MA 732-001 - Ordinary Differential Equations II

Mathematics

The final exam is Friday, May 3, 1 - 4 p.m., in the usual room.

Things you should be able to do:

Use polar coordinates to analyze degenerate equilibria in the plane (Homework 1, problems 1 - 3).
For maps between Banach spaces, use the definition of derivative, and calculate derivatives using the generalized product rule and chain rule (Homework 2, problems 1, 3).
Use the Contraction Mapping Theorem and the Contraction Mapping Theorem with Parameters (Homework 2, problems 4, 5; Homework 4, problem 1).
Use center manifold reduction to analyze nonhyperbolic equilibria (Homework 3, problems 3, 4, 5; Homework 5, problem 4).
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 4, problems 2 - 5.)
Use linearization, nullclines, Bendixson and Dulac Criteria, and the Poincare-Bendixson Theorem to analyze planar ODEs. (Homework 4, problem 4 a-d; Homework 5, problems 1, 2 a-c.)
Use perturbation methods to approximate and study a Poincare map. (Homework 5, problem 3.)
Use center manifold reduction to analyze bifurcations. (Homework 5, problems 5-7.)
Use the variational equation to study a Poincare map. (Homework 6, problems 1 and 3.)
Use the adjoint equation to determine which periodic orbits in a one-parameter family persist under an autonomous perturbation. (Homework 6, problem 2.)

Last modified Tue Apr 30 2012
Send questions or comments to schecter@ncsu.edu