The inhomogeneous Fermi-Pasta-Ulam chain

TL;DR

This paper investigates the resonance phenomena in the inhomogeneous Fermi-Pasta-Ulam chain, analyzing how mass ratios influence the system's dynamics, bifurcations, and stability, with a focus on specific frequency ratios and their effects.

Contribution

It identifies the mass ratios producing prominent resonances and analyzes the resulting dynamical behaviors, including bifurcations and stability, in the inhomogeneous Fermi-Pasta-Ulam chain.

Findings

Certain mass ratios lead to prominent resonances.

A Hamiltonian-Hopf bifurcation causes complex instability.

Normal forms reveal stability characteristics of solutions.

Abstract

The inhomogeneous Fermi-Pasta-Ulam chain is studied by identifying the mass ratios that produce prominent resonances. This is a technically complicated problem as we have to solve an inverse problem for the spectrum of the corresponding linearized equations of motion. In the case of the inhomogeneous periodic Fermi-Pasta-Ulam chain with four particles each mass ratio determines a frequency ratio for the quadratic part of the Hamiltonian. Most prominent frequency ratios occur but not all. In general we find a one-dimensional variety of mass ratios for a given frequency ratio. For the resonance 1:2:3 a small cubic term added to the Hamiltonian leads to a dynamical behaviour that shows a difference between the case that two masses are equal and the more general case of four different masses. For two equal masses the normalized system is integrable and chaotic behaviour is small-scale. In the transition to four different masses we find a Hamiltonian-Hopf bifurcation of one of the normal modes leading to complex instability and Shilnikov-Devaney bifurcation. The other families of short-periodic solutions can be localized from the normal forms together with their stability characteristics. For illustration we use action simplices and the behaviour with time of the H_2 integral of the normal forms.