ÍâÍøÌìÌÃ

Invariant manifolds are landmarks that organize the long term behavior of dynamical systems with complicatedÌý trajectories. ÌýFrom the point of view of applications, it is interesting to knowÌýthat they persist under perturbations or that one can validate theÌýthe numerical calculations finding them. There are two main theoremsÌýin this direction: Normal hyperbolicity and KAM theory. We willÌýpresent numerical explorations and rigorous results on what happensÌýon the boundary of validity of these theorems.ÌýThere are some surprising regularities and scaling relations, some ofÌýwhich can be proved rigorously. This is joint work with manyÌýpeople including R. Calleja, A. Haro, T. Blass.