  We investigate a one-dimensional Bose-Hubbard model in a parametrically
driven global harmonic trap. The delicate interplay of both the local atom
interaction and the global driving allows to control the dynamical stability of
the trapped quantum many-body state. The mechanism is illustrated for weak
interaction by a discretized Gross-Pitaevskii equation within a Gaussian
variational ansatz, yielding to a Mathieu equation for the condensate width.
The parametric resonance condition can be tuned by the atom interaction
strength. For stronger interaction, this mechanism is confirmed by results of
the numerically exact time-evolving block decimation scheme. The global
modulation also induces an effective time-independent inhomogeneous hopping
strength for the atoms.
