Phase fronts and synchronization patterns in forced oscillatory systems (Q5929122)

Summary: 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 \(\pi/n\) where \(n= 1,2,\dots\), hereafter \(\pi/n\)-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 \(\pi\)-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 \(\pi\)-fronts coexist with \(\pi/2\)-fronts. At high forcing strengths the stationary \(\pi\)-fronts are stable and standing two-phase waves, consisting of successive oscillatory domains whose phases differ by \(\pi\), prevail. Upon decreasing the forcing strength the stationary \(\pi\)-fronts lose stability and ecompose into pairs of propagating \(\pi/2\)-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 experiment observations made on the photosensitive Belousov-Zhabotinsky reaction subjected to periodic illumination. Observations not accounted for by the theory are pointed out.