Math 134, Linear and Nonlinear Systems of Differential Equations.

Lectures: MWF 10:00-10:50 in MS 5117.

Instructor: Rowan Killip, 6935 Math Sciences Building.

Office Hours: Mon 1:30--2:30pm and Wed 2:30--3:30pm in 6935 Math Sciences Building.

Exams: In-class midterm: Wednesday, May 1st. Three-hour final: Tuesday, June 11th, 8:00am-11:00am.
Bring student ID to both midterms and the final. There will be no make-up exams.
No calculators, notes, or books will be permitted in any exam.

Homework: There will be weekly homework. It is due in class. Further information is given below.

No late homework will be accepted.
Write your name and ID number at the top of the first page.
Staple your pages!
Homework will be returned in TA section.
The weakest homework score will be omitted.

Grading: Homework 20%; Midterm 20%; Final 60%.

Syllabus: General Course Outline.

Here are the expected topics of lectures; it will be updated as we progress:

Lecture Book Sections Topics
1 1.0-2.1 Introduction to linear/nonlinear (non)autonomous systems, examples.
(Chapter 1 is mostly advertising for the class.)
2 2.1-2, 2.4 Flows on the line. (Section 2.3 will be discussed in TA section.)
3 2.7, 2.6, 2.8 Potentials. Impossibility of oscillations. Numerical methods.
2.8, 2.5 Numerical methods (cont.). Existence and uniqueness.
2.5 Existence and uniqueness.
3.1, 3.2 Bifurcations: saddle-node and transcritical.
3.4 Bifurcations: pitchfork.
3.4, 3.5 Bifurcations: example problems.
3.4, 3.7, 3.5 Hysteresis, nondimensionalization, bead on a rotating hoop.
3.5 Bead on a rotating hoop.
3.6 Imperfect bifurcations, bifurcations with two parameters.
Ch. 4 Flows on the circle.
Ch. 5 Linear systems.
Ch. 2-5 Midterm.
Ch. 5 Linear systems (cont.).
6.2, 6.1 Existence/uniqueness, direction fields.
6.1, 6.3 Drawing phase portraits.
6.3, 5.1 Hartman-Grobman Theorem, Lyapunov stability.
Gradient flows, Hamiltonian flows.
Cf. 6.7 Hamiltonian flow, worked example.
6.8 Gradient flows, worked example. Index theory.
7.1-2 Index theory, limit cycles.
7.2 Lyapunov functions.
7.2, 7.3 Dulac's criterion, Poincare-Bendixson.
7.3, 8.1 Poincare-Bendixson (cont.), saddle-node in 2D.
Review by example.

Homework Problems:
Homework 1. Due Friday, April 12th.

Lecture	Book Sections	Topics
1	1.0-2.1	Introduction to linear/nonlinear (non)autonomous systems, examples. (Chapter 1 is mostly advertising for the class.)
2	2.1-2, 2.4	Flows on the line. (Section 2.3 will be discussed in TA section.)
3	2.7, 2.6, 2.8	Potentials. Impossibility of oscillations. Numerical methods.
	2.8, 2.5	Numerical methods (cont.). Existence and uniqueness.
	2.5	Existence and uniqueness.
	3.1, 3.2	Bifurcations: saddle-node and transcritical.
	3.4	Bifurcations: pitchfork.
	3.4, 3.5	Bifurcations: example problems.
	3.4, 3.7, 3.5	Hysteresis, nondimensionalization, bead on a rotating hoop.
	3.5	Bead on a rotating hoop.
	3.6	Imperfect bifurcations, bifurcations with two parameters.
	Ch. 4	Flows on the circle.
	Ch. 5	Linear systems.
	Ch. 2-5	Midterm.
	Ch. 5	Linear systems (cont.).
	6.2, 6.1	Existence/uniqueness, direction fields.
	6.1, 6.3	Drawing phase portraits.
	6.3, 5.1	Hartman-Grobman Theorem, Lyapunov stability.
		Gradient flows, Hamiltonian flows.
	Cf. 6.7	Hamiltonian flow, worked example.
	6.8	Gradient flows, worked example. Index theory.
	7.1-2	Index theory, limit cycles.
	7.2	Lyapunov functions.
	7.2, 7.3	Dulac's criterion, Poincare-Bendixson.
	7.3, 8.1	Poincare-Bendixson (cont.), saddle-node in 2D.
		Review by example.