Phase fronts and synchronization patterns in forced oscillatory systems

Abstract

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 2 n : 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.