*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

- Motivating examples
- Mathematical ecology: Lotka-Volterra, predator-prey
- Damped and forced pendulum
- Equations on cylinders
- Differential equations on manifolds

- Existence and Uniqueness using Picard iteration; the case of linear equations
- Flows, Invariant Regions, and Attractors; Ważewski principle
- Calculus and the Implicit Function Theorem in Banach Spaces
- Differentiable dependence of solutions on parameters and the variational equation
- Bifurcation theorems for fixed points
- Hopf bifurcations
- Stability through linearization; Stable and Unstable Manifolds
- Stability of periodic orbits; Poincaré maps
- Transverse homoclinic points; homoclinic tangle