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In this paper we investigate the large-amplitude axisymmetric free vibrations of an incompressible nonlinear elastic cylindrical structure. The material behavior is described as orthotropic and hyperelastic using the physically-based invariants proposed by Rubin and Jabareen (2007, 2010). The cylinder is modeled using the theory of a generalized Cosserat membrane, which allows for finite deformations that include displacements along the longitudinal axis of the structure. The bi-dimensional approach represents a significant contribution with respect to most works published in this field, which approach the problem at hand assuming plane strain conditions along the axis of the cylinder. We have carried out a systematic analysis of the parameters that govern the dynamic behavior of the structure, paying specific attention to those describing the orthotropy of the material and the dimensions of the cylinder. Using Poincare maps, we have shown that the motion of the structure can turn from periodic to quasi-periodic and chaotic as a function of the initial conditions, the elastic and kinetic energy supplied to the specimen, the dimensions of the cylinder and the degree of mechanical orthotropy of the material. (C) 2017 Elsevier Ltd. All rights reserved.