Interaction of chemical patterns in coupled layers

Míguez, David G.; Dolnik, Miloš; Epstein, Irving R.; Muñuzuri, Alberto P.

doi:10.1103/physreve.84.046210

Cited by 20 publications

(18 citation statements)

References 19 publications

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“…[16][17][18][19][20][21] In this paper, we investigate the complex pattern formation processes by using the dimensionless two-layer coupled Brusselator model, described as follows:…”

Section: Model and Methodsmentioning

confidence: 99%

“…In recent years, the complex patterns have attracted much attention, and a rich variety of complex patterns have been obtained in experiment research, including the white-eye and black-eye pattern, the honeycomb hexagon pattern, the vibrating hexagonal pattern, and the grid state pattern. [2][3][4][5][6][7][8][9] Since Banrrio et al [10] first proposed that the superlattice pattern may originate from the interaction between two modes in 1999, research on the complex patterns has become a challenging topic by the two-layer coupled model, and scientists have obtained plentiful achievements for the past decades. For example, Yang et al [11] obtained the white-eye and black-eye hexagonal patterns with a two-layer coupled Brusselator model at the same time, and proved the Zhou et al guess [12] about the black-eye hexagonal pattern.…”

Section: Introductionmentioning

confidence: 99%

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A method for estimating eigenfunctions of integral operators of certain classes in unbounded domains

Kaplitskii¹

2011

Izv. Math.

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The resonance interaction between two modes is investigated using a two-layer coupled Brusselator model. When two different wavelength modes satisfy resonance conditions, new modes will appear, and a variety of superlattice patterns can be obtained in a short wavelength mode subsystem. We find that even though the wavenumbers of two Turing modes are fixed, the parameter changes have influences on wave intensity and pattern selection. When a hexagon pattern occurs in the short wavelength mode layer and a stripe pattern appears in the long wavelength mode layer, the Hopf instability may happen in a nonlinearly coupled model, and twinkling-eye hexagon and travelling hexagon patterns will be obtained. The symmetries of patterns resulting from the coupled modes may be different from those of their parents, such as the cluster hexagon pattern and square pattern. With the increase of perturbation and coupling intensity, the nonlinear system will convert between a static pattern and a dynamic pattern when the Turing instability and Hopf instability happen in the nonlinear system. Besides the wavenumber ratio and intensity ratio of the two different wavelength Turing modes, perturbation and coupling intensity play an important role in the pattern formation and selection. According to the simulation results, we find that two modes with different symmetries can also be in the spatial resonance under certain conditions, and complex patterns appear in the two-layer coupled reaction diffusion systems.

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“…[16][17][18][19][20][21] In this paper, we investigate the complex pattern formation processes by using the dimensionless two-layer coupled Brusselator model, described as follows:…”

Section: Model and Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A method for estimating eigenfunctions of integral operators of certain classes in unbounded domains

Kaplitskii¹

2011

Izv. Math.

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“…[26][27][28][29] Most of the experimental patterns are essentially 2D in nature and in cases where patterns are 3D, elucidation of structures from 2D projections can be challenging. The reaction does not lend itself to the techniques that have been exploited with the BZ reaction such as optical tomography and MRI.…”

Section: Discussionmentioning

confidence: 99%

Helical Turing patterns in the Lengyel-Epstein model in thin cylindrical layers

Bánsági

Taylor

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The formation of Turing patterns was investigated in thin cylindrical layers using the Lengyel-Epstein model of the chlorine dioxide-iodine-malonic acid reaction. The influence of the width of the layer W and the diameter D of the inner cylinder on the pattern with intrinsic wavelength l were determined in simulations with initial random noise perturbations to the uniform state for W < l/2 and D $ l or lower. We show that the geometric constraints of the reaction domain may result in the formation of helical Turing patterns with parameters that give stripes (b ¼ 0.2) or spots (b ¼ 0.37) in two dimensions. For b ¼ 0.2, the helices were composed of lamellae and defects were likely as the diameter of the cylinder increased. With b ¼ 0.37, the helices consisted of semi-cylinders and the orientation of stripes on the outer surface (and hence winding number) increased with increasing diameter until a new stripe appeared.

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“…For each scenario, we state generic conditions on the traces and determinants τ 1,2 (q) and 1,2 (q) in Eq. (8). In practice, we enforce these conditions by controlling the global extrema of τ 1,2 (q) and 1,2 (q); see Sec.…”

Section: Primary Bifurcationsmentioning

confidence: 99%

“…This approach with the chlorine dioxide-iodinemalonic acid (CDIMA) reaction has produced superlattice patterns called black-eyed and white-eyed patterns, which involve wavelength ratios of nearly 2:1; other ratios were not feasible for this reaction and experimental configuration [7]. Recent experiments have exploited the photosensitivity of the CDIMA chemical reaction, using an external light source to probe the interaction between different forced patterns [8]. For a broad overview of experimental and numerical results for some multilayer systems, see Ref.…”

Section: Introductionmentioning

confidence: 99%

Instabilities and patterns in coupled reaction-diffusion layers

2012

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We study instabilities and pattern formation in reaction-diffusion layers that are diffusively coupled. For two-layer systems of identical two-component reactions, we analyze the stability of homogeneous steady states by exploiting the block symmetric structure of the linear problem. There are eight possible primary bifurcation scenarios, including a Turing-Turing bifurcation that involves two disparate length scales whose ratio may be tuned via the interlayer coupling. For systems of n-component layers and nonidentical layers, the linear problem's block form allows approximate decomposition into lower-dimensional linear problems if the coupling is sufficiently weak. As an example, we apply these results to a two-layer Brusselator system. The competing length scales engineered within the linear problem are readily apparent in numerical simulations of the full system. Selecting a √ 2:1 length-scale ratio produces an unusual steady square pattern.

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Interaction of chemical patterns in coupled layers

Cited by 20 publications

References 19 publications

A method for estimating eigenfunctions of integral operators of certain classes in unbounded domains

A method for estimating eigenfunctions of integral operators of certain classes in unbounded domains

Helical Turing patterns in the Lengyel-Epstein model in thin cylindrical layers

Instabilities and patterns in coupled reaction-diffusion layers

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