  Spherically symmetric perturbations in AdS-scalar field systems of small
amplitude epsilon approximately periodic on time scales of order 1/epsilon^2
(in the sense that no significant transfer of energy between the AdS normal
modes occurs) have played an important role in considerations of AdS stability.
They are seen as anchors of stability islands where collapse of small
perturbations to black holes does not occur. (This collapse, if it happens,
typically develops on time scales of the order 1/epsilon^2.) We construct an
analytic treatment of the frequency spectra of such quasiperiodic
perturbations, paying special attention to the large frequency asymptotics. For
the case of a self-interacting phi^4 scalar field in a non-dynamical AdS
background, we arrive at a fairly complete analytic picture involving
quasiperiodic spectra with an exponential suppression modulated by a power law
at large mode numbers. For the case of dynamical gravity, the structure of the
large frequency asymptotics is more complicated. We give analytic explanations
for the general qualitative features of quasiperiodic solutions localized
around a single mode, in close parallel to our discussion of the probe scalar
field, and find numerical evidence for logarithmic modulations in the
gravitational quasiperiodic spectra existing on top of the formulas previously
reported in the literature.
