When a closed spherical shell is suddenly enveloped by a uniform impulse, the response is a purely radial motion of each shell element. If the impulse is slightly nonuniform, this membrane “breathing mode” may be dynamically unstable due to the interaction of the membrane stress with flexural curvature. The linear solution for axisymmetric motion divides into two sets of modes. One set, the membrane modes, is essentially extensional; the second set, the composite modes, is neither purely extensional nor purely inextensional. The linear response to an arbitrary radial velocity distribution indicates that breathing mode stability may be considered with respect to perturbations exciting only composite modes. Using this result the nonlinear study shows that a cyclic energy exchange may occur between the breathing mode and rather high‐order composite modes. The ensuing deflections and stresses may be considerably in excess of those determined by consideration of the breathing mode only.