### MA 732-001 Spring 2013 Ordinary Differential Equations II

Mathematics

Ordinary Differential Equations by Schaeffer and Cain | Feedback on text
Ordinary Differential Equations and Dynamical Systems by Gerald Teschl
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### Material on the Final Exam

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.)