  We elaborate a scheme of trapping-expulsion management (TEM), in the form of
the quadratic potential periodically switching between confinement and
expulsion, as a means of stabilization of two-dimensional dynamical states
against the backdrop of the critical collapse driven by the cubic
self-attraction with strength g. The TEM scheme may be implemented, as
spatially or temporally periodic modulations, in optics or BEC, respectively.
The consideration is carried out by dint of numerical simulations and
variational approximation (VA). In terms of the VA, the dynamics amounts to a
nonlinear Ermakov equation, which, in turn, is tantamount to a linear Mathieu
equation. Stability boundaries are found as functions of g and parameters of
the periodic modulation of the trapping potential. Below the usual collapse
threshold, which is known, in the numerical form, as g < 5.85 (in the standard
notation), the stability is limited by the onset of the parametric resonance.
This stability limit, including the setup with the self-repulsive sign of the
cubic term (g < 0), is accurately predicted by the VA. At g > 5.85, the
collapse threshold is found with the help of full numerical simulations. The
relative increase of the critical value of g above 5.85 is ~ 1.5%, which is a
meaningful result, even if its size is small, because the collapse threshold is
a universal constant, which is difficult to change.
