Math 550. Ordinary Differential Equations

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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

Department of Mathematics
University of Illinois at Urbana-Champaign
273 Altgeld Hall, MC-382
1409 W. Green Street, Urbana, IL 61801 USA
Telephone: (217) 333-3350 Fax (217) 333-9576
office@math.uiuc.edu

Syllabus Revised 2/7/01