Math 519–Ordinary Differential Equations

Course Syllabus

Instructor

Sigurd Angenent, 609 Van Vleck Hall.

Lecture times and location

MoWeFr 8:50AM - 9:40AM Van Vleck B119

Prerequisites

Students should know linear algebra or matrix algebra (at the level of either math 320, 340, 341, or 375), as well as some analysis (either math 375+376, 421, or 521).

Textbook:

V.I.Arnold, Ordinary Differential Equations, 3^rd edition, Springer Verlag, 1992.

We will follow the book more or less, filling in gaps where the author assumes more than the prerequisites for this course. The book has problems which range from straightforward to really, really hard. Most homework will be assigned from the additional problems that appear on the Math 519 Problems Page.

Office hours, Email, Piazza

Email is not a good medium for mathematical discussion, and as the enrollment in our class has reached 40, office hours are not an effective way for answering mathematical questions about differential equations. Instead, please direct all your questions to the Piazza page for our course. I will check Piazza daily. You can post questions, answer other students’ questions, chime in (“I had the same question”), and browse old questions and answers. You can post questions anonymously to the other students, if you like.

Exams, homework

There will be two in–class midterm exams and one final exam. Homework will be assigned regularly below on this page. The midterms will be given on the following dates:

Midterm 1 : Friday February 26
Midterm 2 : Friday April 8

Grading scheme

Homework will count for 15%, the two midterms 25% each, and the final for 35%. A letter grade will be determined according to this tentative set of cut offs:

A≥92%, AB≥86%, B≥78%, BC≥70%, C≥60%, D≥50%, F≥0%.

Course description, topics

Direction fields, Vector Fields, and Differential Equations.
One dimensional examples. Stability of fixed points, bifurcation diagrams, and the Implicit Function Theorem.
Existence and Uniqueness theorems for ODE
Phase planes and higher dimensional examples: Lotka–Volterra or Predator–Prey systems, Mechanical systems with one degree of freedom, Spherical Pendulum
“$1+1$ dimensional” examples. Growth equations with periodically changing environment—Poincaré maps.
Return to existence and uniqueness theorems: dependence on initial data and on parameters.
Application to second order linear equations: Sturm oscillation theorems.
General linear theory (a long chapter!)
- Review of eigenvalues, eigenvectors, and matrix norms.
- Power series definition of $e^A$
- Solution of $\dot\vx = A\vx$ when $A$ is diagonalizable.
- Solution of $\dot\vx = A\vx$ when $A$ is nilpotent.
- The case of complex eigenvalues. Arnold’s notions of “complexification” and “realification.”
- Solving inhomogeneous equations $\dot\vx = A\vx + \vf(t)$. Resonance.
- Non autonomous linear equations, i.e. equations of the form $\dot\vx = A(t)\vx$; fundamental solution, Wronskian, Liouville’s theorem.
- Application: the flow of a Hamiltonian system in $\R^2$ is area preserving.
Topological classification of singular points
Other higher dimensional examples:
- Periodically forced pendulum,
- Coupled Springs,
- Negative Feedback Loops,
- Rigid Body Motion.