Title:

Chaotic dynamical systems: main properties and scope of application

Katrin Gelfert

Abstract:

Smale introduced in the 60's the theory of so-called hyperbolic dynamical systems which became the cornerstone of the development of the qualitative theory of differential equations. However this theory does not cover large families of dynamical systems - the Lorenz flow and the Henon map are among the most prominent examples. In fact, from the point of view of concrete physical systems, hyperbolicity seems to be a rather rare property. I will give a brief review of properties of hyperbolic dynamical systems. I will also describe some routes on how to destroy hyperbolicity and will present some properties that continue to hold beyond that realm. Here, frequency and distribution of periodic orbits points will play the leading part.