  We present a theoretical model to describe an instability mechanism in
ultra-cold gases, where long-range interactions are taken into account.
Focusing on the nonlinear coupling between the collective (plasma-like) and the
center-of-mass modes, we show that the resulting dynamics is governed by a
parametric equation of the generalized Mathieu type and compute the
corresponding stability chart. We apply our model to typical ranges of
magneto-optical traps (MOT) parameters and find a good agreement with previous
experimental observations.
