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A reflection-asymmetric deformed oscillator potential is analysed from theclassical and quantum mechanical point of view. The connection betweenoccurrence of shell structures and classical periodic orbits is studied usingthe ''removal of resonances method'' in a classical analysis. In thisapproximation, the effective single particle potential becomes separable andthe frequencies of the classical trajectories are easily determined. It turnsout that the winding numbers calculated in this way are in good agreement withthe ones found from the corresponding quantum mechanical spectrum using theparticle number dependence of the fluctuating part of the total energy. Whenthe octupole term is switched on it is found that prolate shapes are stableagainst chaos whereas spherical and oblate cases become chaotic. An attempt ismade to explain this difference in the quantum mechanical context by looking atthe distribution of exceptional points which results from the matrix structureof the respective Hamiltonians. In a similar way we analyse the modifiedNilsson model and discuss its consequences for nuclei and metallic clusters.Comment: to appear in Physica Scripta., CNLS-94-02, a talk given at the Nobel  sponsored conference SELMA 94 "New Nuclear Phenomena in the Vicinity of  Closed Shell" (Stockholm and Uppsala, 29 Aug.- 3 Sept. 1994

Shell structures and chaos in nuclei and large metallic clusters