Symmetric kicked self-oscillators: iterated maps, strange attractors, and symmetry of the phase locking Farey hierarchy

The stroboscopic map of symmetric self-oscillators driven by pulses of alternating sign is constructed, for the regime of strong relaxation, by means of the so-called phase-transition curve. An ordering for the symmetries of the phase-locked orbits is found which is conjectured to be universal. Both results are illustrated by an exactly solvable example of such a system. When the approximation fails, unusual period doubling from symmetric orbits and strange attractors of two dimensional structure are encountered.