MATH 6170 - Dynamical Systems

Yulij Ilyashenko, fall 2014.

The dramatic history of the development of the theory of dynamical systems will be presented. The paradoxical situation, when a deterministic system exhibits a chaotic behavior will be explained. The main topics are the following.

Introduction. Philosophy of general position.

Generic dynamical systems in the plane. Limit behavior of solutions; Andronov-Pontryagin criterion of structural stability; Poincare-Bendixson theorem; 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 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. Attractors. Lyapunov stability of equilibrium points and periodic orbits; maximal attractors and their fractal dimension; strange attractors; Smale-Williams solenoid; attractors with intermingled basins.

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.

Time permitting, some new results on instability of attractors and on the dynamics on the circle 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 Hasselblat, Introduction to the Modern Theory of Dynamical Systems. Some part will be covered by lecture notes.