We investigate pattern formation in self-oscillating systems forced by anexternal periodic perturbation. Experimental observations and numerical studiesof reaction-diffusion systems and an analysis of an amplitude equation arepresented. The oscillations in each of these systems entrain to rationalmultiples of the perturbation frequency for certain values of the forcingfrequency and amplitude. We focus on the subharmonic resonant case where thesystem locks at one fourth the driving frequency, and four-phase rotatingspiral patterns are observed at low forcing amplitudes. The spiral patterns arestudied using an amplitude equation for periodically forced oscillatingsystems. The analysis predicts a bifurcation (with increasing forcing) fromrotating four-phase spirals to standing two-phase patterns. This bifurcation isalso found in periodically forced reaction-diffusion equations, theFitzHugh-Nagumo and Brusselator models, even far from the onset of oscillationswhere the amplitude equation analysis is not strictly valid. In aBelousov-Zhabotinsky chemical system periodically forced with light we alsoobserve four-phase rotating spiral wave patterns. However, we have not observedthe transition to standing two-phase patterns, possibly because with increasinglight intensity the reaction kinetics become excitable rather than oscillatory.
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