Meeting times: 10:30-11:20 MWF

Meeting place: 218 MLH

Prerequisites: 22M:100 or consent of instructor. A course in differential equations. Some knowledge of computer programming. The language Maple will be introduced.

Instructor: Laurent O. Jay

Office: 225L MLH

Office hours: 9:30-10:20 MWF. I will also be available at other times. Just drop by my office or send me an e-mail to make an appointment.

Telephone: (319)-335-0898

Fax: (319)-335-0627

E-mail address: ljay@math.uiowa.edu

Mailbox: in Mailroom 15 MLH

Course web page: Assignments and other information about the course will be given in http://www.math.uiowa.edu/~ljay/m142_03.html. Students are responsible for checking regularly this course web page.

Textbook (reference): Dynamical Systems with Applications using Maple by Stephen Lynch, Birkhauser, Boston, 2001, 398 pages, ISBN 0-8176-4150-5, Price: $59.95, (ENGINEERING Library QA614.8 .L96 2001).

Additional references are:

Nonlinear dynamics and chaos by Steven H. Strogatz, Westview Press, 1994, ISBN: 0-7382-0453-6, (MATH Q172.5.C45 S767 1994).

Chaos: an introduction to dynamical systems by Kathleen T. Alligood, Tim D. Sauer, James A. Yorke, Springer, New York, Textbooks in mathematical sciences, 1997, 603 pages, ISBN: 0387946772, (MATH QA614.8 .A44 1997).

Stability, instability and chaos by Paul Glendinning, Cambridge University Press, Cambridge texts in applied mathematics, 1994, 388 pages, ISBN: 0-521-42566-2, reprinted 1999.

An introduction to chaotic dynamical systems, Second edition by Robert L. Devaney, Addison-Wesley, Addison-Wesley Studies in Nonlinearity, 1989, 336 pages, ISBN: 0-201-13046-7, (PHYS QA614.8 .D48 1988, MATH QA614.8 .D48 1986).

Goals and objectives of the course: This course is a graduate course and it is assumed that you can work along the course in an independent fashion. This course will cover nonlinear differential equations, one- and two-dimensional flows, stability, phase plane, limit cycles, bifurcations, chaos, fractals, and applications. At the end of the course the student should master essential issues in nonlinear dynamics and chaos.

Class procedures: The majority of each class period will be lecture oriented. It is strongly advised to read the material to be discussed before coming to class.

Computer language: Maple

Computer resources: Computer accounts will be made available on the network in MLH B5. Check the laboratories reservation schedule and the web page of the Division of Mathematics Sciences Educational Laboratories for more information.

Grading procedures: The final grade will be based on one mid-term examination, the final examination, and homework, as follows:

1. The mid-term examination and the final examination will each account for 30% of the course grade.
2. Homework and project assignments will account for 40% of the course grade. Late homework will be accepted only by special permission of the instructor. The grade for your homework will be based on the best 75% of your homework.

The tests are open books and open notes examinations. Bring a scientific calculator. In assigning grades, plus/minus grading will be used.

Teaching assistant: Jasang Yoon, B20E MLH, 335-3807, e-mail: jyoon@math.uiowa.edu.

Final examination: To be held on Monday, May 12, 2:15-4:15 PM in room 218 MLH. Only under exceptional circumstances will a student be permitted to shift the time of this examination. This final examination is an open books and open notes examination. Bring a scientific calculator.

Course outline:

• Linear and nonlinear differential equations;
• Trajectories, phase space, flows, limit sets;
• Derivatives with respect to initial conditions, the resolvent;
• Critical points, invariant manifolds;
• Stable, unstable, and center manifolds;
• Periodic orbits, limit cycles, Poincare maps, Floquet theory;
• Hamiltonian systems, Liapunov functions and stability;
• Modelling interacting species;
• Bifurcation theory;
• Local and global bifurcations;
• Three dimensional autonomous systems and chaos, Lyapunov exponents;
• Cascades, the Feigenbaum number;
• Homoclinic orbits, heteroclinic orbits;
• Strange attractors;
• The second part of David Hilbert's 16'th problem;
• Limit cycles of Lienard systems;
• Linear and nonlinear discrete dynamical systems;
• Complex iterative maps;
• Fractals;
• Multifractals;
• Controlling chaos;

This course plan may be modified during the semester. Such modifications will be announced in advance during class periods and on the course web page; the student is responsible for keeping abreast of such changes.

Notes to student: The Department of Mathematics has offices in 14 MLH. To make an appointment to speak with the Chair of the Department, call 335-0714 or contact the Departmental Secretary in 14 MLH.

I need to hear from anyone who has a disability, which may require some modification of seating, testing or other class requirements so that appropriate arrangements may be made. Please see me after class or during my office hours.