## Math
848 Class Notes

Note: These are informal notes which are to be
corrected and updatedregularly.

## Math 848, Final Examination:

## TBA

## Prerequisite:
Advanced Calculus, Linear Algebra, Analysis, suggested course to be
taken in advance or concurrently: Math 828

## Suggested additional references: J.Hale, Ordinary Differential
Equations, L. Perko, Differential Equations and Dynamical Systems,
P. Hartman, Ordinary Differential Equations

Here is a zip file with some Latex examples
You will have to process them with some version of Latexwhich handles pdf files. I use "pdflatex"
After you have installed a latex system on your computer, put the zip file into an empty folder.
The "Output" subfolder contains the pdf files which pdflatex produces.
The "macros.tex" file is a set of Latex macros that I use to simplify typing Latex source files. You will not be able to process the enclosed source files without it.
Your first set of Exercises is included here. You can see it in the "Output" subfolder.
- 1. Introduction
- 2. Linear
Transformations
- 3. General
Properties of Differential Equations
- 4. Existence
Uniqueness Theorem
- 5. Dependence
of Solutions to Differential Equations onParameters
- 6. Dependence
of Solutions to Differential Equations onParameters--Continued
- 6n. New version of 6 (Dependence
of Solutions to Differential Equations onParameters--Continued)
- 7. Autonomous
Differential Equations
- 8. Limit
Sets
- 9. Two
dimensional systems
- 9a
Differential Equations on a Two dimensional Torus
- 9b
Differential Equations on a Two dimensional Torus--alternate
- 9c
Index for C^2 vector fields
- 10.
Umlaufsatz
- 10a.
Degree of Circle Maps
- 10b. Alternate
version of Umlaufsatz
- 10c. A
summary of the index theory for planar vector fields
- 11. Another
index formula
- 11a. Alternate
treatment of Section 11
- 12. Linear
Differential Equations
- 13. Linear
Differential Equations with Constant Coefficients
- 14. Some
examples
- Note that the following notes (14a, 14b, 14c, 14d) were taken
from other sets of lecture notes, so the page numberings do not have
the number 14 in them. They involve 17 and 18.
- 14a.
Further Examples and Properties-a
- 14b.
Further Examples and Properties-b
- 14c.
Further Examples and Properties-c
- 14d.
Further Examples and Properties-d
- 15. Jordan
Canonical Form
- 16. The
Grobman-Hartman theorems
- 17. Proofs
of the Grobman-Hartman theorems
- 18. Proofs
of the Grobman-Hartman theorems -Continued
- 19. Hyperbolic
Periodic Orbits and Lyapunov Stability
- 20. Hamiltonian
Systems in $R^{2n}$
- 21. Hamiltonian
Systems and Variational Problems

### Exercises