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In this paper, the stability of a cylinder with circular and elliptical section made of compressible and incompressible nonlinear elastic materials under finite perturbations is considered. A set of computational experiments was performed. The permissible boundaries of the region with respect to final and initial perturbations for given parameters of loading and structures are established. Finite sequences of bifurcation points are constructed, confirming, in contrast to stability at small perturbations, the existence of a hierarchy of stable equilibrium states. Classical linearized stability theories are evaluated. New phenomena and characteristic effects are established.