Syllabus for Math 8501 --- Fall 2016

MWF 11:15 - 12:05 --- Vincent Hall 209


This course will be the first semester of a graduate level introduction to differential equations and dynamical systems with emphasis on qualitative, geometrical methods for nonlinear systems. It will be followed in the spring by Math 8502 -- Differential Equations and Dynamical Systems II, which will explore further topics.

Understanding the modern theory of dynamical systems requires a lot of ideas from many parts of mathematics. I hope to cover both the theory itself and these background ideas in a way which can be understood not only by math graduate students but by any mathematically inclined student with a solid knowledge of linear algebra, advanced calculus and elementary differential equations.

Here are some of the topics I would like to cover in the first semester: basic existence and uniqueness theory for ODE's, linear systems, flows and flow boxes, invariant sets, flows on manifolds, alpha and omega limit sets, flows in the plane, Poincare-Bendixson theory, index theory, Poincare maps, variational equations, dynamics near equilibria including the stable manifold theorem. In addition a good deal of time will be spent on interesting examples.

Recommended references:

Although I will not follow any one book, here are some that I like.

Modern approach:

The second one is available as a free PDF through SpringerLink via the Math Library.

"Dynamical Systems: Stability, Symbolic Dynamics and Chaos", by Clark Robinson
"Ordinary Differential Equations with Applications", by C. Chicone
"Ordinary Differential Equations", by V.I. Arnold

Some Classics:

The first three are inexpensive Dover books.

"Lectures on Ordinary Differential Equations", by W. Hurewicz
"Ordinary Differential Equations", by Jack Hale
"Differential Equations, Geometric Theory", by S. Lefschetz
"Ordinary Differential Equations", by Philip Hartman
"Theory of Ordinary Differential Equations", by E. Coddington and N. Levinson


Based on several homework assignments throughout the semester. No exams. The link leads to PDF versions of the homework assignments.

Homework 100 %

Mathematica Notebooks:

 Hill's Equation
 Eigenvalues and Eigenvectors
 Generalize Eigenvectors
 Cayley-Hamilton, Jordan, etc.
 Center Manifolds