Quasiperiodic granular chains and Hofstadter butterflies

Abstract

We study quasiperiodicity-induced localization of waves in strongly precompressed granular chains. We propose three different set-ups, inspired by the Aubry-André (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). We also show, using the chain of cylinders as an example, how to recover the Hofstadter spectrum from the system dynamics. Finally, in a suite of numerical computations, we demonstrate localization and also that there exist regimes of ballistic, superdiffusive, diffusive and subdiffusive transport. Our models provide a flexible set of systems to study quasiperiodicity-induced analogues of Anderson phenomena in granular chains that one can tune controllably from weakly to strongly nonlinear regimes.This article is part of the theme issue 'Nonlinear energy transfer in dynamical and acoustical systems'.

Topological bands and localized vibration modes in quasiperiodic beams

“…The black regions define ranges of frequency populated by the bulk eigenvalues, while the white areas correspond to frequency ranges where no states exist and identify bandgaps. The spectrum has features similar to the Hofstadter butterfly spectrum encountered in quantum mechanics for lattices under a magnetic field [13] and in discrete mechanical QP lattices [16,25]. One can observe the existence of a low frequency bandgap starting at zero, due to the presence of the ground springs, and a number of other gaps associated with Bragg scattering.…”

“…This observation makes these modes different than the localized modes corresponding to eigenstates in the bulk gaps, which are only localized at the edges, and appear independently of the length of the finite structure considered. In the literature of discrete QP lattices, this type of localized modes has been investigated in the context of phase transitions [14][15][16], but a connection to the edge states spanning the gaps as given here is currently missing. This connection identifies an open question regarding the regions where these modes may localize, which may be explained in the context of the smooth edge-interioredge transitions experienced by some of the topological states as a function of α.…”

“…One of the hallmarks of QP media is in the fractal spectra, which was originally observed for electrons in the presence of magnetic fields exhibiting the well-known Hofstadter's butterfly spectrum [13]. In the context of localization, the transition from globally spanning to localized modes originally introduced in [14] has been observed in different QP systems, such as photonic lattices [15] and elastic granular chains [16]. These localized modes are of interest to physical phenomena like disorder enhanced transport [17], or one-way reflection in acoustic waveguides [18].…”

Introduction to a topical issue ‘nonlinear energy transfer in dynamical and acoustical Systems'

“…The work by Porter and co-workers [51] addresses complicated bifurcation patterns observed in quasi-periodic granular chains. Localization of deformations in nonlinear elastic lattices is discussed in the insightful paper of Ruzzene et al [52].…”

Observation of Hofstadter butterfly and topological edge states in reconfigurable quasi-periodic acoustic crystals

“…Nevertheless, the possibility of implementing the operator with quasiperiodic 1D platforms opens up new ways to observe and study the associated topological phenomena. A number of theoretical and experimental works have recently induced quasiperiodicity in a broad variety of systems 7,[21][22][23][24][25][26][27] to explore topological phase transitions and edge states. Even more, topological phases in higher dimensions 23,[28][29][30][31] have been predicted and the those characterized by second class Chern number in 2D quasiperiodic crystals observed experimentally 21,22 .…”