Four-phase patterns in forced oscillatory systems

We investigate pattern formation in self-oscillating systems forced by an e xternal periodic perturbation. Experimental observations and numerical stud ies of reaction-diffusion systems and an analysis of an amplitude equation are presented. The oscillations in each of these systems entrain to rationa l multiples of the perturbation frequency for certain values of the forcing frequency and amplitude. We focus on the subharmonic resonant case where t he system locks at one-fourth the driving frequency, and four-phase rotatin g spiral patterns are observed at low forcing amplitudes. The spiral patter ns are studied using an amplitude equation for periodically forced oscillat ing systems. The analysis predicts a bifurcation (with increasing forcing) from rotating four-phase spirals to standing two-phase patterns. This bifur cation is also found in periodically forced reaction-diffusion equations, t he FitzHugh-Nagumo and Brusselator models, even far from the onset of oscil lations where the amplitude equation analysis is not strictly valid. In a B elousov-Zhabotinsky chemical system periodically forced with light we also observe four-phase rotating spiral wave patterns. However, we have not obse rved the transition to standing two-phase patterns, possibly because with i ncreasing light intensity the reaction kinetics become excitable rather tha n oscillatory.