you according to the following list: A+ (97-100), A (95-96), A- (90-94), B+ (87-89), B (84-86), B- (80-83), C+ (77-79), C (74-76), C- (70-73), D (60-69), F (0-59).

**Course website:**Course information, homework assignments, and documents can be found on the website:

**Blackboard:**I use blackboard to keep track of your grades. You can view your grades there. To access blackboard visit: Solutions to problem sets may be posted to blackboard.

**Homework assignments:** Homework will be assigned approximately once a week and will have a due date.
Homework assignments will be made available on the course website at least one week before the assignment is due.

I encourage you to work in groups on the homework problems, especially if this best suits your learning style. Nonetheless, you should be confident that you understand how to do each problem, and should be able to solve similar problems independently. Failure to ensure that you can solve problems independently will surely have a negative effect on quiz and exam grades.

If you do work jointly with other students, you must acknowledge this. All work turned in must be written in your own words. Failure to follow these guidlines constitutes plagarism; see the discussion of Academic Integrity below.

Problem sets have a mix of assigned homework problems, some assigned for you to do, and some assigned for you to turn in. You are expected to think carefully about all these problems as I anticipate a failure to do so adversely affecting your exam grades. Please only turn in problems which are marked to be turned in.

**Quizzes and Classwork:**At this point, I do not anticipate collecting classwork or giving quizzes, but this could change if the need becomes apparent as we move through the course. There will be no makeups for missed quizzes or classwork.

**Dropped grades:**The lowest 20% of assignments in the “Homework, Classwork, and Quizzes” category will be dropped (rounding up if necessary).

**Late homework:**Late homework will not be accepted for any reason. If you need to miss class, please scan and email your assignment to me as a PDF document before the start of class on the day it is due.

The dropped grades are intended to compensate for missed assignments due to circumstances beyond your control. If however you feel that these dropped grades are insufficient to rectify the negative impact on your grade by your circumstances, please reach out to me.

**Midterm exams:** There will be three midterm exams. They will be held in class on the following dates: September 22, November 3, and December 13. You will be given the full class period to complete each midterm.

**Final Presentations:** In this course, we will work through many examples of dynamical
systems, with the primary goal being to introduce general tools used to understand these systems.
I hope that the final projects will increase the class's perspective of the breadth of the subject
of dynamical systems and connections to other areas of mathematics and the sciences, and give individuals
an opportunity to connect the subject with their interests.

You are expected to give a presentation on a topic in Dynamical Systems which we have not covered in the course. Presentations will be given at the time our Final exam is scheduled and should be 10-15 minutes long. Each individual will be responsible for their own presentation. More details will be provided midway through the semester.

As of this writing, the final presentations will be held on Tuesday, December 20th from 6pm to 8:15pm. For the most up to date information about the time of the final exam, see the Registrar's Final Exam Schedule.

**Exam and project makeups:** A single midterm missed under well-documented and sufficiently compelling circumstance will result in that grade not counting towards your Exam grade

portion of the course grade. (So in this case your Exam grade will be the average of the remaining midterms.)

If two midterms are missed under well-documented and sufficiently compelling circumstances, an offer of a makeup for the second midterm will be made. The makeup must be taken within one week of the scheduled exam, or a zero will be assigned as the exam grade.

Similarly if the final project is missed under well-documented and sufficiently compelling circumstances, an offer of a makeup will be made. The makeup must be within one week of the final exam date, or a zero will be assigned as the project grade.

Exceptions to the one week time period will only be made if a student can prove that they are unable to take the exam that week for compelling reasons.

Notify me ahead of any exam you expect to miss to be sure your circumstances are sufficiently compelling.

At the department’s or the instructor’s discretion, any makeup exam or makeup for the final project may be administered as an oral examination carried out either in-person or using video-conferencing software (such as Zoom).

**Academic Integrity:** All work submitted for this course should be your own unless explicitly stated or acknowledged by you. This course follows the CUNY Policy on Acacdemic Integrity Policy. The college takes academic integrity issues seriously. Violations will be pursued as described in this policy. Punishments for violating this policy include academic pentalties such as zero on the assignment or failure in the course, as well disciplinary sanctions such as suspension and expulsion from the college.

In particular, it is expected that you not plagarize. This has been especially problematic on homework. Your homework must not be copied from another source, and you must cite any sources used when preparing your solutions. Sources can include textbooks, webpages, discussions with other people, and other student work. Citations should be as specific as possible. All submitted work must be written in your own words.

**Pandemic Issues:** This is an in-person class, but policies may change because of safety concerns. Online instruction methods may be required (for example if the instructor tests positive for COVID-19).

Currently examinations are intended to be in-person, but, due to ramifications of the ongoing COVID-19 pandemic, it is possible that other examination methods may be required. This course may use online examination methods, may give some examinations as oral exams, and may require the use of video cameras during exams.

If online examinations are given in this course, the exams will be given synchronously (at the time in which the class meets), likely on Blackboard.

**Video lectures:**Students who participate in video lectures for this class with their camera on or use a profile image are agreeing to have their video or image recorded solely for the purpose of creating a record for students enrolled in the class to refer to, including those enrolled students who are unable to attend live. If you are unwilling to consent to have your profile or video image recorded, be sure to keep your camera off and do not use a profile image. Likewise, students who un-mute during class and participate orally are agreeing to have their voices recorded. If you are not willing to consent to have your voice recorded during class, you will need to keep your mute button activated and communicate exclusively using the “chat” feature, which allows students to type questions and comments live.