## Nonlinear analysis

### Prof. Andrej Agrachev

Number of cycles: 2

First semester: Thu 9-10:30, Fri 9-10:30 (starting from October 13)

Content:

- Regular and critical point of smooth maps. The notion of transversality.
- Sard's lemma. Generic properties.
- Whitney embedding theorem.
- Topological degree of a continuous map.
- The Leray-Schauder degree.
- Intersection number and linking number.
- Index of a vector field on a smooth manifold.
- Linearization of a vector field at the equilibrium. Phase portraits of linear systems.
- Asymptotic stability of the equilibrium and Lyapunov functions.
- Structural stability of the hyperbolic equilibria; the theoremGrobman.
- Asymptotic behavior of the solutions to two-dimensional systems; the Poincaré-Bendixson theorem.
- Structurally stable two-dimensional phase portraits.