Nonlinear analysis

Prof. Andrej Agrachev

Number of cycles: 2

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


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