  We study the stability of anti-de Sitter (AdS) spacetime to spherically
symmetric perturbations of a real scalar field in general relativity. Further,
we work within the context of the "two time framework" (TTF) approximation,
which describes the leading nonlinear effects for small amplitude
perturbations, and is therefore suitable for studying the weakly turbulent
instability of AdS---including both collapsing and non-collapsing solutions. We
have previously identified a class of quasi-periodic (QP) solutions to the TTF
equations, and in this work we analyze their stability. We show that there
exist several families of QP solutions that are stable to linear order, and we
argue that these solutions represent islands of stability in TTF. We extract
the eigenmodes of small oscillations about QP solutions, and we use them to
predict approximate recurrence times for generic non-collapsing initial data in
the full (non-TTF) system. Alternatively, when sufficient energy is driven to
high-frequency modes, as occurs for initial data far from a QP solution, the
TTF description breaks down as an approximation to the full system. Depending
on the higher order dynamics of the full system, this often signals an imminent
collapse to a black hole.
