Math 617 — Fall 2001 Dynamical Systems

Introduction. Determinism and chaos in dynamical systems

Attractors. Lyapunov stability of fixed points of maps and flows. Stability of periodic orbits. Strange atractors. Smale–Williams solenoid. Elements of symbolic dynamics. Maximal attractors and their fractal dimension. Concept of a minimal attractor.

Elements of hyperbolic theory. Structural stability. Structurally stable flows in the plane. Hadamard–Perron and Grobman–Hartman theorems. Morse–Smale systems. Smale horseshoe. Anosov diffeomorphisms of a torus. 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 tirus: density, unifirm distribution.

Introduction to KAM theory. Rapidly converging iteration method. Diffeomorphisms of the circle close to the rigid rotation. Arnold's theorem on the equivalence of a rigid rotation of a circle and its small perturbation.

Elements of Hamiltonian mechanics. Symplectic structure as an invariant of the Hamiltonian flow. Origin of the sympectic geometry. Completely integrable systems and Liouvillian tori.

Basic analytic theory. Existence and uniqueness theorem in the complex domain. Normal forms near a fixed point. Resonances. Linear systems with complex time. Monodromy. Riemann-Hilbert problem. Solvability for the plane and nonsolvability for the Riemann sphere.

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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 Dinamical Systems.