When a thin film of active, nematic microtubules and kinesin motor clusters is confined on the surface of a vesicle, four +1/2 topological defects oscillate in a periodic manner between tetrahedral and planar arrangements. Here a theoretical description of nematics, coupled to the relevant hydrodynamic equations, is presented here to explain the dynamics of active nematic shells. In extensile microtubule systems, the defects repel each other due to elasticity, and their collective motion leads to closed trajectories along the edges of a cube. That motion is accompanied by oscillations of their velocities, and the emergence and annihilation of vortices. When the activity increases, the system enters a chaotic regime. In contrast, for contractile systems, which are representative of some bacterial suspensions, a hitherto unknown static structure is predicted, where pairs of defects attract each other and flows arise spontaneously.