MATH 6170: Dynamical Systems (Fall 2011)

Instructor: Yulij Ilyashenko

Course Web Site: www.math.cornell.edu/~web6170/

Introduction: philosophy of general position.

Generic dynamical systems in the plane: limit behavior of solutions; Andronov-Pontryagin criterion of structural stability; Poincare-Bendixson theorem.

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.

Attractors: Lyapunov stability of equilibrium points and periodic orbits; maximal attractors and their fractal dimension; strange attractors; Smale-Williams solenoid.

Dynamical systems in low dimension: Diffeomorphisms of a circle; rotation number, periodic orbits; conjugacy to rigid rotation; flows on a torus; density; uniform distribution.

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.

Time permitting, some new results on attractors will be presented.

About 2/3 of the course will be covered by the books of Arnold — Geometric Methods in the Theory of Ordinary Differential Equations — and Katok and Husselblat — Introduction to the Modern Theory of Dynamical Systems. Some part will be covered by lecture notes.