Course Description:

This is a beginning graduate level course on dynamical systems. It covers basic results for
(i) linear systems;
(ii) local theory for nonlinear systems (existence and uniqueness of solutions, dependence on parameters, flows, linearization, stable manifold theorem);
(iii) global theory for nonlinear systems (global existence, limit sets, periodic orbits, Poincare maps);
(iv) further topics (time and interests permitting): bifurcations, averaging, asymptotics.

Instructor:

Aaron Nung Kwan Yip

Department of Mathematics

Purdue University

Contact Information:

Office: MATH 432

Email

Lecture Time and Place:

54300-001 (17540) TR 12:00pm - 1:15pm, SCHM 309

(Class Zoom Room for online meetings)

Office Hours:

Tue: 4:00pm-5:00pm, Wed: 3:00pm-4:00pm, or by appointment.

Prerequisites:

One (undergraduate) course in each of the following topics:
linear algebra (for example, MA 265, 351, 511),
differential equation (for example, MA 266, 366),
and some familiarity in analysis (for example, MA 341, 440, 504).

Textbooks and References:

Main text:
[M] Differentiable Dynamical Systems (Revised edition, 2017), J.D. Meiss

References:
[P] Differential Equations and Dynamical Systems, I. Perko
[B] Stability Theory of Differential Equations, R. Bellman
[T] Ordinary Differential Equations and Dynamical System, G. Teschl
[GH] (Best of the best!) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, J. Guckenheimer, P. Holmes
[HSD] Differential Equations, Dynamical Systems: An Introduction to Chaos, M. W. Hirsch, S. Smale, R. L. Devaney
[S] Nonlinear Dynamics and Chaos, with Applications to Physics, Biology, Chemistry, and Engineering, S. H. Strogatz
[A] Mathematical Methods of Classical Mechanics, V. I. Arnold

Some texts at the undergraduate level:
[Br] Differential Equations and Their Applications, M. Braun
[BD] Elementary Differential Equations and Boundary Value Problems, Boyce and DiPrima (textbook of Purdue MA366, on reserve in libray) ,

Course Policy:

Your grade is based on:

(50%) homeworks (five or six problem sets, roughly once every two weeks, normally due on Thursdays, in Gradescope);

(5%) abstract of paper: due at 11:59pm, Friday, Mar 27th, 2026, in Brighspace;

(25%) paper (roughly ten pages, details TBA): due at 11:59pm, Friday, Apr 24th, 2026, in Brightspace;

(20%) presentation (15 min, details TBA) on the submitted paper during the final exam week (schedule TBA).

Guidelines about submitting work:

For each of the above course components (homework, paper, presentation), you have the option of submitting as a group which can consist of up to three people. You can of course discuss with more people, but the submitted materials must be distinct between groups. Each member of a group will receive the same grade. The group members can change for different homework but those for the paper and presentation must be the same.

For the homework, sufficient work must be shown to explain your answer.
Staple your homework to prevent 5% penalty.

Progressive deduction will be imposed for late submissions.

At our discretion, points will be deducted if you use methods and notations not covered in class. You have the right to contest, but you will be asked to explain using the method covered in class at the point of time the homework is assigned.

You are allowed to utilize extra information from other (online) resources, such as Wikipedia, plotting routines and so forth. However, getting a solution completely from online (such as ChatGPT, chegg.com and so forth) is not permitted.

My MOTTO on the use of technology (which I use often) is:
IF TECHNOLOGY HELPS YOU UNDERSTRAND, BY ALL MEANS USE IT. OTHERWISE, USE IT AT YOUR OWN RISK!

You are expected to observe academic honesty to the highest standard. Any form of cheating will automatically lead to an F grade, plus any other disciplinary action, deemed appropriate.

Nondiscrimination Statement:

This class, as part of Purdue University's educational endeavor, is committed to maintaining a community which recognizes and values the inherent worth and dignity of every person; fosters tolerance, sensitivity, understanding, and mutual respect among its members; and encourages each individual to strive to reach his or her own potential.

Student Rights:

Any student who has substantial reason to believe that another person is threatening the safety of others by not complying with Protect Purdue protocols is encouraged to report the behavior to and discuss the next steps with their instructor. Students also have the option of reporting the behavior to the Office of the Student Rights and Responsibilities. See also Purdue University Bill of Student Rights and the Violent Behavior Policy under University Resources in Brightspace.

Accommodations for Students with Disabilities and Academic Adjustment:

Purdue University strives to make learning experiences accessible to all participants. If you anticipate or experience physical or academic barriers based on disability, you are also encouraged to contact the Disability Resource Center (DRC) at: drc@purdue.edu or by phone at 765-494-1247.

If you have been certified by the DRC as eligible for accommodations, you should contact me to discuss your accommodations as soon as possible. See also Courses: ADA Information for further information from the Department of Mathematics.

Campus Emergency:

In the event of a major campus emergency or circumstances beyond the instructor's control, course requirements, deadlines and grading percentages are subject to change. Check your email and this course web page for such information.

See also Emergency Preparedness and Planning for campus wide updates.

More information on University Policies:

See your MA543 course homepage in Brightspace.
Content (tab at upper left corner): Student Support and Resources, and University Policies and Statements.

Course Progress and Announcement:

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Week 1: Jan 13, 15

[M, Chapter 1]
introduction, notations, conversion between higher order equation and first order system;
examples (from biology and mechanics);
existence and uniqueness of solutions:

explicit formula for solutions, integrating factor, variation of parameters,

finite time blow-up of nonlinear super-linear equations;

non-uniqueness of solutions.

Note: What are ODEs?
Note: Examples of ODEs
Note: Basic concepts of ODEs
Ref: Mathematical models : mechanical vibrations, population dynamics, and traffic flow : an introduction to applied mathematics, Haberman
Ref: An excellent paper on Poincare, celestial mechanics and chaos
(Everytime I read this paper, I get something "more" out of it while at the same time I forgot most of what I read the last time....)

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Week 2: Jan 20, 21

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Week 3: Jan 27, 29

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Week 4: Feb 3, 5

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Week 5: Feb 10, 12

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Week 6: Feb 17, 19

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Week 7: Feb 24, 26

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Week 8: Mar 3, 5

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Week 9: Mar 10, 12

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(Spring Break: Mar 16-20)

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Week 10: Mar 24, 26
(Abstract of paper due at 11:59pm, Friday, Mar 27th, in Brighspace)

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Week 11: Mar 31, Apr 2

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Week 12: Apr 7, 9

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Week 13: Apr 14, 16

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Week 14: Apr 21, 23
(Paper due at 11:59pm, Friday, Apr 24th, in Brighspace)

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Week 15: Apr 28, 30

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Week 16: May 4-8: Pressentation week (schedule TBA)