  We consider scalar fields which are coupled to Einstein gravity with a
negative cosmological constant, and construct periodic solutions
perturbatively. In particular, we study tachyonic scalar fields whose mass is
at or above the Breitenlohner-Freedman bound in four, five, and seven spacetime
dimensions. The critical amplitude of the leading order perturbation, for which
the perturbative expansion breaks down, increases as we consider less massive
fields. We present various examples including a model with a self-interacting
scalar field which is derived from a consistent truncation of IIB supergravity.
