MATH 8430 - Fundamental Theory of ODEs

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Course objectives

Study the theoretical basis of differential equations.
We will be interested in the careful construction of objects and the rigorous proof of results which are for most of them already known to students with background in ordinary differential equations (ODE). We will consider results from the general theory of ODE. We will then study linear systems, emphasizing their role in the study of nonlinear systems (which we will not cover). If time permit, other topics will be chosen from closely related subjects (delay differential equations, linear control theory, differential equations in a more general setting, etc.).

Schedule

We will meet Tuesdays and Thursdays, in MH 503, from 16:00 to 17:30.

Text

Lecture notes will be handed out, as there is no single textbook covering adequately all the material. Several books that are not too expensive and that belong to the library of someone interested in ODE:
Ordinary Differential Equations, P. Hartman, SIAM (2004)
Principles of Differential Equations, N.G. Markley, Wiley (2004)
Basic Theory of Ordinary Differential Equations, P.-F. Hsieh and Y. Sibuya, Springer (1999)

Evaluation

4 or 5 marked assignments and a take-home final evaluation.

Tentative program

General theory of ODEs 3 weeks Existence and uniqueness of solutions
Continuation of solutions
Continuous dependence on initial values, on parameters
Nonuniqueness.
Homogeneous linear systems 2 weeks Asymptotic behavior of solutions
Linearization 2 weeks Hartman-Grobman theorem
General linear systems 4 weeks Floquet theory
Fredholm alternative
Exponential dichotomy 2 weeks Exponential dichotomy
First approximate theory (a.k.a. Hartman-Grobman for nonautonomous systems)

Material

Course announcement.

Lecture notes (posted 7 February 2007): whole file.
Assignments and final examination for a somewhat similar course taught at McMaster in 2003: I will post solutions to these once we have done the corresponding assignments (which will not be like these, but in the same "philosophy").


Last modified: Wed Feb 7 16:31:07 CST 2007