Math 554 : Ordinary differential equations
This course will be organized as a collection of (reasonably)
self-contained and independent modules. In the first part of class, we
will review the basic theory of ODE's (modules 1 and 2). The topics
for the later part of the class will be decided based on the interests
of the class. I have listed some of the possible topics in modules 3-6.
The class should be broadly accessible. I will not assume any prior
knowledge on this subject, and all the neccessary ``technology'' will
be developed in the class.
Module 1 : Basics of ODEs
(a) Equivalance between differential equations with boundary
conditions
and integral equations, Picard iteration, Gronwall's lemma, the
existence/uniqueness for solutions to ODEs.
(b) Classification of ODEs: Linear vs. nonlinear, autonomous vs.
non-autonomous, Initial, boundary and eigen-value problems.
(c) Qualitative behavior: 2-D autonomous systems, phase plane
analysis,
Poincare-Bendixon theory and limit cycles.
(d) Smale's horseshoe, Chaos in $d >= 3$
Module 2 : Linear equations
(a) General theory of autonomous linear systems, Floquet theory.
(b) Second order linear equations, Boundary value and eigenvalue
problems, Strum-Liouville theory, qualitative properties of solutions,
the
maximum principle and comparison theorems for solutions.
(c) The Rayleigh-Ritz criterion and orthogonality of eigenfunctions,
generalized Fourier series.
(d*) (optional) Special functions, solutions in the complex plane,
Gauss' hypergeometric function.
Module 3 : Local theory for nonlinear systems.
(a) Fixed points, Linearization and stability, stable manifolds and
the
Hartman-Grobman theorem.
(b) Normal form analysis for local bifurcations.
(c) The common types of bifurcations: pitchfork, saddle-node, Hopf
etc.
(d) Bifurcations from periodic orbits -- Poincare maps.
Module 4 : Perturbation theory and asymptotics
(a) The method of multiple scales, the WKB method.
(b) Some rigorous analysis of ODE's and the limiting behavior of
solutions as a parameter goes to zero.
Module 5 : Calculus of Variations/classical mechanics
(a) The Euler-Lagrange equations and some applicationst; the
principle of
least action; the Lagrangian and Hamiltonian formulations of classical
mechanics.
Module 6 : Numerical methods.
(a) Various numerical methods for ODEs.