  We construct time-periodic solutions for a system of self-gravitating
massless scalar field, with negative cosmological constant, in d+1 spacetime
dimensions at spherical symmetry, both perturbatively and numerically. We
estimate the convergence radius of the formally obtained perturbative series
and argue that it is greater then zero. Moreover, this estimate coincides with
the boundary of the convergence domain of our numerical method and the
threshold for the black-hole formation. Then we confirm our results with a
direct numerical evolution. This also gives strong evidence for nonlinear
stability of the constructed time-periodic solutions.
