Gonchenko S.V. (Nizhny Novgorod, Russia):On dynamical systems with mixed dynamics

 We say that a dynamical system possesses "mixed dynamics" if:

(i) it has simultaneously infinitely many hyperbolic periodic orbits
of all possible types (stable, saddle of all indexes of stability,
completely unstable) and

(ii) these orbits are not separated as a whole, i.e. the closures
of sets of orbits of different types have nonempty intersections.

The phenomenon of mixed dynamics was discovered in our paper
[Gonchenko,Shilnikov,Turaev, 1997, Proc.Steklov Math.Inst] where
it was shown that it can be generic, i.e. systems with mixed
dynamics fill residual subsets from open regions
(in fact, Newhouse regions) of systems.

In the talk we give a rewiev of the corresponding results, discuss
related problems and consider in more detail the phenomenon of mixed
dynamics in reversible systems for which it has a fundamental meaning.