Syllabus for
Math 550. Ordinary Differential Equations
Textbooks
- Coddington and Levinson, Theory of Ordinary Differential Equations, Krieger Publ., 1984
- Arrowsmith and Place, An Introduction to Dynamical Systems, Cambridge University Press, 1990
The exam topics are based on material that is covered in the texts.
- Existence, Uniqueness
- vector fields and flows
- existence and uniqueness theorems
- examples of non-uniqueness
- continuation of solutions
- Equilibrium and Linearization
- equilibrium and stability
- linearization about a solution and the equation of variation
- solution of linear systems via exponential map
- almost linear systems and linear stability
- hyperbolic fixed points, stable and unstable manifolds
- Liapunov functions and Liapunov stability
- Geometric Methods for Nonlinear Equations
- limit sets and asymptotic behavior
- phase portrait methods in 2 and 3 dimensions
- periodic orbits and the Poincaré-Bendixon theorem
- Hamiltonian Systems
- Hamiltonian flows
- first integrals
- symplectic matrices
- Poincaré maps
- Bifurcation Theory
- bifurcation of equilibria
- Hopf bifurcation theorem
- Lorenz system
- Area Preserving Maps
- area preserving maps
- elliptic fixed points
- Poincaré-Birkhoff theorem