Dynamical Systems
Further information will appear shortly.
Course Description
This course is an introduction to the field of dynamical systems, with detailed proofs and minimal prerequisites, following the book of Zehnder fairly closely. The course will be in three parts:
1) First we will discuss several typical problems and phenomena in dynamics as they arise in simple models. Then we will begin to study the important role that invariant measures play in dynamics, in particular we will prove and illustrate the Birkhoff ergodic theorem.
2) The second part of the course will be concerned with the dynamics in a neighborhood of a hyperbolic fixed point. In particular we will prove the theorem of Hartman-Grobman and the local and global invariant manifold theorems.
3) The third part of the course will be devoted to the study of hyperbolic invariant sets and some of their remarkable properties: with the help of the shadowing lemma we will explore the complicated orbit structure and the presence of chaos through the embedding of Bernoulli systems. The existence of hyperbolic invariant sets will be illustrated through the example of the perturbed pendulum. Finally we will prove some important structural stability properties of systems on hyperbolic invariant sets.
Prerequisites
Analysis I, II, Linear Algebra I, II. The concepts of a topology, the Lebesgue integral, and the flow associated to an ordinary differential equation on euclidean space will also be useful.
Literature
Eduard Zehnder: ''Lectures on Dynamical Systems'', published by the European Mathematical Society.