The Fermi-Pasta-Ulam problem: periodic orbits, normal forms and resonance overlap criteria

TL;DR

This paper investigates the transition from localization to delocalization in the Fermi-Pasta-Ulam chain, comparing approaches based on periodic orbits and normal forms, and identifying resonance overlaps as key to this transition.

Contribution

It demonstrates the quantitative agreement of different theoretical approaches in estimating the localization-delocalization threshold in the Fermi-Pasta-Ulam problem.

Findings

Approaches based on periodic orbits and normal forms agree on the localization threshold.

Resonances of overtones are crucial for the transition.

Threshold values depend on control parameters exceeding certain limits.

Abstract

Fermi, Pasta and Ulam observed, that the excitation of a low frequency normal mode in a nonlinear acoustic chain leads to localization in normal mode space on large time scales. Fast equipartition (and thus complete delocalization) in the Fermi-Pasta-Ulam chain is restored if relevant intensive control parameters exceed certain threshold values. We compare recent results on periodic orbits (in the localization regime) and resonant normal forms (in a weak delocalization regime), and relate them to various resonance overlap criteria. We show that the approaches quantitatively agree in their estimate of the localization-delocalization threshold. A key ingredient for this transition are resonances of overtones.