Math 848 Class Notes

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

Math 848, Final Examination:


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