  We classify the local bifurcations of one dov quantum billiards, showing that
only saddle-center bifurcations can occur. We analyze the resulting planar
system when there is no coupling in the superposition state. In so doing, we
also consider the global bifurcation structure. Using a double-well potential
as a representative example, we demonstrate how to locate bifurcations in
parameter space. We also discuss how to approximate the cuspidal loop using
AUTO as well as how to cross it via continuation by detuning the dynamical
system. Moreover, we show that when there is coupling, the resulting
five-dimensional system--though chaotic--has a similar underlying structure.
  We verify numerically that both homoclinic orbits and cusps occur and provide
an outline of an analytical argument for the existence of such homoclinic
orbits. Small perturbations of the system reveal homoclinic tangles that typify
chaotic behavior.
