Th 8/21 General 1st-order equations: existence-uniqueness, examples (KP 1.1, 1.2)

Due 8/28: (KP 1.6) 1.2, 1.8/1.16, 1.12, 1.13(i), 1.18

See also: 1.3, 1.4, 1.9, 1.19, 1.23

Honors: 1.14, 1.15

Tu 8/26 Autonomous equations: geometric analysis (KP 1.3)/ Autonomous systems, phase-plane diagrams (KP 3.1, 3.2)

Geometric analysis of autonomous equations

MATLAB m-files for ODE: dfield and pplane

If you have Matlab installed on your computer, you can use this software without knowing anything about Matlab.

Download it from this URL:

http://math.rice.edu/~dfield/

As an encouragement: students turning in with their HW a relevant MATLAB plot for one

of the problems (with a caption describing what it shows) will get an extra 1/2 point per HW set.

Th 8/28 Autonomous systems: existence/uniqueness; bounded solutions are globally defined. Equilibrium points.

Phase-plane diagrams: nullclines. Examples: predator-prey, a second-order equation. Conserved quantities.

Tu 9/2 Hamiltonian systems: conserved quantities. Ex: cubic 2nd order, pendulum. Phase plane pictures.

Bounded invariant regions. Potentials and hamiltonians, relation with Mechanics. Level sets of proper functions.

Ref: KP 3.1, W2.9

HW2 (due 9/9): KPsect 3.9: 3.2 (i),(ii), 3.8(i) (ii), 3.13 (competing species), 3.15 (chemostat), 3.37 (Hamiltonian)

Hon: (KP sec 3.9): 3.7 (dipole), 3.14 (epidemic model)

Th 9/4 Predator-prey revisited. Phase plane pictures and stability for linear systems.(KP 3.4, W 2.3, 2.4)

Tu 9/9 Handout: Diagonalizable linear systems and stability.

Linear systems in 2D: eigenspaces, reduction to standard form, trace-det diagram

Ref: [KP 3.3, W 2.3, 2.4]

Th 9/11 Linear systems in 2D: complex eigenvalues, real standard form. Stable/unstable/neutral subspaces

for diagonalizable systems in n dimensions. Case of zero real part. Example: coupled harmonic oscillators.

Ref: handout, [W 1.12]

HW3 [KP 3.9]: 9,10, 11, 12 (only item (i) in each); Waltman 1.12: 1.

Hon: the exercises proposed in the handout; Waltman 1.12: 2

(due Th 9/18)

Tu 9/16 Qualitative methods for NL autonomous systems: flow of a vector field (semigroup property), stability of equilibria,

omega- and alpha-limit sets (examples). Liapunov functions for equilibria. Ref: [KP 3.4, w 2.6]

Th 9/18 Liapunov functions (cont.) Liapunov's theorem on stability/asymptotic stability, sublevel sets are invariant, LaSalle invariance theorem.

Examples (inc. pendulum with friction.)

Handout: Stability of equilibria and Liapunov functions

(7 pages, final version posted 9/30)

HW4 (due Tu 9/30) [KP sect 3.9]: 16, 17(i), 21, 23, 28(i)

Hon: [W, 2.6] 3ab, 5

Tu 9/23: class canceled

Th 9/25: linearization at an equilibrium (ref: [KP 3.5, W 2.5])

Tu 9/30: Examples: sum of periodic functions (from HW), gradient systems, Predator-prey systems (start) [ref: [W, 2.8]).

HW5 (due 10/7): KP sect 3.9: 25, 35, 36, 39, 44

Sum of periodic functions (HW solution)

Th 10/2: Predator-prey systems (cont.), periodic solutions (start): KP 3.6, W 2.7

Some examples of autonomous systems in the plane

(7 pages, posted 10/7)

Tu 10/7: Review/questions

HW5 solutions

Th 10/9: Discussion of examples in handout/ Poincare'-Bendixson theorem

Tu 10/14: First test

Exam 1(with solutions)

Th 10/16: fall break (no classes)

Tu 10/21: existence/nonexistence of periodic solutions: Poincare-Bendixson, closed orbits must enclose an equilibrium,

index of a vector field along a closed curve and local index at a singularity (amazing theorem)/Bendixson-Dulac criterion.

