Phase fronts and synchronization patterns in forced oscillatory systems

This is a review of recent studies of extended oscillatory systems that are subjected to periodic temporal forcing. The periodic forcing breaks the continuous time translation symmetry and leaves a discrete set of stable uniform phase states. The multiplicity of phase states allows for front structures that shift the oscillation phase by π/n where n = 1, 2, …, hereafter π/n-fronts. The main concern here is with front instabilities and their implications on pattern formation. Most theoretical studies have focused on the 2 : 1 resonance where the system oscillates at half the driving frequency. All front solutions in this case are π-fronts. At high forcing strengths only stationary fronts exist. Upon decreasing the forcing strength the stationary fronts lose stability to pairs of counter-propagating fronts. The coexistence of counter-propagating fronts allows for traveling domains and spiral waves. In the 4 : 1 resonance stationary π-fronts coexist with π/2-fronts. At high forcing strengths the stationary π-fronts are stable and standing two-phase waves, consisting of successive oscillatory domains whose phases differ by π, prevail. Upon decreasing the forcing strength the stationary π-fronts lose stability and decompose into pairs of propagating π/2-fronts. The instability designates a transition from standing two-phase waves to traveling four-phase waves. Analogous decomposition instabilities have been found numerically in higher 2n : 1 resonances. The available theory is used to account for a few experimental observations made on the photosensitive Belousov–Zhabotinsky reaction subjected to periodic illumination. Observations not accounted for by the theory are pointed out.