RESUMO
Oscillatory cluster patterns are studied numerically in a reaction-diffusion model of the photosensitive Belousov-Zhabotinsky reaction supplemented with a global negative feedback. In one- and two-dimensional systems, families of cluster patterns arise for intermediate values of the feedback strength. These patterns consist of spatial domains of phase-shifted oscillations. The phase of the oscillations is nearly constant for all points within a domain. Two-phase clusters display antiphase oscillations; three-phase clusters contain three sets of domains with a phase shift equal to one-third of the period of the local oscillation. Border (nodal) lines between domains for two-phase clusters become stationary after a transient period, while borders drift in the case of three-phase clusters. We study the evolving border movement of the clusters, which, in most cases, leads to phase balance, i.e., equal areas of the different phase domains. Border curling of three-phase clusters results in formation of spiral clusters-a combination of a fast oscillating cluster with a slow spiraling movement of the domain border. At higher feedback coefficient, irregular cluster patterns arise, consisting of domains that change their shape and position in an irregular manner. Localized irregular and regular clusters arise for parameters close to the boundary between the oscillatory region and the reduced steady state region of the phase space.
RESUMO
Oscillatory clusters are sets of domains in which nearly all elements in a given domain oscillate with the same amplitude and phase. They play an important role in understanding coupled neuron systems. In the simplest case, a system consists of two clusters that oscillate in antiphase and can each occupy multiple fixed spatial domains. Examples of cluster behaviour in extended chemical systems are rare, but have been shown to resemble standing waves, except that they lack a characteristic wavelength. Here we report the observation of so-called 'localized clusters'--periodic antiphase oscillations in one part of the medium, while the remainder appears uniform--in the Belousov-Zhabotinsky reaction-diffusion system with photochemical global feedback. We also observe standing clusters with fixed spatial domains that oscillate periodically in time and occupy the entire medium, and irregular clusters with no periodicity in either space or time, with standing clusters transforming into irregular clusters and then into localized clusters as the strength of the global negative feedback is gradually increased. By incorporating the effects of global feedback into a model of the reaction, we are able to simulate successfully the experimental data.
