  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. Remarkably, 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 they can
lead to local maxima of transmitted force at its fixed boundary. In addition,
we identify and discuss minima of the transmitted force ("antiresonances") in
between these resonances. Representative one-parameter complex bifurcation
diagrams involve period doublings, 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 an unprecedented example of 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.
