Differential equations and Dynamical Systems: Math 645
Meeting : TuTh 1:00 LGRT 111
Instructor : Luc Rey-Bellet
Office : 1423 J LGRT
Phone : 545-6020
E-Mail : lr7q@math.umass.edu
Office Hours :
Tuesday 2:30--3:45, Thursday 2:30--3:45, or by appointment.
Text: There is no official textbook for the
class, since I will not follow any particular one.
There are very many books on the subject, and buying one which suits you best
is a good investment. Here are some recommendations with some comments.
Some of them will be put on reserve at the library.
The material covered in the class can be found, roughly speaking, in a
combination of Hurewicz (basic theory of ODE's)
with Verhulst or Devaney, Hirsch and Smale (more oriented toward
dynamical systems apsects).
I am some writing class notes. Although they won't replace a textbook they
will be posted here. This is a preliminary version. If you find typos,
mistakes, unclear statements, please tell me.
Class notes: Chapters 1 to 5 -- PDF file
References
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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) Poincare-Bendixson 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 and aged extremely well.
-
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.
-
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.
Among the subjects to be treated:
-
1) Existence-uniqueness theory for differential equations. Local and global
existence. Differential equation as a dynamical system.
-
2) General theory of linear differential equations, Floquet theory
-
3) Dependence on the initial conditions and parameters.
-
4) Stability theory, linearization, Liapunov method, stable/unstable
manifolds.
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5) Periodic orbits and Poincare-Bendixson theory.
Grade: There will be one midterm exam (take-home)
and one final exam. Homework will be assigned regularly, some will be graded.
- 40% Midterm
- 40% Final Exam
- 20% Homework, attendance and participation
Exams:
Homework :
Homework #1 (due on Thursday September 23):
HW#1 PDF file
Homework #2 (due on Thursday October 7):
HW#2 PDF file
Homework #3 (due on Tuesday October 19):
HW#3 PDF file
Homework #4 (due on Thursday October 28):
HW#4 PDF file
Homework #5 (due on Tuesday November 23):
HW#5 PDF file
Homework #6 (due on Thursday December 2):
HW#6 PDF file
Homework #7 (due on Monday December 13):
HW#7 PDF file