Detecting chaos, determining the dimensions of tori and predicting slow diffusion in Fermi--Pasta--Ulam lattices by the Generalized Alignment Index method

TL;DR

The paper introduces an efficient GALI method based on SVD for detecting chaos, identifying tori dimensions, and predicting slow diffusion in multi-dimensional Hamiltonian systems like FPU lattices.

Contribution

It proposes a novel SVD-based computation of GALI indices, enabling rapid chaos detection, torus dimensionality determination, and diffusion prediction in complex Hamiltonian systems.

Findings

GALI indices effectively detect chaos and regular motion.

The method accurately identifies low-dimensional tori in FPU lattices.

It predicts weak diffusion long before it manifests in oscillations.

Abstract

The recently introduced GALI method is used for rapidly detecting chaos, determining the dimensionality of regular motion and predicting slow diffusion in multi--dimensional Hamiltonian systems. We propose an efficient computation of the GALI$_k$ indices, which represent volume elements of $k$ randomly chosen deviation vectors from a given orbit, based on the Singular Value Decomposition (SVD) algorithm. We obtain theoretically and verify numerically asymptotic estimates of GALIs long--time behavior in the case of regular orbits lying on low--dimensional tori. The GALI$_k$ indices are applied to rapidly detect chaotic oscillations, identify low--dimensional tori of Fermi--Pasta--Ulam (FPU) lattices at low energies and predict weak diffusion away from quasiperiodic motion, long before it is actually observed in the oscillations.