Instructor: Chris Vales, chris.vales@dartmouth.edu, Kemeny 209

Lectures: TuTh 2.25-4.15pm, Kemeny 004

X-hour: W 5.30-6.20pm, Kemeny 004

Office hours: F 5-7pm, Kemeny 229

Canvas webpage: canvas.dartmouth.edu

Course description. Dynamical systems is a branch of mathematics studying the properties of time-dependent phenomena, with applications throughout science and engineering. This is an introductory course, focusing primarily on dynamical systems that evolve in continuous time, described by ordinary differential equations.

The material is naturally applied and interdisciplinary, and should be interesting and accessible to students from a broad variety of majors. We will consider applications in science and engineering throughout the course. For students familiar with a programing language (such as Python or Matlab), there will also be opportunities for computational projects.

Contrary to traditional courses on differential equations, our primary focus will be on qualitative properties of the studied equations, such as the kinds of solutions they admit, how these solutions change with changing equation parameters, and how that affects the kinds of models that can be built with these equations. The course will culminate in the study of chaotic dynamical systems and their properties.

Prerequisites. Familiarity with ordinary differential equations and linear algebra, especially the calculation of eigenvalues and eigenvectors of matrices. The courses Math 22 and 23 or similar ones will suffice. If you are unsure about prerequisites, contact the instructor to discuss your background.

The course can also be extended with more advanced reading and computational assignments to serve as an entry level graduate course. If you are a graduate student interested in this course, contact the instructor to discuss your background and interests.

Textbook. Nonlinear dynamics and chaos, Steven Strogatz (CRC Press, 2nd ed, 2018).
PDF version available from the Dartmouth Library: library.dartmouth.edu

Teaching method. We will meet for two lectures each week, using the X-hour as required. Before each lecture, I will ask you to read a few sections of the textbook before attending the lecture. During the lecture, we will cover parts of the material in more detail, consider applications and address student questions.

Grading

The weekly homework assignments will consist of problems from the textbook, and are meant to give you an opportunity to test your understanding of the material covered each week. They will be due on Sunday night at the end of eack week, and will be submitted on Canvas.

The two midterm exams will consist of problems similar to the ones featured in the weekly homework assignments. They are meant to act as checkpoints during the course, to help you make sure that you understand the main concepts of the covered material before we move on. Students that complete the homework assignments and attend class regularly will find both exams straightforward.

The final project will be on a topic of your choosing related to the course material, and it can have both analytical and computational components. The purpose of the final project is to give you an opportunity to study in more detail topics of the course that align with your interests. It is also a chance to apply some of the concepts learned in the course to problems in science and engineering.

Exams

Tentative schedule. Below is a tentative schedule for the course, listing the topics scheduled for each week and the textbook sections they correspond to.

  1. Intro, 1D flows (1, 2)
  2. 2D flows (5.0-5.2, 6.0-6.3)
  3. 2D flows (6.4-6.7, 7.0-7.3)
  4. 1D bifurcations & Midterm 1 (3.0-3.2, 3.4, 3.6)
  5. 2D bifurcations (8.0-8.4, 8.7)
  6. Intro to chaos & Midterm 2 (9.0, 9.2-9.5)
  7. 1D maps (10)
  8. Fractal sets (11)
  9. Chaotic attractors (12.0-12.4)
  10. Deadline to submit final project

Class attendance. Class attendance is not mandatory but strongly encouraged. For the health and safety of others, do not attend class if you are experiencing symptoms of illness.

Academic integrity. We will enforce Dartmouth's Academic Honor Policy. Violations will be referred to the Committee on Standards. In particular, please be aware of rules regarding cheating and unauthorized collaboration.

Refer to the Student Handbook for additional information.

Schedule conflicts. Students anticipating schedule conflicts with course activities due to religious observances or participation in athletic events should inform the instructor as soon as possible to discuss appropriate adjustments.

Disabilities. Students who want to request disability accommodations approved by the Student Accessibility Services office should inform the instructor as soon as possible.