### MATH 6307 Ordinary Differential Equations 1

### Fall 2009

### TTh 4:35-5:55, Skiles 154

### Office Hours: TTr 3-4, Skiles 224.

### Textbook

The class will be based on the lecture notes by prof. Jack Hale. I'll
distribute the notes in class.

The notes are an update and extension of the book:

Ordinary Differential Equations.

Jack K. Hale

Dover

### Syllabus

See th online syllabus.

### Grading

There
will be two midterms and one final. The first midterm will be September
22. I will assign HW and collect them every 2 or 3 weeks. The final
grade will be based on HW (15%), midterms (40%) and final (45%).

**First week**

- Material covered: Existence and uniqueness of solutions.
- Notes: 1.1,
1.2
- Book: I.1, I.2, I.3

**Second week**
- Material covered: Continuos dependence. Differential
Inequalities.
- Notes: 1.3, 1.4
- Book: I.4, I.6

**Third week**

- Material covered: Differential Inequalities (continued).
Linear and Autonomous Systems.
- Notes: 1.5, 1.6
- Book: I.7, I.8

**Fourth week**

- Material covered: Limit set, locally attracting set and
attractor.
- Notes: 1.7

**Fifth week**

- Material covered: Liapunov function.
- Notes: 1.8, 1.9

#### Midterm

Solve these three exercises.
Write clear and mathematically precise answers. The solution must be
turned in by Thursday October 1st. You are more than welcome to ask me
clarifications and expalnations on the exercises.

**Sixth week**

- Material covered: The principle of Wazewsky. Discrete systems.
- Notes: 1.9, 1.10

**Seventh week**

- Material covered: The Poincare-bendixon Theorem
- Notes: 1.11

**Eighth week**

- Material covered: Linear and Linear Perturbed Systems: General properties. Liouville's Theorem
- Notes: 2.1, 2.2

**Nineth week**

- Material covered: Stability. Linear Periodic Systems.
- Notes: 2.3, 2.4

**Tenth week**

- Material covered: Nonhomogenous Linear Systems. Stability and Perturbation.
- Notes: 2.5, 2.6

**Eleventh week**

- Material covered: Fredholm Alternative. Affine Maps.
- Notes: 2.7, 2.8.2

**Twelfth week**

- Material covered: Poincare-Andropov-Hopf Bifurcation
- Notes: 2.8.7
- Book: VIII.1

Final Exam: the
final exam will be on December Friday 11 at 2:50. You will have 30/40
minutes each to expose an argument to the rest of the class. Tofether
with the exposition you should submit a written paper on the argument.
A list of possible argumnet follows.

- Invariant Manifold for Hyperbolic Fixed Point. See for example Chapter 4 of the "Ordinary Differential Equation with Application" by Carmen Chicone.
- Simple oscillatory Phenomena and the Method of Averaging. See for example Capter V of "Ordinary Differential Equation" by Jack Hale.
- Elliptic Periodic Orbit in a Biliard. The result is contained in this paper. Sorry fot the bad copy.
- Keplerian two bodies problem and perturbations. See for example Sections 3.2.2, 3.2.3 and 3.2.4 of "Ordinary Differential Equation with Application" by Carmen Chicone.
- Forced Dumped Oscillations: the Anchor Escapment System. See for expamle the book by G. Gallavotti section 2.14 to 2.18.
- Motion of the Rigid Body: Euler Angles and Integrability. Any book in Classical Mechanics
- Bessel Function and Spherical Harmonics. Plenty of reference. I have several of them if interested.