Pattern selection in the generalized Swift-Hohenberg model

Hilali, M. F.; Métens, Stephane; Borckmans, Pierre; Dewel, Guy

doi:10.1103/physreve.51.2046

Cited by 92 publications

(98 citation statements)

References 31 publications

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“…The bifurcation diagrams shown in Figure 1.1, which have been discussed, for instance, in [29,44,8,3,14,32,33,34,15,16,24,27], exhibit several intriguing features, which are commonly referred to as snaking. There are two intertwined wiggly solution branches that correspond to localized patterns whose width, measured by the L 2 -norm, increases as we move along each branch.…”

Section: Introductionmentioning

confidence: 99%

Snakes, Ladders, and Isolas of Localized Patterns

Beck¹,

Knobloch²,

Lloyd³

et al. 2009

SIAM J. Math. Anal.

162

260

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Abstract. Stable localized roll structures have been observed in many physical problems and model equations, notably in the 1D Swift-Hohenberg equation. Reflection-symmetric localized rolls are often found to lie on two "snaking" solution branches, so that the spatial width of the localized rolls increases when moving along each branch. Recent numerical results by Burke and Knobloch indicate that the two branches are connected by infinitely many "ladder" branches of asymmetric localized rolls. In this paper, these phenomena are investigated analytically. It is shown that both snaking of symmetric pulses and the ladder structure of asymmetric states can be predicted completely from the bifurcation structure of fronts that connect the trivial state to rolls. It is also shown that isolas of asymmetric states may exist, and it is argued that the results presented here apply to 2D stationary states that are localized in one spatial direction.

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Section: Introductionmentioning

confidence: 99%

Snakes, Ladders, and Isolas of Localized Patterns

Beck¹,

Knobloch²,

Lloyd³

et al. 2009

SIAM J. Math. Anal.

162

260

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“…Since this pioneering work, the Swift-Hohenberg ͑SH͒ model equation has been derived for various nonequilibrium systems, for instance, in optics, 4,5 chemical reactions with diffusion, 6 and in biology. 7 As a result, it is one of the most studied nonlinear equations not only in its field of origin, hydrodynamical systems, but in most domains of the natural sciences.…”

Section: Introductionmentioning

confidence: 99%

Nonvariational real Swift-Hohenberg equation for biological, chemical, and optical systems

Kozyreff

Tlidi

2007

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We derive asymptotically an order parameter equation in the limit where nascent bistability and long-wavelength modulation instabilities coalesce. This equation is a variant of the SwiftHohenberg equation that generally contains nonvariational terms of the form ٌ 2 and ٌ͉ ͉ 2 . We briefly review some of the properties already derived for this equation and derive it on three examples taken from chemical, biological, and optical contexts. Near the critical point associated with nascent bistability and close to long-wavelength regime, the dynamics of many natural spatially extended systems can be described by a single real partial differential equation. This approximation is very useful for the study of out of equilibrium dissipative structures that can be either periodic or localized in space. In this contribution, we derive a real order parameter equation that includes nonvariational effects and which is capable of describing a very wide class systems. Examples taken from biology, chemistry, and optics are considered together with a general derivation that shows that the obtained real order parameter equation is universal and has a larger spectrum of space-time dynamical behaviors compared with the usual variational Swift-Hohenberg equation.

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“…In the Brusselator this requires that 0.564 < Aη < 2.418. Outside this range higher-order contributions must be considered [12]. Second, the spatial derivatives must be changed inton 1 …”

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confidence: 99%

Selection and competition of Turing patterns

Peña¹,

Pérez–García²

2000

Europhys. Lett.

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PACS. 47.54.+r -Pattern selection; pattern formation. PACS. 82.40.Bj -Oscillations, chaos, and bifurcations in homogeneous nonequilibrium reactors. PACS. 47.20.Ky -Nonlinearity (including bifurcation theory).Abstract. -We examine the selection and competition of patterns in the Brusselator model, one of the simplest reaction-diffusion systems giving rise to Turing instabilities. Simulations of this model show a significant change in the wave number of stable patterns as the control parameter is increased. A weakly nonlinear analysis makes it possible to obtain the amplitude equations for the concentration fields near the instability threshold. Together with the linear diffusive terms, these equations also contain nonvariational spatial terms. When these terms are included, the stability diagrams and the thresholds for secondary instabilities are heavily modified with respect to the usual diffusive case. The results obtained from the numerical simulations fit very well into the calculated stability regions.Several systems out of equilibrium exhibit pattern formation. The amplitude formalism introduced by Newell, Whitehead and Segel [1] allows spatial modulations of these patterns to be described close to a supercritical bifurcation point. Recently, several authors have discussed the necessity of including nonlinear spatial terms into generalized amplitude equations (GAE) at the leading orders for subcritical bifurcations [2,3]. The role and the weight of these terms is still under discussion. The main aim of this paper is to determine the GAE for some kind of Turing patterns arising in reaction-diffusion systems. These patterns result from a coupling between nonlinear kinetic and diffusion of reactants in which two opposed mechanisms are involved: autocatalysis (activation) and an inhibitor process [4]. In recent years, the interest in Turing patterns has been renewed with the experimental evidences of the CIMA (ChloriteIodide Malonic Acid) and CDIMA reaction (Chlorine Dioxide-Iodine Malonic Acid) [5,6]. These oscillatory reduction-oxidation reactions consist of the oxidation of I 2 by Cl or ClO 2 and the iodination of malonic acid. The difference between the diffusion coefficients of the activator (I − ) and the inhibitor species (Cl − or ClO − 2 ) necessary to have Turing patterns is reached by using the starch as color indicator, because it decreases the diffusivity of the activator [7]. Several kinds of steady pattern have been observed: stripes, hexagons, rhombs, mixed modes, black eyes, etc. [8].Realistic reactive schemes are so complicated that analytical results become unattainable. Therefore, we analyse the simple Brusselator model [9], which exhibits Turing patterns similar c EDP Sciences

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Pattern selection in the generalized Swift-Hohenberg model

Cited by 92 publications

References 31 publications

Snakes, Ladders, and Isolas of Localized Patterns

Snakes, Ladders, and Isolas of Localized Patterns

Nonvariational real Swift-Hohenberg equation for biological, chemical, and optical systems

Selection and competition of Turing patterns

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