Uniform approximations for non-generic bifurcation scenatios including bifurcations of ghost orbits

Details Uniform approximations for non-generic bifurcation scenatios including bifurcations of ghost orbits Uniform approximations for non-generic bifurcation scenatios including bifurcations of ghost orbits

1999-08-18

arxiv.org/abs/chao-dyn/9908019v1

Annals of Physics 277 (1999) 19-73

Nonlinear Sciences

Chaotic Dynamics

56 pages, 21 figure, LaTeX2e using amsmath, amssymb, epsfig, and rotating packages. To be published in Annals of Physics

Scientific paper

10.1006/aphy.1999.5961

Gutzwiller's trace formula allows interpreting the density of states of a classically chaotic quantum system in terms of classical periodic orbits. It diverges when periodic orbits undergo bifurcations, and must be replaced with a uniform approximation in the vicinity of the bifurcations. As a characteristic feature, these approximations require the inclusion of complex ``ghost orbits''. By studying an example taken from the Diamagnetic Kepler Problem, viz. the period-quadrupling of the balloon-orbit, we demonstrate that these ghost orbits themselves can undergo bifurcations, giving rise to non-generic complicated bifurcation scenarios. We extend classical normal form theory so as to yield analytic descriptions of both bifurcations of real orbits and ghost orbit bifurcations. We then show how the normal form serves to obtain a uniform approximation taking the ghost orbit bifurcation into account. We find that the ghost bifurcation produces signatures in the semiclassical spectrum in much the same way as a bifurcation of real orbits does.

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