Elliptic tori in FPU non-linear chains with a small number of nodes

TL;DR

This paper adapts an algorithm for constructing elliptic tori to FPU chains, enabling semi-analytical stability analysis of these nonlinear systems and comparing results with numerical methods.

Contribution

It introduces a novel adaptation of an elliptic tori construction algorithm specifically for FPU chains, validated through computational and numerical comparisons.

Findings

Successful construction of elliptic tori for N=4,8 FPU chains.

Identification of stability regions around elliptic tori.

Agreement between semi-analytical and numerical stability results.

Abstract

We revisit an algorithm constructing elliptic tori, that was originally designed for applications to planetary hamiltonian systems. The scheme is adapted to properly work with models of chains of $N+1$ particles interacting via anharmonic potentials, thus covering also the case of FPU chains. After having preliminarily settled the Hamiltonian in a suitable way, we perform a sequence of canonical transformations removing the undesired perturbative terms by an iterative procedure. This is done by using the Lie series approach, that is explicitly implemented in a programming code with the help of a software package, which is especially designed for computer algebra manipulations. In the cases of FPU chains with $N=4,\, 8$, we successfully apply our new algorithm to the construction of elliptic tori for wide sets of the parameter ruling the size of the perturbation, i.e., the total energy of the system. Moreover, we explore the stability regions surrounding 1D elliptic tori. We compare our semi-analytical results with those provided by numerical explorations of the FPU-model dynamics, where the latter ones are obtained by using techniques based on the so called frequency analysis. We find that our procedure works up to values of the total energy that are of the same order of magnitude with respect to the maximal ones, for which elliptic tori are detected by numerical methods.