The basins associated with different attractors can be arranged in many topologically different ways. In the simplest case, that of gradient dynamics, the different basins are separated by piecewise regularly embedded hypersurfaces. The considerations apply even more too vague attractors of Hamiltonian dynamics, where the authors can scarcely talk of basins and the dominance of an atrractor is never certain. This hazy and fluctuating aspect of the situation in Hamiltonian dynamics is reminiscent of the probabilism of quantum mechanics. The only existing universal model known occurs in the case of gradient dynamics, and thus in principle well-defined local models leading to ordinary catastrophes arise only from gradient dynamics with previously polarized domains. The other cases can give rise to essential catastrophe points leading to the appearance of generalized catastrophes. It is quite conceivable that some natural processes approach equilibrium through a superposition of the two methods with a reversible component of Hamiltonian character and an irreversible gradient component.