The dynamical behaviors are studied for a sphere with a micro-void at the center under periodic perturbation loads, where the sphere is composed of a class of radially transversely isotropic incompressible neo-Hookean materials. A strongly nonlinear nonautonomous ordinary differential equation describing the radially symmetric motion of the micro-void is derived in terms of the equilibrium differential equation and initial-boundary conditions. Through qualitatively analyzing the solutions of the differential equation, some interesting qualitative behaviors of the micro-void are discussed. (1) For constant loads, the effects of material parameters and structural parameters on equilibrium points of the system are discussed, and the bifurcation behaviors, especially the secondary turning bifurcation of the micro-void are analyzed. By analyzing the well potentials, the phenomena of period and amplitude jump of the micro-void are discussed. (2) For periodic perturbation loads, the quasiperiodic and chaotic motions of the micro-void are discussed in terms of the secondary turning bifurcation by using the time response curves, Poincaré sections and the maximal Lyapunov characteristic exponents, the existence conditions of chaos are given, and the effects of periodic perturbation loads on the chaotic motions of the micro-void are further analyzed.