Dynamical Systems

The Theory of Dynamical systems is a branch of mathematics, based on the idea of iterated functions. Concepts of contraction, ergodicity, structural stability, and others may be helpful for the investigations related to the theory of Differential Equations, and (to some extent) the Number Theory, the Mathematical Physics and the Probability Theory. Substantial contributions here 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 theorems are proved; the key ideas are presented.

Prerequisites: An introductory course to Ordinary Differential Equations.


  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.


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