title: Nonlinear resonances and antiresonances of a forced sonic vacuum.
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.
authors.affiliation: Department of Chemical and Biological Engineering and PACM, Princeton University, Princeton, New Jersey 08544, USA.
authors.forename: D
authors.initials: D
authors.lastname: Pozharskiy
authors.affiliation: Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 West Green Street, Urbana, Illinois 61822, USA.
authors.forename: Y
authors.initials: Y
authors.lastname: Zhang
authors.affiliation: Department of Chemical and Biological Engineering and PACM, Princeton University, Princeton, New Jersey 08544, USA.
authors.forename: M O
authors.initials: MO
authors.lastname: Williams
authors.affiliation: Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61822, USA.
authors.forename: D M
authors.initials: DM
authors.lastname: McFarland
authors.affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA.|Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87544, USA.
authors.forename: P G
authors.initials: PG
authors.lastname: Kevrekidis
authors.affiliation: Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 West Green Street, Urbana, Illinois 61822, USA.
authors.forename: A F
authors.initials: AF
authors.lastname: Vakakis
authors.affiliation: Department of Chemical and Biological Engineering and PACM, Princeton University, Princeton, New Jersey 08544, USA.
authors.forename: I G
authors.initials: IG
authors.lastname: Kevrekidis