<p>This research offers a novel mechanical system made up of a cylinder-shaped shell with a unique structure, a hybrid nanocomposite sandwich cylindrical shell with an auxetic nanocomposite origami core. A hybrid sandwich shell exposed to preloading in the axial direction and an external excitation force. Auxetic origami shell dynamics are investigated with the Galerkin’s approach and the Runge–Kutta technique; the equations are the result of Donnell’s modified shell theory. The chaotic and nonlinear dynamic response of the hybrid auxetic sandwich shells was observed. Numerous parametric examples are provided and explored to further illustrate the effect of these factors on the three-layer spherical shell’s oscillation. The suggested approach is compared to previous efforts in the literature to ensure the correctness and precision of the numerical findings. This article aims to further investigate the performance and behavior of these shells reinforced with auxetic cores, considering the effect of porosity by examining vibrations, nonlinear dynamic analysis, and the effect of the chaos phenomenon, so this approach will enable us to utilize these enhanced structures in practical and industrial applications while improving these characteristics.</p>