Dynamical Systems

Math V3030, Spring 1999

Syllabus

Class Meetings: Tuesday and Thursday 9:10-10:25 AM, Mathematics Building 417.

Prerequisites: preferably Math V1205 (Calculus IIIS) but at least Math V1201 (Calculus IIIA) or the equivalent. Also Math 2010 (Linear Algebra) and Math 3027 (Ordinary Differential Equations).

Required Texts:

• 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 Library.

• 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 covering.
• 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 chapter 7
• 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 Chemistry majors.

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 class participation.

Homework: Assignments 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 office hours.

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