Uniform Approximation for Period-Quadrupling Bifurcations

We derive a uniform approximation for semiclassical contributions of periodic orbits to the spectral density which is valid for generic period-quadrupling bifurcations in systems with a mixed phase space. These bifurcations involve three periodic orbits which coalesce at the bifurcation. In the vicinity of the bifurcation the three orbits give a collective contribution to the spectral density while the individual contributions of Gutzwiller's type would diverge at the bifurcation. The uniform approximation is obtained by mapping the action function onto the normal form corresponding to the bifurcation. The present article is a continuation of previous work in which uniform approximations for generic period-$m$-tupling bifurcations with $m \neq 4$ were derived.