MATH 617
Dynamical Systems
(Fall 2003)
Instructor:
Yulij Ilyashenko
Meeting
Time & Room
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.
Last modified:
August 13, 2003
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