Syllabus for MA 573 Qualitative Theory of Ordinary Differential Equations

Syllabus for MA 573 Qualitative Theory of Ordinary Differential Equations, Fall 2022

Course Information

Lecturer: Ryan Goh

Email: rgoh@bu.edu
Office: MCS 243
Office Hours: Tu 2:30p-3:30p (in person), W 11a - 12:10p (in person), Thu 11-12p(zoom) or by appointment (please provide your availability when requesting an appointment).
Web page: http://math.bu.edu/people/rgoh/

Lectures:

Time: Monday,Wednesday,Friday, 10:10am-11:00 am;
Location: SOC B59

Textbook:

"Differential Equations, Dynamical Systems, and an Introduction to Chaos" by Morris W. Hirsch, Stephen Smale, Robert L. Devaney; 3rd Edition. ISBN: 978-0123820105. A copy has been put on reserve at the BU Science and Engineering Library. An electronic copy can be found through the BU library website here .

Course Webpage:

math.bu.edu/people/rgoh/teaching/ma573-fs22/course-page.html

Course Description:

An introductory course in nonlinear dynamical systems. We shall focus on how to qualitatively analyze solutions of ordinary differential equations, even when solutions cannot be explicitly written down. I will assume that you have a background in Calculus, up to and including Multivariable Calculus (BU's MA 123-124-225 or 230), linear algebra (BU's MA 242 or 442) and that you have had a "Sophomore level" differential equations class (at the level of BU's MA 226 or MA 231). Necessary topics and tools from these fields will be reviewed, and sometimes extended as needed. If you want any extra references to help review these topics please do not hesitate to contact me!

Our course will be divided into roughly 3 parts.
  1. 1-D systems, 2-D linear systems, linear algebra, and solving/analyzing general linear systems
  2. Nonlinear systems, bifurcations, closed orbits, and limit sets
  3. Chaos, applications, and interesting areas of Dynamical Systems
Roughly our course schedule, subject to student interests and class pace, will be as follows

Numerical exploration

We will also utilize computer/numerical simulations to help us explore and analyze differential equations throughout the course. In my examples, I will mainly use MATLAB, which can be downloaded here for free by any BU student. See this quick tutorial on how to get started with the software if you haven't used it, and here or here for a tutorial on solving differential equations with it. You are of course welcome to use another software (such as Python, Julia, Mathematica, etc...) and while I may be familiar with the specific language you are using, I cannot guarentee I will be able to help debug/troubleshoot code written in all languages.

Applications

One of the beautiful things about the theory of dynamical systems is how it has been applied in many scientific disciplines. We aim to highlight as many applications as possible throughout this course. If you have a specific field which you are interested in, please contact me and I will try to work related examples into my lectures! Some possible areas of application include:

Additional References

Here are some other useful references which can be found at the BU Science & Engineering Library. More references and links to specific papers relevant to lectures will be posted on the course webpage.
  • "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering," by Steven H. Strogatz, Second Edition. An online copy of the textbook can be found via the BU library here.
  • Hale, Jack K., and Kocak, H.. Dynamics and bifurcations. Vol. 3. Springer Science & Business Media, 2012.
  • Arnold, V. I. Ordinary differential equations (translated by RA Silverman). MIT Press, Cambridge, Massachusetts 2 (1973): 196-213.

    Homework

    Homework and associated readings will be posted on the course webpage. Please neatly write out or type your solutions, turn them into a PDF (scanning if you write them), and email them to me. They will be due at the beginning of class on Fridays. No late work will be accepted. Your lowest two scores will be dropped.

    Doing exercises and problems is the best way to learn mathematics. As such we will have weekly homework assignments. You are welcome (and encouraged!) to work together, but please make sure to write up solutions by yourself and in your own words. You must understand how to solve the problems yourself. If it is suspected that you copied down your answers from another source of any kind, you may be asked to come to my office and explain your solution(s), without any books or notes.

    In your solution sets, explain your work clearly, concisely, and using complete sentences. Homework should be legible and organized. If you are worried about the legibility of your handwriting, please contact me about using the mathematical typesetting software LaTeX. I will pick 4 problems to grade from each problem set. The majority of the homework problems will be graded using the following 6-point system adapted from Prof. Chad Topaz, of Williams College: Solution sets will be emailed out along with your graded solutions one to two weeks after the due date.

    Midterm

    There will be two take home midterms during the semester. View these as a kind of larger homework assignment which span material from more than one chapter or section of the text.
    The problems will be roughly similar to examples done in class and in homework. Material covered will be outlined at the week before the exam. You will have at least 48 hours to complete the exam. The exam will be open book and open note. The use of the internet or other resources, unless explicitly mentioned, will not be allowed. You must do the exam on your own and will not be allowed to communicate with others about it. The tentative starting dates for the midterms are October 14th and November 18th.

