  We consider theories with time-dependent Hamiltonians which alternate between
being bounded and unbounded from below. For appropriate frequencies dynamical
stabilization can occur rendering the effective potential of the system stable.
We first study a free field theory on a torus with a time-dependent mass term,
finding that the stability regions are described in terms of the phase diagram
of the Mathieu equation. Using number theory we have found a compactification
scheme such as to avoid resonances for all momentum modes in the theory. We
further consider the gravity dual of a conformal field theory on a sphere in
three spacetime dimensions, deformed by a doubletrace operator. The gravity
dual of the theory with a constant unbounded potential develops big crunch
singularities; we study when such singularities can be cured by dynamical
stabilization. We numerically solve the Einstein-scalar equations of motion in
the case of a time-dependent doubletrace deformation and find that for
sufficiently high frequencies the theory is dynamically stabilized and big
crunches get screened by black hole horizons.
