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MA 732-001 Spring 2009

Ordinary Differential Equations II

Mathematics

### Material on Test 1

Test 1 is Friday Feb. 27. In the Meiss text, it will cover what we have done in Sections 4.11 - 4.12 and Chapter 5.

Things you should be able to do:

- For a periodic differential equation of the form x' = f(t,x), with x in R and f 2pi-periodic in t, use the graph of the Poincare map and linearization to show existence and stability of 2pi-periodic solutions. (Homework 1, problems 1 - 3.)
- Show that an alleged norm is actually a norm, and that a vector space with this norm is complete. Show that a map between Banach spaces is bounded linear. Show that a map between Banach spaces is continuous. Show that a map between Banach spaces is continuously differentiable. Use the definition of derivative. Calculate a derivative. (Homework 2; Homework 3, problems 1 and 2; Homework 5, problems 3 b, c.)
- Use the Contraction Mapping Theorem with Parameters (Homework 4)
- Use center manifold reduction (Homework 5, problems 4 - 6).

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Last modified Tue Apr 21 2009

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