Georgia Institute of Technology College of Sciences

The long term behavior of dynamical systems can be understood by studying invariant manifolds that act as landmarks that anchor the orbits. It is important to understand which invariant manifolds persist under modifications of the system. A deep mathematical theory, developed in the 70's shows that invariant manifolds which persist under changes are those that have sharp expansion (in the future or in the past) in the the normal directions. A deep question is what happens in the boundary of these theorems of persistence. This question requires to understand the interplay between the geometric properties and the functional analysis of the functional equations involved.In this talk we present several mechanisms in which properties of normal hyperbolicity degenerate, so leading to the breakdown of the invariant manifold. Numerical studies lead to surprising conjectures relating the breakdown to phenomena in phase transitions. The results have been obtained combining numerical exploration and rigorous reasoning.