## Math 848 Class Notes

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

## 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