Pattern Formation in the Belousov−Zhabotinsky Reaction with Photochemical Global Feedback

Vanag, Vladimir K.; Zhabotinsky, and Anatol M.; Epstein, Irving R.

doi:10.1021/jp002390h

Cited by 72 publications

(80 citation statements)

References 52 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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.…”

supporting

confidence: 55%

“…In addition to localization, these models display the basic features of the modified Oregonator models studied in [6] and [5] to reproduce the experimental results: shape of the nullclines, a limit cycle created in a supercritical Hopf bifurcation, and display of canards among others.…”

Section: Discussionmentioning

confidence: 96%

“…In particular, the existence of localized oscillatory clusters has been reported in [4,5] for the BZ reaction with global inhibitory feedback. 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.…”

mentioning

confidence: 92%

“…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:…”

mentioning

confidence: 97%

See 3 more Smart Citations

A Canard Mechanism for Localization in Systems of Globally Coupled Oscillators

Rotstein¹,

Epstein²,

Kopell³

2003

SIAM J. Appl. Math.

Self Cite

View full text Add to dashboard Cite

Abstract. Localization in a discrete system of oscillators refers to the partition of the population into a subset that oscillates at high amplitudes and another that oscillates at much lower amplitudes. Motivated by experimental results on the Belousov-Zhabotinsky reaction, which oscillates in the relaxation regime, we study a mechanism of localization in a discrete system of relaxation oscillators globally coupled via inhibition. The mechanism is based on the canard phenomenon for a single relaxation oscillator: a rapid explosion in the amplitude of the limit cycle as a parameter governing the relative position of the nullclines is varied. Starting from a parameter regime in which each uncoupled oscillator has a large amplitude and no other periodic or other stable solutions, we show that the canard phenomenon can be induced by increasing a global negative feedback parameter γ, with the network then partitioned into low and high amplitude oscillators. For the case in which the oscillators are synchronous within each of the two such populations, we can assign a canard-inducing critical value of γ separately to each of the two clusters; localization occurs when the value for the system is between the critical values of the two clusters. We show that the larger the cluster size, the smaller is the corresponding critical value of γ, implying that it is the smaller cluster that oscillates at large amplitude. The theory shows that the above results come from a kind of self-inhibition of each cluster induced by the local feedback. In the full system, there are also effects of interactions between the clusters, and we present simulations showing that these nonlocal interactions do not destroy the localization created by the self-inhibition.

show abstract

supporting

confidence: 55%

Section: Discussionmentioning

confidence: 96%

mentioning

confidence: 92%

mentioning

confidence: 97%

See 2 more Smart Citations

A Canard Mechanism for Localization in Systems of Globally Coupled Oscillators

Rotstein¹,

Epstein²,

Kopell³

2003

SIAM J. Appl. Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Many studies have been undertaken of feedback control for CSTRs, where the flow rate is altered in response to the concentrations in the reactor; (4) represents an analogous form of feedback control for a reaction-diffusion cell, see [3,24,25]. Experimentally this can be achieved using a diffusive gel coupled to a CSTR, which represents the reservoir; [13] considered a BZ reaction while [14,15] considered a glycolytic reaction.…”

Section: Model Equationsmentioning

confidence: 99%

Non-smooth feedback control for Belousov–Zhabotinskii reaction–diffusion equations: semi-analytical solutions

2016

View full text Add to dashboard Cite

The Belousov-Zhabotinskii reaction is considered in one and two-dimensional reaction-diffusion cells. Feedback control is examined where the feedback mechanism involves varying the concentrations in the boundary reservoir, in response to the concentrations in the centre of the cell. Semi-analytical solutions are developed, via the Galerkin method, which assumes a spatial structure for the solution, and is used to approximate the governing delay partial differential equations by a system of delay ordinary differential equations. The form of feedback control considered, whilst physically realistic, is non-smooth as it has discontinuous derivatives. A stability analysis of the sets of smooth delay ordinary differential equations, which make up the full non-smooth system, allows a band of Hopf bifurcation parameter space to be obtained. It is found that Hopf bifurcations for the full non-smooth system fall within this band of parameter space. In the case of feedback with no delay a precise semi-analytical estimate for the stability of the full nonsmooth system can be obtained, which corresponds well with numerical estimates. Examples of limit cycles and the transient evolution of solutions are also considered in detail. Abstract The Belousov-Zhabotinskii reaction is considered in one and two-dimensional reactiondiffusion cells. Feedback control is examined where the feedback mechanism involves varying the concentrations in the boundary reservoir, in response to the concentrations in the centre of the cell. Semi-analytical solutions are developed, via the Galerkin method, which assumes a spatial structure for the solution, and is used to approximate the governing delay partial differential equations by a system of delay ordinary differential equations. The form of feedback control considered, whilst physically realistic, is non-smooth as it has discontinuous derivatives. A stability analysis of the sets of smooth delay ordinary differential equations, which make up the full non-smooth system, allows a band of Hopf bifurcation parameter space to be obtained. It is found that Hopf bifurcations for the full non-smooth system fall within this band of parameter space. In the case of feedback with no delay a precise semi-analytical estimate for the stability of the full non-smooth system can be obtained, which corresponds well with numerical estimates. Examples of limit cycles and the transient evolution of solutions are also considered in detail.

show abstract

Recognition of Spatial Finiteness in Meniscus Splitting Based on Evaporative Interface Fluctuations

Wu,

Saito,

Hongo

et al. 2023

Adv Materials Inter

View full text Add to dashboard Cite

The geometric deformation of viscous fingering is useful for understanding natural multiscale patterns and designing dissipative structures in materials. Although the spatio‐temporal patterns in soft materials are reported previously, there is a lack of research on the spatial finiteness and boundary effects. In this study, the recognition of spatial finiteness in “meniscus splitting phenomena” in aqueous polymer dispersions during water evaporation is demonstrated. By providing heat energy to polymer dispersions in a Hele‐Shaw cell, an interface fluctuation with concentration unevenness is induced to split the evaporative interface. The spatial finiteness of the interface causes asynchronous nucleation, which is demonstrated using polysaccharide dispersions. The results of the quasi‐natural experiments revealed that the nonequilibrium drying/wetting period for repositioning polymer clusters allows for considerable changes in Reynolds number in a low range (<10−6) to form multiple nuclei. This splitting method will be universally useful in various fields, including fluid dynamics, biology, and microfluidics, as well as non‐equilibrium, colloid, interface, polymer, and materials sciences.

show abstract

Pattern Formation in the Belousov−Zhabotinsky Reaction with Photochemical Global Feedback

Cited by 72 publications

References 52 publications

A Canard Mechanism for Localization in Systems of Globally Coupled Oscillators

A Canard Mechanism for Localization in Systems of Globally Coupled Oscillators

Non-smooth feedback control for Belousov–Zhabotinskii reaction–diffusion equations: semi-analytical solutions

Recognition of Spatial Finiteness in Meniscus Splitting Based on Evaporative Interface Fluctuations

Contact Info

Product

Resources

About