We investigate numerically and analytically the dynamics of a rigid rotator under the action of kicks with slowly varying strength and period. We derive a discrete map for this model and use the map to study three effects. The first of them is the dynamic autoresonance, which can lead to a significant regular acceleration or deceleration of the rotator. We find conditions for which the effect occurs, including the condition for an unlimited acceleration. The second effect is the transition to global chaos that arises due to the increase of the stochasticity parameter with time. We find that this transition occurs through bifurcation of the main island (the bifurcation parameter being simply time) and the development of a complicated separatrix structure. Also, slowly evolving global chaos is studied numerically, and a time-dependent analog of the quasilinear diffusion equation is shown to describe quite accurately the numerical simulations.