Math V3030, Spring 1999
Class Meetings: Tuesday and Thursday 9:10-10:25 AM,
Mathematics Building 417.
Prerequisites: preferably Math V1205
IIIS) but at least Math V1201
IIIA) or the equivalent. Also Math 2010
Algebra) and Math 3027
- Differential Equations and Dynamical Systems (Second
Edition) by Lawrence Perko, published by Springer (1996);
- Nonlinear Dynamics and Chaos with Applications to Physics,
Biology, Chemistry and Engineering by Steven H. Strogatz,
published by Addison Wesley (1994).
Both books will be available at
Labyrinth Books, 536 W
112th Street. You will need to have access to both books on a regular
basis. The book by Perko presents the material in a mathematically
rigorous way; the book by Strogatz gives a great deal of insight into
the material together with many applications. One copy of each will
be on reserve in the Mathematics Library.
Recommended Texts: Here are a few other good books that you
may want to consult. Most will be on reserve in the Mathematics
- Ordinary Differential Equations by V.I. Arnold (MIT
Press). This is a very beautiful treatment of the material covered
in a first course in Ordinary Differential Equations. It is more
mathematical than the books typically used in a first course, and
also has many interesting examples from mathematical physics. I
recommend it for a general review of ODE and also for the material
in the first three sections of this course (dimension 1, linear
systems, dependence on initial conditions and flows).
- Dynamical Systems by D.K. Arrowsmith and C.M. Place
(Chapman and Hall 1992). Again this is an entry level book, thus a
bit elementary for this course. Besides the elementary material
you are already supposed to know, it has a good chapter on higher
dimensional systems, plus a chapter on examples and bifurcations.
- Order within chaos by Pierre Berge, Yves Pomeau and
Christian Vidal (John Wiley 1984). An advanced book written by 3
physicists about chaos. Many interesting examples, and a possible
source for special projects.
- An introduction to Chaotic Dynamical Systems by Robert
Devaney ((Addison-Wesley 1989). A more detailed presentation than
Strogatz of the chaos exhibited in one-dimensional maps.
- Nonlinear Physics with Maple for Scientists and
Engineers by Richard H. Enns and George C. McGuire. Similar to
Strogatz, but more on the physics side. A good resource for
students who know and use Maple.
- Introduction to Ordinary Differential Equations with
Mathematica by Alfred Gray, Michael Mezzino and Mark A.
Pinsky (Telos/Springer 1997). Besides being a good entry level ODE
text, this book shows how to use Mathematica as a tool for
studying the kinds of equations that will come up in our course.
- Dynamics and Bifurcations by J. Hale and H. Kocak
(Springer 1991) This book is about half way between Perko and the
Strogatz: it is organized by dimension like Strogatz, but with
fewer examples, and is not quite as mathematical as Perko.
- Differential Equations, Dynamical Systems and Linear
Algebra by Morris W.Hirsch and Stephen Smale, (Academic Press
1975). A great classic. In principle an entry level book both for
Ordinary Differential Equations and Linear Algebra, it goes fast
and deep and covers much of the material we will be covering.
- A First Course in Discrete Dynamical Systems (Second
Edition) by Richard A. Holmgren (Springer 1996). A very elementary
presentation of discrete dynamical systems. A good complement to
chapter 10 of Strogatz.
- Differential Equations: A Dynamical Systems Approach, Parts
I and II by J.H. Hubbard and B.H. West (Springer 1995). Part I
is an entry level text; Part II covers much of what we will be
- Nonlinear Dynamics and Chaos by J.M.T. Thompson and
H.B. Stewart (John Wiley 1986). Very similar to Strogatz, but at a
more advanced level.
Course Objectives and Topics: This second course on
differential equations will focus on qualitative techniques for
solving non-linear equations. Here are the topics I hope to cover:
- First a short review of qualitative techniques for non-linear
equations in dimension 1, principally using the examples in Part I
of Strogatz; the book by Arnold is another good source.
- Next a quick review of linear systems, using chapter 1 of
Perko; again Arnold is a good reference.
- Dependence on initial conditions and the local behavior near
critical points. Perko, chapter 2.
- Non linear equations in the plane, culminating in the
Poincaré-Bendixson theorem. Perko, chapter 3 and Strogatz
- Bifurcations. Perko, chapter 4; Strogatz, chapter 8.
- Chaos. Strogatz, part III.
Student Population: This is an elective for Mathematics and
Applied Mathematics Majors, as well as for the Math/Stat and
Econ/Math majors. This is also a very useful course for Physics and
Test dates: Midterm 1: Thursday, February 25. Midterm 2:
Thursday, March 25. Final: Thursday, May 13.
Last day to drop classes: Thursday, March 25. Last day of class:
Thursday, April 29
Grading Policy: Each midterm will count 25% of the grade
and the final 40%. The remaining 10% will be based on homework and
will be due every Thursday, except on exam weeks. Assignments will be
handed out a week in advance. No late homework will be
accepted. Several of the assigned problems will be randomly
chosen for grading; written solutions to most of the assigned
problems will be handed out. You should attempt all the suggested
homework problems. I will also answer specific questions by
e-mail and during my
Software: DEGraph is a piece
of software for graphing and the solution curves to differential
equations, and for solving them numerically. It only runs on
Macintoshes . Just click to download it.
Last Updated on January 20, 1999