The course will give an introduction to both the qualitative and quantitative aspects of the theory of systems of ordinary differential equations. A classical example is the damped simple pendulum, governed by a differential equation of the form x'' + c x' + sin x = 0, which can be rewritten as a system of two first order differential equations x' = y, y'= -c y -sin x. The picture above shows pieces of solutions in the x-y plane.

Qualitative methods of study will include equilibrium point analysis and linearization. One highlight of the course will be the Poincaré-Bendixson Theorem, which completely describes the long-term behavior of solutions in the plane. Another topic of study are linear systems and their significance in analyzing non-linear systems. Also addressed will be questions of sensitive dependence on initial conditions and stability.

Prerequisites. The course material is accessible to beginning graduate students and advanced undergraduate students, who have taken an introductory differential equations course (Math 3226), know some linear algebra (Math 3323 or Math 3426), and have a strong background in real analysis (at least Math 3341).

Software. Students will make some use of computer software (Mathematica) to visualize the results presented. Knowledge of Mathematica is not a prerequisite; instead, students are expected to aquire some basic knowledge during the course.

Textbook. Ferdinand Verhulst. Nonlinear Differential Equations and Dynamical Systems. Springer-Verlag, 2nd edition,1996. $32.50?

For further information, please contact me at 747-7002, or send email.

May 3, 1997.