Week 1: January 24 - 28
Monday:
Separable equations.
(
lecture,
tex file,
slides).
Wednesday:
Logistic equation. Homogeneity trick for separable equations. Exact
equations (if time).
(
lecture,
tex file,
slides).
Friday:
Exact equations. Integrating factors.
(
lecture,
tex file,
slides).
Week 2: January 31 - February 4
Monday:
First-order linear. Linear homogeneous constant coefficients.
(
lecture,
tex file,
slides).
Wednesday:
Bernoulli equation motivation. Linear homogeneous constant coefficients:
complex roots and repeated roots. Method of undetermined coefficients.
(
lecture,
tex file,
slides).
Friday:
Special second-order equations.
(
lecture,
tex file,
slides).
Week 3: February 7 - 11
Monday:
Matrix exponentiation.
(
lecture,
tex file,
slides).
- Reading: Sections 1.1–1.3.
(If you are not using edition 3, make sure you are reading the correct
sections by consulting the TOC—there is a link at the top of this
webpage.)
- Turn in: HW2
(Overleaf template)
Wednesday:
Fundamental theorem for linear systems.
(
lecture,
tex file,
slides).
Friday:
Fundamental theorem for linear systems. Linear systems in \(\mathbb{R}^2\).
(
lecture,
tex file,
slides).
Week 4: February 14 - 18
Monday:
Linear systems in \(\mathbb{R}^2\).
(
lecture,
tex file,
slides).
Wednesday:
Jordan form.
(
lecture,
tex file,
slides).
- Reading: Sections 1.6–1.8.
Friday:
Exponentiating Jordan matrices. Algorithm for computing Jordan form.
(
lecture,
tex file,
slides).
- Reading: Sections 1.6–1.9.
Week 5: February 21 - 25
Monday:
Stability theory. Linear systems in \(\mathbb{R}^3\). Nonhomogeneous equations.
(
lecture,
tex file,
slides).
Wednesday:
Nonhomogeneous equations.
(
lecture,
tex file,
slides).
- Reading: Section 1.10 (and the lecture notes).
Friday:
Higher-order homogeneous linear equations with constant coefficients.
(
lecture,
tex file).
Week 6: February 28 - March 5
Monday:
Higher-order homogeneous linear equations with constant coefficients.
(
lecture,
tex file,
slides).
Wednesday:
Existence and uniqueness for non-linear systems.
(
lecture,
tex file,
slides).
- Reading: Sections 2.1 and 2.2.
Friday:
Existence and uniqueness for non-linear systems.
(
lecture,
tex file,
slides).
- Reading: Sections 2.1 and 2.2.
Week 7: March 8 - 11
Monday:
Existence and uniqueness for non-linear systems.
(
lecture,
tex file,
slides).
Wednesday:
Linearization.
(
lecture,
tex file,
solutions,
solutions tex file,
awesomeness).
- Reading: Sections 2.1 and 2.2.
Friday:
Dependence on parameters, maximal interval. Begin stable manifold theorem.
(
lecture,
tex file,
slides).
Week 8: March 14 - 18
Monday:
Stable manifold theorem.
(
lecture,
tex file,
slides).
Wednesday:
Stable manifold theorem.
(
lecture,
tex file,
solutions).
Friday:
Hartman-Grobman theorem.
(
lecture,
tex file,
slides).
- Reading: Sections 2.7 and 2.8.
Week 9: March 28 - April 2
Monday:
Stability and Liapunov functions.
(
lecture,
tex file,
slides).
Wednesday:
No class today
Friday:
Liapunov functions.
(
lecture,
tex file,
slides).
Week 10: April 4 - 8
Monday:
Planar systems.
(
lecture,
tex file,
slides).
- Reading: Sections 2.10 and 2.11.
- Turn in: HW9
(Overleaf template)
- Example due to Peron (example 5 in Perko reading). A node turns into
a focus upon the addition of nonlinear terms: node, focus.
Wednesday:
Global theory for nonlinear systems: index theory.
(
lecture,
tex file,
slides).
Friday:
Global theory for nonlinear systems: index theory.
(
lecture,
tex file).
Week 11: April 11 - 15
Monday:
Global theory for nonlinear systems: index theory.
(
lecture,
tex file,
slides).
Wednesday:
Critical points at infinity, and global phase portraits.
(
lecture,
tex file,
slides).
- Reading: Reading: Section 3.12.
Friday:
Critical points at infinity, and global phase portraits.
(
lecture,
tex file,
slides).
Week 12: April 18 - 22
Monday:
Critical points at infinity, and global phase portraits.
(
lecture,
tex file,
slides).
Wednesday:
Resolution of singularities.
(
lecture,
tex file,
no slides).
Friday:
Hamiltonian systems.
(
lecture,
tex file,
slides).
Week 13: April 25 - 29
Monday:
Gradient systems.
Wednesday:
Evaluations.
(
slides).
Friday:
No class.
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