Resonances and Adiabatic Invariants for the FPUT System

We review our investigations into the possible existence of adiabatic invariants in the FPUT system, in the framework of classical perturbation methods as the main analytical tools. Early theoretical approaches based on the search for nonlinear normal modes proved unable to account for the lack of ergodicity observed in the original FPUT experiment. More recently, some authors have proposed studying the existence of a single adiabatic invariant by modeling the chain as a collection of identical oscillators with the same frequency, perturbed by both linear and nonlinear interactions between neighboring particles. The resulting analyses suggest that complete resonance allow us to identify an adiabatic invariant that persists independently of the number of particles.