  Considerable effort has been dedicated to study the nonlinear instability of
Anti-de Sitter (AdS) within spherical symmetry, but little is known about this
nonlinear instability in the purely gravitational sector, where spherical
symmetry is necessarily broken. In \cite{Bizon:2011gg} the onset of such
nonlinear instability was associated with the existence of irremovable secular
resonances at third order in perturbation theory. Furthermore, it was also
conjectured in \cite{Bizon:2011gg} that certain very fine tuned initial data
would not collapse. Such solutions, upon linearisation, correspond to
individual normal modes of AdS, which can be consistently backreacted to all
orders in perturbation theory. However, the analysis of \cite{Bizon:2011gg} was
restricted to spherical symmetry. The perturbative arguments of
\cite{Bizon:2011gg} were then generalised to gravitational perturbations in
\cite{Dias:2011ss}, and in particular certain time-periodic solutions were also
conjectured to exist - these were coined geons. However, in \cite{Dias:2011ss},
only a certain class of perturbations was considered, for which the
perturbative analysis considerably simplifies. In this manuscript we present
details of the systematic computational formalism and an exhaustive and
complementary analysis of physical properties of the geons and gravitational
AdS instability that were absent in our companion Letter \cite{Dias:2016ewl}.
In particular, we find that, unlike in spherical symmetry, a (single)
gravitational normal mode of AdS can be backreacted to generate a nonlinear
solution only in very exceptional circumstances. We also show that weak
turbulent perturbative theory predicts the existence of direct and inverse
cascades, and give evidence suggesting that the former dominates the latter for
equal energy two-mode seeds.
