Syllabus for MA 573 Qualitative Theory of Ordinary Differential Equations

Syllabus for MA 573 Qualitative Theory of Ordinary Differential Equations, Fall 2019

Course Information

Lecturer: Ryan Goh

Email: rgoh@bu.edu
Office: MCS 243
Office Hours: Wednesday 11:00am - 12pm, Thursday 10am-12pm or by appointment.
Web page: http://math.bu.edu/people/rgoh/

Lectures:

Time: Monday,Wednesday,Friday, 10:10am-11:00 am;
Location: CAS 221

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 is on reserve at the BU Science and Engineering library

Course Webpage:

math.bu.edu/people/rgoh/teaching/ma573-fs19/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-D systems, 2-D linear systems, linear algebra and solving/analyzing general linear systems,
Nonlinear systems, bifurcations, closed orbits, and limit sets.
Chaos, applications, and interesting areas of Dynamical Systems

Roughly our course schedule, subject to student interests and class pace, will be as follows

Week 1: Chapter 1, Fundamental concepts in one-dimensional ODEs
Week 2: Chapter 2, Planar linear systems
Week 3: Chapter 3, Classification of linear systems, trace-determinant plane
Week 4: Chapter 4 and 5, Dynamical systems
Week 5: Chapter 5, General concepts from linear algebra
Week 6: Chapter 6, Solving general linear systems
Week 7: (No class on Monday 10/14, subsitute with class on Tuesday 10/15) Chapter 7, Intro to nonlinear systems, existence and uniqueness
Week 8: Chapter 7, More on existence, uniqueness, dependence on initial data
Week 9: Chapter 8, Fundamentals of local nonlinear analysis (sources and sinks, linearization, invariant manifolds, stability,)
Week 10: Chapter 8, More nonlinear analysis (bifurcations, examples, exploration), start Chapter 9, global nonlinear analysis
Week 11: Chapter 9, Global techniques, dynamics away from equilibria
Week 12: Chapter 10, Closed orbits and limit sets
Week 13: (Thanksgiving week, no class on 11/27 or 11/29) Exploration into Chemical Oscillations and patterns (or another topic such as invasion fronts, political opinion dynammics, numerical continuation, or dynamics and PDE's)
Week 14: Class preference: Possible topics include chaotic systems, and intro to discrete dynamical systems, one of the applications listed below, or current areas of interest in dynamical systems,
Week 15: Continue with Week 14 topic, last day of class is 12/11

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, 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:

Pattern formation in nature
Chemical oscillations
Singular pertubations
Quantum systems
Celestial mechanics
Climate tipping points
Fluid convection rolls
Soliton waves
Coupled oscillators and biological systems
Models for infections disease
Chaos and Cryptography
Opinion and voting dynamics in populations

Additional References

Here are some other useful references. All of these have been put on reserve 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 and Quizzes

Homework and associated readings will be posted on the course webpage. They will be due at the beginning of class on Fridays. No late work will be accepted, but it is ok if you have a classmate turn in a weeks assignment. You may also turn in your assignment to me before the due date. Your lowest two scores will be dropped.

Doing exercises and problems is the best way to learn mathematics (and it is the best way to study for the midterms and exams). 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 turned in on stapled pages. If using notebook paper please remove any frills. 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 at random 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:

Mathematical content (4 points)
- 4 points - The solution 100% correct.
- 3 points - The solution shows a good understanding of the problem but there are some minor mistakes and/or typos present.
- 2 points - The solution is incorrect but has some good ideas in it.
- 1 points - A serious effort has been made on the problem
- 0 points - Only a trivial, or no effort has been made
Communication (2 points) These will only be assigned if you get 2 points or higher on the mathematical content portion.
- 2 points - The solution is clearly written, well-organized, and justified with sufficient detail.
- 1 point - The solution is readable, but is missing details or is hard to follow in places.
- 0 points - The solution is unreadable, or has is missing many details.

Midterm

There will be one in-class midterm during the semester. It will be held on Tuesday, October 15, during the class period. The problems will be roughly similar to examples done in class and those done in the homework. Material covered will be outlined at least a week before the exam.

Final Exam

The final will be take-home, with problems being emailed out on the morning of Monday, December 16th. Problems will be similar to the those of the midterm, homeworks, and in-class examples. Logistics of the final will be discussed a few weeks prior. No collaboration, or use of the internet is allowed. You are allowed to consult your class notes, textbook, and any course handouts, as well as use any computer codes for plotting phase portraits you may have written up over the span of the semester.

Grades

Your course grade will assigned as follows

Homework scores (50% - two lowest scores dropped)

Midterm 1 (20%)

Final exam (30%)

Records for assignment grades will be kept on the course blackboard website.

Classroom policies

Questions are always welcome during lecture. Please raise your hand. 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, 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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