Arnold tongues and subharmonics in the forced oscillations of a mechanical clock

We consider a simple model of a mechanical clock and its response to periodic disturbances. The model consists of a damped linear oscillator subjected to impulses whenever the system achieves a prescribed state (position and velocity). The unforced system possesses a limit cycle which, due to the piecewise linear nature of the model, is known in an explicit form. The response of this system to periodic excitation depends in a delicate manner on the ratio of the limit cycle period to the forcing period. Arnold tongues emerge, as the driving amplitude is increased form zero, from points where this ratio is a rational number and correspond to the appearance of subharmonic motions. We present explicit formulae for some of these bifurcation curves and for some secondary bifurcation curves where the subharmonics lose their stability, these bifurcations are either period doubling or Hopf bifurcations.