    Final Project

    You will be required to do a final project which investigates or explains a topic related to ordinary differential equations and dynamical systems, but is not covered by the course. The project will consist of a 20-30 page scientific report and a class presentation. The goal is to give you experience in independently learning about and investigating a research topic and composing a coherent scientific document (with an organized bibliography) and presentation which communicates your findings. The report should describe and synthesize the topic in you own words, summarizing relevant and related literature, important results, and working out example calculations, exercises, or applications. Wrote copying or paraphrasing of results from a textbook will not be sufficient.
    There are many ways one could approach this project, several general ideas would be to
  • Read research and review papers in a topic, explain them in your own words, and then work out, reproduce, and/or expand upon calculations and numerical simulations done in the works.
  • Apply the theoretical and numerical methods discussed in the course to a mathematical model or dynamical system of interest to you.
  • Read through chapter(s) in the textbook not covered (or another textbook) as well as related mathematical literature, synthesize what you've learned, and solve several exercises in the texts. I encourage you to pick a topic of interest to you or related to what you wish to go on to work on or study after BU. Please do contact me if you need help finding a topic and we can discuss potential ideas!

    Near the end of the semester you will also be required to give a short class presentation on your project. More details and a schedule of talks will be determined later in the semester.

    You are welcome to work by yourself, or with one other classmate on the project. This must be decided when you get your topic approved by me. If working in a pair, the scope of the project will be expected to be proporationally larger (for example: maybe one person describes the theory of a subject, while the other does numerical investigations, and each write half of the report). You will also be required to jointly compose and sign a short statement describing each persons contributions to the project.

    You must meet with me (in office hours or by appointment) at least once to discuss the suitability of your project and get it approved by me. Of course, you are more than welcome to meet with me more than once!

    Project Deadlines

    Missing any of these will result in a 2% grade penalty for the project grade each day you are late.
  • September 28th: Meet with me at least once by this date (ideally earlier), and get project topic and plan approved.
  • November 28th: First draft of report due. I will read through and give feedback on the report, so you can make revisions. The draft will not be graded but turning a significant draft (i.e. not just a page or two) on time is required.
  • December 12th: Final draft of project is due to me by email.

    • A few project ideas

    (of course feel free to come up with one of your own!):
  • Computer assisted proofs in dynamical systems: a way to do numerical simulations in a mathematically rigorous ways
  • Data-based methods in dynamical systems: (such as Dynamic Mode Decomposition, Machine-learning, Deep learning, resevoir computing, or data assimilation)
  • Numerical continuation, and how its used to numerically study how solutions (equilibria, periodic orbits, heteroclinic orbits, invariant manifolds) vary with parameters.
  • Numerically computing invariant manifolds
  • Complex differential equations
  • Some possible application areas: Infectious disease dynamics, opinion dynamics, mathematical neuroscience, fluid dynamics, chemical systems, mathematical ecology, climate models, celestial mechanics, pattern formation in nature, financial models, nonlinear/condensed matter physics, water waves and coherent structures in nature, forced nonlinear oscillators, dynamics on networks.
  • Some possible theoretical areas: Dynamics and control problems, Dynamical systems with noise/randomness, large deviation theory, invariant manifold theory, dynamical systems and symmetry theory, bifurcation theory, discrete dynamical systems, Hamiltonian dynamical systems,
  • You could also take a look through recent issues of the following journals to get some ideas: SIAM Journal on Applied Dynamical Systems , SIAM Journal on Applied Mathematics , SIAM Review .

    Grades

    Your course grade will assigned as follows
  • Homework scores (45% - two lowest scores dropped)
  • Midterms (15% each )
  • Final project (25%)
    • Records for assignment grades will be kept on the course blackboard website.

    Classroom policies

  • Questions are always welcome during lecture. Please raise your hand or interupt me if I'm facing the board. Sometimes, due to time restraints or flow of the lecture, I may have to delay a question till another section, or after class.
  • The use of cellphones, laptops, or the internet, when not part of class activities (i.e. to take notes, or share the result of a computer simulation), is forbidden. Such activities can be very disruptive, distracting, and disrespectful to those around you and are not conducive to a productive learning environment.
  • Throughout the semester we will break up into small groups to work on explorational problems and examples. I expect all students to be kind, encouraging, respectful, and engaged participants during these activities.
  • Attendance Policy : Students are expected to attend each class. We will follow the BU attendance policy . The class periods will be a key component of the course. While math is mostly learned by doing, I view lectures as a guide which helps one more efficiently and effectively learn a topic.

    Excused absenses and make-up exams

    Please let me know about all religious observances at the beginning of the semester. As mentioned above, there will be no make-ups for homeworks. In extreme circumstances (religious observance, death in the family, emergency) there can be make-ups for exams. Please tell me during the first week of school if you will need a make-up exam.

    Students with disabilities

    Please contact me as soon as possible. I am happy to work with you and the BU office of Dissability and Access Services.

    Academic code of conduct

    Please do not cheat. Furthermore, copying of answers from a friend, solution manual, or online solution set is detrimental to your learning. You will be held to the BU academic code of conduct.

    Extra Help

    If you feel you are falling behind in the course please do not hesitate to contact me! It is always easier to address misunderstandings sooner than later. In addition to working with me, I can also put you in touch with TAs in the     tutoring room who are knowledgeable in dynamical systems theory.



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