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

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

2007-05-07

arxiv.org/abs/0705.0804v1

Nonlinear Sciences

Pattern Formation and Solitons

14 pages, 4 figures

Scientific paper

10.1016/j.physd.2007.11.017

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.

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