DSNT Seminar: On the applicability of KAM theory to realistic physical problems: effective stability and computer-assisted proofs

Abstract: Many physical problems can be described by nearly-integrable Hamiltonian systems and KAM theorem is one of the main results in this field. Since it ensures the existence of invariant solutions, it can be used to prove stability. However, the strict hypotheses (in particular the smallness of the perturbation parameter) make this instrument most of the time unusable in realistic models. In this seminar, I want to show how the problem of the applicability can be overcome with the use of
computer-assisted techniques. I am going to present a simple example about how to rigorously compute a lower bound for the time of stability around an elliptic equilibrium point (joint work with U. Locatelli). The outcome is the so-called effective stability, that is a stability for times longer than the characteristic time of the system, in the spirit of Nekhoroshev theorem (where the stability time is exponentially long w.r.t. the inverse of the small parameter). Similar estimates can be used to bound the time of stability around KAM tori or lower dimensional elliptic tori.

See all upcoming seminars in this seminar series.