Colloquium

The talk is a brief historical account of the development of the theory that deals with the phenomenon widely known as "deterministic chaos" — the appearance of irregular chaotic motions in purely deterministic dynamical systems on compact phase spaces.

The hyperbolic theory of dynamical systems provides a mathematical foundation for this paradigm and thus serves as a basis for the theory of chaos. The hyperbolic behavior can be interpreted in various ways and the weakest one is associated with dynamical systems with non-zero Lyapunov exponents.

I will describe various types of hyperbolicity, outline some examples of systems with hyperbolic behavior and discuss the still-open problem of whether chaotic dynamical systems are generic. This genericity problem is closely related to two other important problems in dynamics on whether systems with non-zero Lyapunov exponents exist on any compact phase space and whether chaotic behavior can coexist with a regular (non-chaotic) one in a robust way.