<p>This work investigates the nonlinear static and dynamic behavior of a spherical dielectric hyperelastic membrane (SDHM) subjected to combined mechanical pressure and electric voltage. The formulation is based on finite deformations, assuming incompressibility and axisymmetric kinematics. A single physical material is considered, but described by three constitutive models–neo-Hookean (NH), Mooney–Rivlin (MR), and Ogden (O), to assess the influence of the constitutive law on the predicted response. The static analysis reveals a limit-point instability for all models. The considered NH and MR models provide nearly identical equilibrium paths, whereas the O model predicts qualitatively different behavior, including bistability and the emergence of a second potential well. The applied voltage reduces both the limit pressure and the natural frequencies. In the dynamic regime, for the material parameters investigated, the MR model exhibits predominantly softening behavior, while O model presents a richer response, including softening–hardening transitions and inter-well dynamics. Under harmonic excitation, the system displays hysteresis, multi stability, and dynamic jumps, with significantly more complex behavior for the Ogden model. The results demonstrate that, although the same material is considered, the predicted nonlinear response is highly sensitive to the constitutive description. In particular, advanced models such as the Ogden formulation are essential to capture key phenomena that are not predicted by simpler models.</p>