  As part of the quest to uncover universal features of quantum dynamics, we
study catastrophes that form in simple many-body wave functions after a quench,
focusing on two-mode systems that include the two-site Bose Hubbard model, and
under some circumstances optomechanical systems and the Dicke model. When the
wave function is plotted in Fock space plus time certain characteristic
structures generically appear that we identify as cusp caustics. In the
vicinity of such a catastrophe the wave function takes on a universal form
described by the Pearcey function and obeys scaling relations which depend on
the total number of particles $N$. In the thermodynamic limit ($N \rightarrow
\infty$) the cusp becomes singular, but at finite $N$ it is decorated by an
interference pattern. This pattern contains an intricate network of
vortex-antivortex pairs, initiating a theory of topological structures in Fock
space. In the case where the quench takes the form of a $\delta$-kick we show
how to analytically map the wave function onto the Pearcey function and hence
obtain the scaling exponents for the size and position of the cusp, as well as
those for the amplitude and characteristic length scales of its interference
pattern.
