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Dynamical cavitation for an incompressible hyper-elastic sphere subjected to periodic radial tensile boundary loading is examined within the framework of finite elasto-dynamics.The displacement response curves, the power spectrum curves, the phase portrait and the Poincare maps are given by numerical computation.It is shown that there exists a critical value for the mean load of the periodic load.When the mean load is less than this critical value, the sphere remains undeformed while stressed.However, when the mean load is larger than this critical value, a spherical cavity will suddenly be formed at the center of the sphere.It is proved that the evolution of the cavity radius with time follows that of nonlinear quasi-periodic oscillations.