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Classical periodic orbits responsible for emergence of the superdeformedshell structures for single-particle motions in spheroidal cavities areidentified and their relative contributions to the shell structures areevaluated. Both prolate and oblate superdeformations (axis ratio about 2:1) aswell as prolate hyperdeformation (axis ratio about 3:1) are investigated.Fourier transforms of quantum spectra clearly show that three-dimensionalperiodic orbits born out of bifurcations of planar orbits in the equatorialplane become predominant at large prolate deformations, while butterfly-shapedplanar orbits bifurcated from linear orbits along the minor axis are importantat large oblate deformations.Comment: 18 pages including 21 PS figures, REVTeX with epsf.st

Semiclassical Origin of Superdeformed Shell Structure in the Spheroidal Cavity Model