By varying the forcing frequency and amplitude of a periodically forced planar oscillator, the author obtains a rich variety of responses. Whenever the resonance regions that are known to exist for small amplitudes of forcing terminate, he shows that a fixed-point Hopf bifurcation must be involved. The main tool, whose properties he discusses in detail, is a self-rotation number for orbits in the plane. He illustrates his theorems with a numerical model.