Differential equations and Dynamical Systems: Math 645
Meeting : TuTh 2:303:45 Hasbrouck Lab Add room 109
Instructor : Luc ReyBellet
Office : LGRT 1423 J
Phone : 5456020
EMail : luc@math.umass.edu
Office Hours :
Tu 9:4510::45, Th 1:302:30, or by appointment.
Text: There is no official textbook for the
class, but classnotes are provided. There are many books on the subject,
buying one which suits you best is a good investment.
I am some writing class notes. They will be posted here. This is a preliminary version. Check for updates. If you find typos, mistakes, unclear statements, please tell me.
Class notes PDF file, revised version 10/08
References

Differential Dynamical Systems,
by James D. Meiss, SIAM.
This is a new book whose first 6 chapters cover the same material and is quite close in spirit
to the class lectures notes.

Nonlinear Differential Equations and Dynamical Systems,
by Ferdinand Verhulst, Universitext, Springer.
This is a good book oriented toward applied mathematics. It contains a lot of
examples and the material is organized very efficiently.
However the general theory (existence, uniqueness, etc...) is assumed to be
known already.

Lectures on ordinary Differential Equations,
by Witold Hurewicz, Dover Phoenix Editions.
This beautiful little book (122 small pages) contains the general theory of
differential equations up to (and included) PoincareBendixson theory.
It was written in 1943 (!) and is a beautiful piece of mathematical writing.
Some aspects I want to discuss are not included and there are very few
examples
and no exercises.

Ordinary Differential Equations,
by Philip Hartman, Classic in Applied Mathematics 38, SIAM.
This book was originally written in 1963, a classic.

Differential Equations and Dynamical Systems,
by Lawrence Perko, Text in Applied Mathematics 7, Springer.
This is standard book oriented toward applications to dynamical systems.
It treats a lot of results usually not found in textbooks, but many proofs,
even elementary one are omitted. Excellent as a reference, not as a textbook.

Ordinary Differential Equations,
by V.I. Arnold, MIT Press.
This is a very good book, written with a more geometrical point of view.
It is is difficult though, but highly recommended for a second reading.

Differential Equations, Dynamical Systems, and an Introduction to Chaos.
by Robert L. Devaney, Morris Hirsch, Stephen Smale, Academic Press
This is a rather elementary and very clear and informative
introduction to dynamical systems. (I am referring here to the the recent new edition which is rather different from the previous ones).

Theory of Ordinary Differential equations,
by Earl A. Coddington and Norman Levinson, Krieger Publishing Company.
Older standard textbook on ODE's. Very complete and very good.

Ordinary Differential Equations,
by Wolfgang Walter, Graduate Texts in Mathematics, Springer.
This is a very complete and detailed exposition of the general theory of
differential equations. For the analytically oriented student.

Solving Ordinary Differential Equations I: Nonstiff Problems
by Ernst Hairer, S.P. Norsett, and gerhard Wanner, Springer in Computational
Mathematics, Springer.
The standard reference if you are interested in numerical methods for ODE's. The first
chapter contains a brief and efficient summary of the general theory.
Syllabus:
This course is an introduction to differential equations and dynamical systems.
Among the subjects to be treated:

1) Existenceuniqueness theory for differential equations. Local and global
existence.

2) General theory of linear differential equations, Floquet theory

3) Differential equation as a dynamical system.
Dependence on the initial conditions and parameters.

4) Stability theory, linearization, Liapunov method, stable/unstable
manifold Theorem, center manifold theorem.

5) Periodic orbits and PoincareBendixson theory.
Grade: There will be one midterm exam (takehome)
and one final exam (in class). Homework will be assigned regularly.
 40% Midterm
 40% Final Exam
 20% Homework, attendance and participation
Exams:
 Midterm: Take home. Closed books, but you can use
the classnotes.
 Final: Room:
Homework :
Homework #1 (due on September 23):
HWK #1
Homework #2 (due on October 7):
HWK #2
Homework #3 (due on October 21 ):
HWK #3
Homework #4 (due on November 4):
HWK #4
Homework #5 (due on November 25):
HWK #5
Homework #6 (due on December 12):
HWK #6