Bifurcations in one degree-of-vibration quantum billiards

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