Instructor Yu Ilyashenko
Fall 2013
None
Yu Ilyashenko (e-mail: yulij@math.cornell.edu): Wednesday, 5pm - 6pm in Malott 511
You can download the homeworks here:
Homework 1
Introduction. Philosophy of generic 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.
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 in attractors 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.
There will be biweekly take-home assignments, that will count for 80% of the entire grade.
The last homework will be a take-home final exam counting 20% for the grade.