Development of Standing-Wave Labyrinthine Patterns

Yochelis, Arik; Hagberg, Aric; Meron, Ehud; Lin, Anna L.; Swinney, Harry L.

doi:10.1137/s1111111101397111

Cited by 55 publications

(59 citation statements)

References 23 publications

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“…Note that Ising fronts destabilize to Bloch fronts as the forcing strength is increased, unlike earlier studies of the NIB bifurcation, where Bloch fronts appear as the forcing strength is decreased. 17 The reverse nature of the spatial NIB bifurcation in the LE model can be understood, at least partly, by deriving an amplitude equation for stripe patterns in the LE model (assuming an infinite system). Approximating a solution of the LE model as in (6), the amplitude A satisfies the equation…”

Section: A Subcritical Front Bifurcation In the Le Modelmentioning

confidence: 99%

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Fronts and patterns in a spatially forced CDIMA reaction

Haim

Hagberg

Nagao

et al. 2014

Phys. Chem. Chem. Phys.

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We use the CDIMA chemical reaction and the Lengyel-Epstein model of this reaction to study resonant responses of a pattern-forming system to time-independent spatial periodic forcing. We focus on the 2 : 1 resonance, where the wavenumber of a one-dimensional periodic forcing is about twice the wavenumber of the natural stripe pattern that the unforced system tends to form. Within this resonance, we study transverse fronts that shift the phase of resonant stripe patterns by p. We identify phase fronts that shift the phase discontinuously, and pairs of phase fronts that shift the phase continuously, clockwise and anti-clockwise. We further identify a front bifurcation that destabilizes the discontinuous front and leads to a pair of continuous fronts. This bifurcation is the spatial counterpart of the nonequilibrium Ising-Bloch (NIB) bifurcation in temporally forced oscillatory systems. The spatial NIB bifurcation that we find occurs as the forcing strength is increased, unlike earlier studies of the NIB bifurcation. Furthermore, the bifurcation is subcritical, implying a range of forcing strength where both discontinuous Ising fronts and continuous Bloch fronts are stable. Finally, we find that both Ising fronts and Bloch fronts can form discrete families of bound pairs, and we relate arrays of these front pairs to extended rectangular and oblique patterns.

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Section: A Subcritical Front Bifurcation In the Le Modelmentioning

confidence: 99%

“…6,10,11,[15][16][17][18][19] There are two main mechanisms by which periodic forcing can induce new patterns. The first is a new pattern-forming instability of the original uniform state.…”

Section: Introductionmentioning

confidence: 99%

Fronts and patterns in a spatially forced CDIMA reaction

Haim

Hagberg

Nagao

et al. 2014

Phys. Chem. Chem. Phys.

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“…There we saw that a reduction of these three equations is possible if the lateral modes have equal amplitude. Here we will analyze the set of coupled amplitude equations (11,12).…”

Section: Transitions Between Stripes and Hexagons At 1:1 Resonancementioning

confidence: 99%

“…Some of the contributions in the study of pattern formation induced by temporal forcing are Refs. [5][6][7][8][9][10][11][12][13][14][15][16][17]. We are not addressing cases in which the forcing is itself the origin of the pattern-forming instability, such as for instance in Faraday waves [1,18].…”

Section: Introductionmentioning

confidence: 99%

Theory of pattern forming systems under traveling-wave forcing

et al. 2007

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The response of pattern forming systems to external forcing either spatial or temporal has received much attention for several decades. Combined spatio-temporal forcing has only been introduced recently, in particular in the form of a spatially resonant traveling-wave forcing [S. Rüdiger, D.G. Míguez, A.P. Muñuzuri, F. Sagués, J. Casademunt, Phys. Rev. Lett. 90 (2003) 128301]. Since then, both a series of experiments, in the context of Turing patterns in reaction-diffusion systems, and the development of the corresponding generic theory, have unveiled a wealth of new and unexpected phenomena. In this article we review these phenomena, we provide a unified and comprehensive description of them, and extend the theoretical analysis to new situations. We formulate the generic amplitude equations for different orders of spatial resonance for 1d and 2d patterns (stripes and hexagons). We identify and describe in detail the autonomous dynamical system which underlies the phenomenon of traveling-stripe resonance. For 1d we focus on localized solutions (kinks and pulses), their dynamics and their interaction for both 1:1 and 2:1 resonance. Specifically, we discuss the effect of wave-number mismatch (inexact resonance) combined with the non-gradient dynamics induced by the motion of the forcing, resulting in non-trivial interactions and complex spatio-temporal dynamics. Analytical results in the phase approximation are obtained, while numerical techniques are used to study the complete problem. We show that defect interaction is oscillatory with distance, allowing for the existence of locked chaotic wave trains. In 2d we discuss the modulational instabilities of striped patterns and focus mostly on the emergence of hexagons induced by traveling stripe forcing, for exact 1:1 resonance. In this case, we examine in detail the complex bifurcation scenario beyond primary instabilities. We finally discuss the problem of traveling-wave forcing in the context of a specific system, the photosensitive CDIMA reaction, where most experiments have been carried out. We compare experiments and theoretical predictions and propose other experimental systems where the study could be extended. Finally, we review related work by other authors and discuss possible further developments and open questions which hold the promise of new interesting findings.

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“…There the spontaneous oscillations, which do not arise in the Faraday system, and their competition with phase-locked patterns driven by the forcing may lead to additional complexity. For single-frequency forcing above the Hopf bifurcation, labyrinthine stripe patterns arise from the oscillations through front instabilities and stripe nucleation [54]. It is unknown what happens if the stripes are unstable to the more complex patterns discussed here.…”

Section: Resultsmentioning

confidence: 92%

Superlattice Patterns in the Complex Ginzburg–Landau Equation with Multiresonant Forcing

Conway¹,

Riecke

2009

SIAM J. Appl. Dyn. Syst.

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Motivated by the rich variety of complex patterns observed on the surface of fluid layers that are vibrated at multiple frequencies, we investigate the effect of such resonant forcing on systems undergoing a Hopf bifurcation to spatially homogeneous oscillations. We use an extension of the complex Ginzburg-Landau equation that systematically captures weak forcing functions with a spectrum consisting of frequencies close to the 1:1-, the 1:2-, and the 1:3-resonance. By slowly modulating the amplitude of the 1:2-forcing component we render the bifurcation to subharmonic patterns supercritical despite the quadratic interaction introduced by the 1:3-forcing. Our weakly nonlinear analysis shows that quite generally the forcing function can be tuned such that resonant triad interactions with weakly damped modes stabilize subharmonic patterns comprised of four or five Fourier modes, which are similar to quasi-patterns with 4-fold and 5-fold rotational symmetry, respectively. Using direct simulations of the extended complex Ginzburg-Landau equation we confirm our weakly nonlinear analysis. In simulations domains of different complex patterns compete with each other on a slow time scale. As expected from energy arguments, with increasing strength of the triad interaction the more complex patterns eventually win out against the simpler patterns. We characterize these ordering dynamics using the spectral entropy of the patterns. For system parameters reported for experiments on the oscillatory Belousov-Zhabotinsky reaction we explicitly show that the forcing parameters can be tuned such that 4-mode patterns are the preferred patterns.

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Development of Standing-Wave Labyrinthine Patterns

Cited by 55 publications

References 23 publications

Fronts and patterns in a spatially forced CDIMA reaction

Fronts and patterns in a spatially forced CDIMA reaction

Theory of pattern forming systems under traveling-wave forcing

Superlattice Patterns in the Complex Ginzburg–Landau Equation with Multiresonant Forcing

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