Nonlinear resonances and antiresonances of a forced sonic vacuum

D. Pozharskiy, Y. Zhang, M. O. Williams, D. M. McFarland, P. G. Kevrekidis, A. F. Vakakis, I. G. Kevrekidis

Research output: Contribution to journalArticle

Abstract

We consider a harmonically driven acoustic medium in the form of a (finite length) highly nonlinear granular crystal with an amplitude- and frequency-dependent boundary drive. Despite the absence of a linear spectrum in the system, we identify resonant periodic propagation whereby the crystal responds at integer multiples of the drive period and observe that this can lead to local maxima of transmitted force at its fixed boundary. In addition, we identify and discuss minima of the transmitted force ("antiresonances") between these resonances. Representative one-parameter complex bifurcation diagrams involve period doublings and Neimark-Sacker bifurcations as well as multiple isolas (e.g., of period-3, -4, or -5 solutions entrained by the forcing). We combine them in a more detailed, two-parameter bifurcation diagram describing the stability of such responses to both frequency and amplitude variations of the drive. This picture supports a notion of a (purely) "nonlinear spectrum" in a system which allows no sound wave propagation (due to zero sound speed: the so-called sonic vacuum). We rationalize this behavior in terms of purely nonlinear building blocks: apparent traveling and standing nonlinear waves.

Original languageEnglish (US)
Article number063203
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume92
Issue number6
DOIs
StatePublished - Dec 23 2015

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Condensed Matter Physics

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    Pozharskiy, D., Zhang, Y., Williams, M. O., McFarland, D. M., Kevrekidis, P. G., Vakakis, A. F., & Kevrekidis, I. G. (2015). Nonlinear resonances and antiresonances of a forced sonic vacuum. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 92(6), [063203]. https://doi.org/10.1103/PhysRevE.92.063203