ABSTRACT
Natural systems are often subjected to external periodic forcing. The entrainment of cardiac cells in the atrioventricular node to signals generated at the sinoatrial node, circadian rhythms entrained by the 24h day-night periodicity, and phytoplankton oscillations under seasonal forcing are examples of temporal periodic forcing of oscillatory systems [106, 110, 137]. There also exists a spatial counterpart for this class of systems, namely, periodic spatial structures, such as crystals or self-organized patterns, that are subjected to periodic spatial modulation. The modulation can be induced by an intrinsic instability, e.g., charge density waves in a 1d crystal [306, 10], or externally applied. Examples of the latter case include periodic spatial modulations of an electric field in nematic liquid-crystal convection [182, 127], periodic temperature modulation of the lower boundary of a Rayleigh-Be´nard system [279, 96], and periodic illumination of a photo-sensitive chemical reaction [69, 68, 217]. An ecological example, discussed earlier in Section 3.3.3, is the restoration of degraded vegetation bands by means of parallel embankments. The ability of periodically forced systems to yield to the forcing in a synchronized manner is the subject of this Section. Yielding to temporal forcing by frequency
locking will be analyzed in section 8.1.1, while yielding to spatial forcing by wavenumber locking will be analyzed in Section 8.1.2.
