Dynamical Systems

The Theory of Dynamical systems is a branch of the theory of Differential Equations. It was founded as a separate discipline by Poincare, and developed by Birkhoff and Smale. Substancial contributions were made by mathematicians of the Moscow school, e.g. Kolmogorov, Arnold, Alekseev, Anosov, Sinai, and others. The proposed course is an introduction to the subject. It contains a survey of the field. The basic theorem is proved; the ideas of the proofs of more difficult results are explained without technical details.

Prerequisites: An introductory course to Ordinary Differential Equations.


Curriculum:

  1. Introduction. Philosophy of general position
  2. Generic dynamical systems in the plane. Limit behavior of solutions; Andronov-Pontryagin criterion of structural stability; Poincare-Bendixson theorem
  3. Elements of hyperbolic theory. Hadamard – Perron theorem; Smale horseshoe; elements of symbolic dynamics; Anosov diffeomorphisms of a torus and their structural stability; Grobman-Hartman theorem; normal hyperbolicity and persistence of invariant manifolds; structurally stable DS are not dense.
  4. Attractors. Lyapunov stability of equilibrium points and periodic orbits; maximal attractors and their fractal dimension; strange attractors; Smale-Williams solenoid.
  5. Dynamical systems in low dimension. Diffeomorphisms of a circle; rotation number, periodic orbits; conjugacy to rigid rotation; flows on a torus; density; uniform distribution.
  6. Elements of ergodic theory. Survey of measure theory; invariant measures of dynamical systems; Krylov-Bogolyubov theorem; Birkhoff-Khinchin ergodic theorem; ergodicity of nonresonant shifts and Anosov diffeomorphisms of a torus; geodesic flows; mixing.

Textbooks

  • V. I. Arnold, Geometric theory of ordinary differential equations.
  • B. Hasselblat, A. Katok, Introduction to the modern theory of dynamical systems, Cambridge Univ. Press.