Quasiperiodic granular chains and Hofstadter butterflies

TL;DR

This paper investigates how quasiperiodicity causes wave localization in granular chains, comparing different models inspired by the Aubry–André model, and analyzes the resulting fractal spectra and transport regimes.

Contribution

It introduces three quasiperiodic granular chain models, explores localization transitions, and connects Hofstadter spectra with wave transport properties in nonlinear lattices.

Findings

Localization transition in spherical particle chains with on-site quasiperiodicity.

No localization transition observed in cylindrical particle chains with quasiperiodic contact angles.

Fractal dimension of Hofstadter spectrum decreases near localization transition.

Abstract

We study quasiperiodicity-induced localization of waves in strongly precompressed granular chains. We propose three different setups, inspired by the Aubry--Andr\'e (AA) model, of quasiperiodic chains; and we use these models to compare the effects of on-site and off-site quasiperiodicity in nonlinear lattices. When there is purely on-site quasiperiodicity, which we implement in two different ways, we show for a chain of spherical particles that there is a localization transition (as in the original AA model). However, we observe no localization transition in a chain of cylindrical particles in which we incorporate quasiperiodicity in the distribution of contact angles between adjacent cylinders by making the angle periodicity incommensurate with that of the chain. For each of our three models, we compute the Hofstadter spectrum and the associated Minkowski--Bouligand fractal dimension, and we demonstrate that the fractal dimension decreases as one approaches the localization transition (when it exists). Finally, in a suite of numerical computations, we demonstrate both localization and that there exist regimes of ballistic, superdiffusive, diffusive, and subdiffusive transport. Our models provide a flexible set of systems to study quasiperiodicity-induced analogs of Anderson phenomena in granular chains that one can tune controllably from weakly to strongly nonlinear regimes.