  We study the properties of quantum cusp and butterfly catastrophes from an
algebraic viewpoint. The analysis employs an interacting boson model
Hamiltonian describing quantum phase transitions between specific quadrupole
shapes by interpolating between two incompatible dynamical symmetry limits. The
classical properties are determined by using coherent states to construct the
complete phase diagrams associated with Landau potentials exhibiting such
catastrophes.The quantum properties are determined by analyzing the spectra,
transition rates and symmetry character of the eigenstates of critical
Hamiltonians.
