Content: This course will introduce and develop the notions underlying the geometric theory of dynamical systems and ordinary differential equations. Particular attention will be paid to ideas and techniques that are motivated by applications in a range of the physical, biological and chemical sciences. In particular, motivating examples will be taken from chemical reaction network theory, climate models, fluid motion, celestial mechanics and neuronal dynamics.
The module will be structured around the following topics:
- Review of basic theory: flows, notions of stability, linearization, phase portraits, etc.
- `Solvable' systems: integrability and gradient structure, applications in celestial mechanics and chemical reaction networks.
- Invariant manifold theorems: stable, unstable and center manifolds.
- Bifurcation theory from a geometric perspective.
- Compactification techniques: flow at infinity, blow-up, collision manifolds.
- Chaotic dynamics: horsehoes, Melnikov method and discussion of strange attractors.
- Singular perturbation theory: averaging and normally hyperbolic manifolds.