Resonant normal form for even periodic FPU chains

TL;DR

This paper demonstrates that even periodic FPU chains near equilibrium can be simplified into a completely integrable resonant normal form of order four, explaining their observed numerical behavior and revealing hyperbolic dynamics.

Contribution

It introduces a resonant Birkhoff normal form for even FPU chains that is completely integrable, providing new insights into their dynamics and stability properties.

Findings

Normal form is completely integrable for even chains.

Hyperbolic dynamics are typical in the integrable approximation.

KAM theorem applies to chains with Dirichlet boundary conditions.

Abstract

In this paper we investigate periodic FPU chains with an even number of particles. We show that near the equilibrium point, any such chain admits a \emph{resonant} Birkhoff normal form of order four which is \emph{completely integrable} - an important fact which helps explain the numerical experiments of Fermi, Pasta, and Ulam. We analyze the moment map of the integrable approximation of an even FPU chain. Unlike in the case of odd FPU chains these integrable systems (generically) exhibit hyperbolic dynamics. As an application we prove that any FPU chain with Dirichlet boundary conditions admits a Birkhoff normal form up to order four and show that a KAM theorem applies.