This is a Qualifying Eligible (QE) course for the Math PhD with regular, graded HW and a comprehensive final exam.

#### Prerequisites

Linear algebra (the equivalent of 221); real analysis (the equivalent of 531 or 431) an introductory course on differential equations (the equivalent of 353 or 356) strongly encouraged by not strictly required.#### Syllabus

- Background material: linear systems of differential equations -matrix exponentials, diagonization, Jordan form -phase portraits for 2-dimensional linear systems
- Existence/uniqueness: -contraction mapping, Picard iteration -Gronwall's inequality -local and global existence -continuous dependence and differentiability with respect to data
- Non-dimensionalization and scaling
- Dynamical systems: -flows, orbits, invariant sets -stability, Lyapunov functions -stable manifold theorem -Hartman-Grobman theorem
- Oscillations: -Floquet theory -periodic orbits, limit cycles -Poincare-Bendixon theorem -Poincare map, stability of periodic orbits
- Hamiltonian dynamics -integrability of the 2 body problem
- Bifurcation of equilibria -basic bifurcation theorem for pitchfork, transcritical, saddlenode
- Discrete dynamical systems -Introduction to chaos -period doubling, Sharkovskii's theorem

#### References

- Gerald Teschl.
*Ordinary Differential Equations and Dynamical Systems* - David G. Schaeffer and John W. Cain.
*Ordinary Differential Equations: Basics and Beyond*