Oscillatory clusters in a model of the photosensitive Belousov-Zhabotinsky reaction system with global feedback

Abstract

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.

A Canard Mechanism for Localization in Systems of Globally Coupled Oscillators

“…Simulations performed on the Oregonator model [5] and a modified Oregonator model [6] of the BZ reaction, both with global inhibitor feedback, reproduce the experimental findings. However, the mechanism by which localized cluster formation occurs remains unclear from both the mathematical and chemical points of view.…”

“…The MFHN model is a simplification of the modified version of the Oregonator model used in [6]; it allows an easier qualitative dynamical understanding by reproducing important aspects of the BZ dynamics and keeping some of its features, including the "N" shape of the nullcline corresponding to the first equation in (1), its asymptotic approach to the w axis, its qualitative behavior as a function of the global feedback parameter, and an inhibitor dynamics described by a sigmoid function rather than a line. The motivation for using the MFHN system instead of more classical versions of the FHN system is that, by changing the global feedback parameter γ, we can find small amplitude limit cycles with smaller amplitude in the v direction than for the FHN equations.…”

“…In the FHN-type models with global feedback presented here, as well as in the BZ model with global feedback used in [6], the intersection point between nullclines remains fixed as γ is increased. When γ = 0 (no global feedback) the uncoupled oscillators are in an LAO regime; localization for these models is a consequence of the global coupling.…”

“…The localized cluster patterns found in the experiments and simulations on the BZ reaction with global inhibitory feedback [4,5,6] present two main features that might seem counterintuitive:…”

“…Although they are not precise as descriptions of chemical phenomena, they display the localization phenomenon and are easier to study analytically in order to give some insight into the dynamical mechanisms that produce localized solutions. In addition, they display some of the relevant qualitative features of the modified Oregonator model studied in [6], such as the shape of the nullclines, the fact that the limit cycle is created in a supercritical Hopf bifurcation, and the relaxation nature of the oscillator. In a forthcoming paper we will address the questions related to the mechanism of localization in the modified version of the Oregonator used in [6,7].…”