MATH 716 spring 2021

Graduate Course on Ordinary Differential Equations

Prerequistes

Two semesters of undergraduate real analysis (math 521&522) or graduate real analysis (math 721).

Course outline

We will not follow any particular textbook. I will write notes as we go along. The topics to be covered are

  1. Motivating examples
  2. Existence and Uniqueness using Picard iteration; the case of linear equations
  3. Flows, Invariant Regions, and Attractors; Ważewski principle
  4. Calculus and the Implicit Function Theorem in Banach Spaces
  5. Differentiable dependence of solutions on parameters and the variational equation
  6. Bifurcation theorems for fixed points
  7. Hopf bifurcations
  8. Stability through linearization; Stable and Unstable Manifolds
  9. Stability of periodic orbits; Poincaré maps
  10. Transverse homoclinic points; homoclinic tangle