MATH 6170 - Dynamical Systems

Yulij Ilyashenko, fall 2015.

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 program is renewed, in comparison with the past years. The main topics are the following.

Philosophy of general position. Sard lemma and Thom’s transversality theorem

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

Diffeomorphisms of a circle. Rotation number, periodic orbits; conjugacy to rigid rotation; flows on a torus; density; uniform distribution. Appendix: main ideas of the KAM theory.

Elements of the bifurcation theory in the plane. Local, nonlocal and global bifurcations. Together with the classical, some brand new results will be presented in this section.

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.

Dynamics beyond the uniform hyperbolicity. Newhouse phenomena and Lyapunov instability of attractors

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.