Th 10/23: equations of Lienard and van der Pol[KP, p.131]

Notes on Lienard's equation

HW6 (due 10/30): 45, 47, 52, 53, 54 (i), 57

HW6 solutions

PLAN for the second part of the course.

1. 2nd order equations: existence-uniqueness for the IVP, the Wronskian, self-adjointness (intro): [KP 4.1]

2. Oscillation and comparison theorems, disconjugacy and uniqueness [W 4.3, KP 5.5]

3. Non-homogeneous linear BVPs, Green's functions. [W 4.2, KP 5.9]

4. Sturm-Liouville eigenvalue problems: existence of eigenvalues, properties, eigenfunction expansions

[W 4.4, 4.5, 4.6, 4.8; KP 5.4]

Tu 10/28: E/U for the IVP, Wronskian, self-adjointness (KP 5.1)

Problems: KP 5.1, 5.2, 5.3, 5.4, 5.8

HW7 (due 11.11): 5.1 (iii), (iv), 5.8

Th 10/30: disconjugacy, Green's functions, existence for non-hom BVP

Problems: KP 5.62, 5.65, 5.67, 5.68, 5.69

HW7; (due 11/11): 5.62(iii), 5.65, 5.68 (iii), (iv)

Green's functions (summary)

(version date: 11/4)

take-home test given.

Tu 11/4: Green's functions: examples; Liapunov's criterion for disconjugacy.

Take-home test collected.

solutions (take-home test)

Th 11/6: Liapunov criterion-examples; Oscillation/comparison theorems.

DISCUSSION PROBLEMS: from now until the end of the course I will list a small number

of more challenging problems, with written solutions to be handed in on the last day of classes.

You may work on those with a classmate, or ask questions about them in class or during office hours.

The grade on this total assignment will enter the average with the same weight as a test.

The weights will be: HW=20%, high two of three tests: 20% each, Discussion Problems: 20%, Final:20%

(This is optional: if you decide not to turn it in, or if it would lower your average, I will compute the average with

the originally announced weights instead.)

First list: KP 5.4, 5.69, W 4.3.1

Tu 11/11: Oscillatory properties of solutions

HW7 solutions

HW 8 (due 11/18): KP 5.16 (iii), (v); 5.17, 5.29; W, p.224 no. 4.

Th 11/13: Existence of SL eigenvalues/Properties of SL eigenfunctions/ eigenfunction expansions

Tu 11/18: Definitions/examples: metric space, normed vector space, completeness, Banach space, Hilbert space.

Contractions in complete metric spaces have unique, globally attracting fixed points.

HW8 solutions

Th 11/20: Picard's existence/uniqueness theorem; Lipschitz condition, Picard iteration, error estimate.

Take-home test given.

Exam 3

Exam 3 solutions

Problems (from [W, Ch, 3.3]: 2,3,4,5/ Ch 3.4: 1(e)(g)

[KP, 8.11]: 9,10,11,12,13

Discussion problems: [W, Ch 3.2]: 3,7,8

HW9: problems in boldface listed above.

Tu 11/25: extensions of Picard's theorem: global existence in globally Lipschitz or linear cases/ Nonlinear BVPs [W, 3.6]

Take-home test collected

Existence theory (summary)

Discussion problems: [W, 3.6]: 4,6

Solutions to two discussion problems

Tu 12/2 Discussion of problems/ examples. Gronwall's inequality

HW 9 due

HW9 solutions

Final Exam: Monday, 12/8: 10:15-12:15.

Suggestions: The entries in the course log give an idea of which topics I think are important; for a summary of the theory,

read the online handouts. To review for the final: try to solve the problems on the three tests and the homework, without looking at the

solutions. The final will be based on the problems found in the tests and the homework sets, with small variations.

Final Exam