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.