We present the results of an investigation of a novel dynamical system in which one, two or three solid spheres are free to move in a horizontal rotating cylinder which is completely filled with a highly viscous fluid. At low rotation rates, steady motion is found where the balls adopt stable equilibrium positions rotating adjacent to the rising wall at a speed which is in agreement with available theory. At higher cylinder speeds time-dependent motion sets in via Hopf bifurcations, and when one or two balls are present the motion is strictly periodic. Perhaps surprisingly, it is found that three balls are required to produce low-dimensional chaos.