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 byπ/nwheren=1,2,…,hereafterπ/n-fronts. The main concern here is with front instabilities and their implications on pattern formation. Most theoretical studies have focused on the2:1resonance where the system oscillates at half the driving frequency. All front solutions in this case areπ-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 the4:1resonance stationaryπ-fronts coexist withπ/2-fronts. At high forcing strengths the stationaryπ-fronts are stable and standing two-phase waves, consisting of successive oscillatory domains whose phases differ byπ,, prevail. Upon decreasing the forcing strength the stationaryπ-fronts lose stability and decompose into pairs of propagatingπ/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 higher2n:1resonances. 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.
Highlights
A conspicuous property of systems driven far from equilibrium is the possible appearance of persistent oscillations [1,2,3,4,5]
In spatially extended reactions it often involves spatial phase variations that lead to traveling wave phenomena. Biological rhythms provide another manifestation of persistent oscillatory dynamics
Numerical studies of amplitude equations for higher resonances suggest the existence of --front decomposition instabilities in 2n resonances with n > 1
Summary
A conspicuous property of systems driven far from equilibrium is the possible appearance of persistent oscillations [1,2,3,4,5]. In spatially extended reactions it often involves spatial phase variations that lead to traveling wave phenomena Biological rhythms provide another manifestation of persistent oscillatory dynamics. Most theoretical studies of periodically forced oscillatory systems have focused on frequency locking phenomena and the onset of chaos in single oscillator models (or circle maps) [4,6,7,8,9,10]. The fractional frequencies a forced system can realize follow the Farey rule: between the tongues col: co n m and cof CO k there exists the (n + k):(m + l) tongue Another property of forced systems is the coexistence of multiple stable phase states, corresponding to uniform oscillations with different fixed phases.
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