Cornell Math - MATH 617, Fall 2003
MATH 617
Dynamical Systems
(Fall 2003)
Instructor: Yulij Ilyashenko
Introduction. Philosophy of general position.
Structural stability and robust properties of vector fields. Andronov-Pontryagin criterion of structural stability for planar vector fields. Hadamard-Perron theorem. Smale horseshoe. Elements of symbolic dynamics.
Attractors. Lyapunov stability of fixed points of maps and flows. Stability of periodic orbits. Strange attractors. Smale-Williams solenoid. Maximal attractors and their fractal dimension. Concept of a minimal attractor. Invariant measures. Krylov-Bogolyubov theorem.
Elements of hyperbolic theory. Grobman-Hartman theorem. Morse-Smale systems. Anosov diffeomorphisms of a torus and their structural stability. Hyperbolic sets. Persistence of invariant manifolds. Homoclinic web.
Dynamical systems in low dimensions. Poincaré-Bendixson theorem. Attractors of planar differential equations. Diffeomorphisms of the circle: rotation number, periodic orbits, conjugacy to the rigid rotation. Flows on a torus: density, uniform distribution. Polynomial dynamical systems.
Elements of ergodic theory. Birkhoff-Hinchin ergodic theorem. Ergodicity of nonresonant shifts of a torus. Geodesic flows on surfaces with the negative curvature. Mixing. Survey on regular and chaotic dynamical systems.
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.