  Two generically different but universal dynamical quantum many-body behaviors
are discovered by probing the stability of trapped fragmented bosonic systems
with strong repulsive finite/long range inter-particle interactions. We use
different time-dependent processes to destabilize the systems -- a sudden
displacement of the trap is accompanied by a sudden quench of the strength of
the inter-particle repulsion. A rather moderate non-violent evolution of the
density in the first "topology-preserved" scenario is contrasted with a
highly-non-equilibrium dynamics characterizing an explosive changes of the
density profiles in the second scenario. The many-body physics behind is
identified and interpreted in terms of self-induced time-dependent barriers
governing the respective under- and over-a-barrier dynamical evolutions. The
universality of the discovered scenarios is explicitly confirmed in 1D, 2D and
3D many-body computations in (a)symmetric traps and repulsive finite/long range
inter-particle interaction potentials of different shapes. Implications are
briefly discussed.
