Differential equations and Dynamical Systems: Math 645

Meeting : TuTh 2:30--3:45    Hasbrouck Lab Add room 109

Instructor : Luc Rey-Bellet

 Office :  LGRT 1423 J
 Phone :  545-6020
 E-Mail :  luc@math.umass.edu
 Office Hours :   Tu 9:45--10::45,   Th 1:30--2:30,   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) 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, 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) Existence-uniqueness 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 Poincare-Bendixson theory.

Grade: There will be one midterm exam (take-home) 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