Abstract. The numerical experiment of Fermi, Pasta and Ulam (FPU) in 1954 aimed to probe ergodicity in an one-dimensional chain of N weakly nonlinearly coupled oscillators, however led to an unexpected integrable-like behaviour. It is noteworthy that FPU was the first system which was solved numerically by a computer and it is linked with the birth of integrable systems through the discovery of solitons in the Korteweg-de Vries equation. In recent years there is a growing interest regarding the FPU model as perturbed Toda lattice, the latter of which is completely integrable. This idea goes back in a work of Flaschka, while recent studies suggest that the Toda integrals are the relevant dynamical observables for studying the FPU model. In the present talk I will discuss and compare the stages of dynamics in the FPU model for different classes of initial conditions, and propose a simple method to determine slow diffusion to estimate the two fundamental timescales, namely: i) the time of stability, where FPU behaves as Toda, and ii) the time to equilibrium.
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