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Hyperelastic membranes are found in many engineering fields. A key step in their mathematical modelling is the choice of an appropriate constitutive law and, subsequently, the determination of the associated material parameters, usually obtained from experimental results, with the occurrence of multiple sets of optimal material parameters for the same data sets, depending on the fitting process. The influence of the constitutive law of a hyperelastic material on its nonlinear behavior is well known and both static and dynamic responses can vary greatly, depending on the parameter values. Thus, the application of the uncertainty propagation analysis, capturing the expected outcome in a probabilistic sense, constitutes an interesting tool in the analysis of hyperelastic structures, particularly in the dynamic case. Here it is applied to the analysis of a pressure-loaded spherical hyperelastic membrane with constitutive uncertainty. Different hyperelastic constitutive models are addressed, comparing the static and dynamic nonlinear response under parametric uncertainty. Finally, the global stability is developed by expanding a previous concept of (Q, Λ)-attractors to the dual space where basins are defined. Specifically, the inclusion of parametric uncertainty is responsible for the diffusion of attractors and basins boundaries, drastically changing the phase-space topology. A novel methodology of phase-space discretization is also considered to represent the phase-space topology under uncertainty. A global dynamical analysis framework in the context of parametric uncertainty is yet to be addressed, specifically for hyperelastic constitutive models …