How stable is chaos? From Smale to the Hénon attractor.

One of the main questions in dynamical systems is to understand the long-term behavior of a system and how this behavior changes when the system is slightly perturbed. Often, the long-term dynamics is organized around an attractor, a set towards which nearby orbits evolve. Some attractors can have very complicated, chaotic dynamics, while still possessing a strong form of stability known as hyperbolicity.

I will give an overview of the history that led Smale to conjecture that hyperbolicity should describe the typical behavior of dynamical systems. While this picture works well in many one-dimensional settings, it is known to fail in dimension two. I will discuss what can happen beyond the hyperbolic setting, focusing on chaotic attractors and, in particular, on the Hénon attractor. I will discuss as well the case of physical systems coming from biological observations. 

This brings us back to the question of stability: if hyperbolicity is no longer present, what, if anything, survives under small perturbations? I will discuss some results in this direction and ask to what extent Smale's original picture can still survive in a weaker form.