  The well known phenomenon of avoided level crossing (ALC) can be perceived as
a quantum analogue of the Thom's catastrophes in classical dynamical systems.
One-dimensional Schr\"{o}dinger equation is chosen for illustration. In
constructive manner, a family of confining polynomial potentials is considered,
characterized by the presence of an $N-$plet of high barriers separating the
$(N+1)-$plet of deep valleys. The bifurcations of the long-time classical
equilibria are shown paralleled by the ALCs in the quantum low-lying spectra.
Every tunneling-controlled fine-tuned switch of dominance between the valleys
is finally interpreted as a change of the topological structure of the
probability density representing a genuine quantum relocalization catastrophe.
