Bifurcations in one degreeofvibration quantum billiards We classify the local bifurcations of one dov quantum billiards, showing that only saddlecenter 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 doublewell 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 fivedimensional systemthough chaotichas a similar underlying structure.  :101,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.

Analytical approximation of cuspidal loops using a nonlinear time transformation method Abstract   In this work we consider cuspidal loops, i.e., homoclinic orbits to cuspidal singular points. We develop an iterative procedure, founded on the nonlinear time transformation method, to estimate such codimensionthree global bifurcations up to any wanted order, not only in the space of parameters but also in the phase plane. As far as we know, this is the first time in the literature that this theoretical result is achieved for these global connections. The existence and uniqueness of the perturbed solution obtained are proved. To illustrate the effectiveness of the method we study cuspidal loops in two normal forms of degenerate Takensxe2x80x93Bogdanov bifurcations. Excellent agreement is found between our analytical predictions and the corresponding numerical continuations.