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

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.

14 pages, 4 figures

References in corpus (6)

• The Fermi-Pasta-Ulam problem revisited: stochasticity thresholds in nonlinear Hamiltonian systems
• q-Breathers in Fermi-Pasta-Ulam chains: existence, localization and stability
• Low energy chaos in the Fermi-Pasta-Ulam problem
• Tail resonances of FPU q-breathers and their impact on the pathway to equipartition
• q-breathers in finite two- and three-dimensional nonlinear acoustic lattices
• Scaling properties of $q$-breathers in nonlinear acoustic lattices

Cited by in corpus (12)

• Bridging coupling bandgaps in nonlinear acoustic metamaterials
• Energy localization on q-tori, long term stability and the interpretation of FPU recurrences
• The two-stage dynamics in the Fermi-Pasta-Ulam problem: from regular to diffusive behavior
• Low-dimensional q-Tori in FPU Lattices: Dynamics and Localization Properties
• Boundary effects on quantum q-breathers in a Bose-Hubbard chain
• q-breathers in Discrete Nonlinear Schroedinger lattices
• Elliptic tori in FPU non-linear chains with a small number of nodes
• Prethermalization in Fermi-Pasta-Ulam-Tsingou chains
• Extensive packet excitations in FPU and Toda lattices
• Nonlinear dynamics and chaos in multidimensional disordered Hamiltonian systems
• Computational efficiency of numerical integration methods for the tangent dynamics of many-body Hamiltonian systems in one and two spatial dimensions
• Stages of dynamics in the Fermi-Pasta-Ulam system as probed by the first Toda integral