Traveling-Stripe Forcing Generates Hexagonal Patterns

Míguez, David G.; Nicola, Ernesto M.; Muñuzuri, Alberto P.; Casademunt, Jaume; Sagués, Francesc; Kramer, Lorenz

doi:10.1103/physrevlett.93.048303

Cited by 48 publications

(40 citation statements)

References 14 publications

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“…We will thus revisit mostly the theoretical parts of Refs. [40,[42][43][44][45]47] but we will also add new analytical and numerical results and new insights into the physical significance of the results. We will thus focus our discussion on amplitude equations.…”

Section: Introductionmentioning

confidence: 99%

“…[40] and followed up by Refs. [42][43][44][45][46][47], all of them with a direct interplay with experiments on Turing patterns in reaction-diffusion systems. Following the above references, we will focus on the simplest case, namely that in which the external forcing takes the form of a periodic modulation of an external control parameter, which is traveling with a finite velocity.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Theory of pattern forming systems under traveling-wave forcing

et al. 2007

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“…The well-characterized photosensitivity of the CDIMA reaction also has the advantage of being sensitive to light [14,15]. Its photosensitivity has been used to force transitions in the system in many different contexts [16][17][18][19]. In principle, homogeneous constant external intensity of illumination induces changes in the spatial configuration of the pattern, from a hexagonal configuration to a labyrinth pattern of stripes and a honeycomb-like arrangement of spots [20].…”

Section: Introductionmentioning

confidence: 99%

Interaction of chemical patterns in coupled layers

et al. 2011

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We investigate the interaction between reaction-diffusion systems coupled by diffusion. The photosensitive CDIMA (chorine dioxide-iodine-malonic acid) reaction allows us to study experimentally the mutual influence of two layers of Turing patterns coupled via a diffusive interaction. By illuminating each of the layers with different intensities of homogeneous external light, the chemical conditions in each layer can be shifted, allowing us to study the result of diffusive interaction between Turing patterns with different spatial configurations. Our experiments suggest a complex scenario for the interaction between different patterns, strongly dependent on the spatial characteristics of the interacting patterns. Numerical simulations are also reported in full agreement with experimental observations.

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“…DOI: 10.1103/PhysRevLett.99.028302 PACS numbers: 82.40.Ck, 05.45.Yv, 47.54.ÿr The effects of external forcing on pattern forming systems exhibit fascinating nonlinear behavior from a fundamental point of view and at the same time provide a valuable tool for the control of pattern forming systems. While the cases of purely temporal [1][2][3] and spatial forcing [4 -6] have been considered for many years, the spatiotemporal modulation of control parameters has been introduced only recently [7][8][9][10][11][12][13]. Simultaneously, both fundamental questions [14] and interesting applications of pattern control have arisen for possible information processing devices based on nonequilibrium patterns [15,16].…”

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confidence: 99%

Dynamics of Domain Walls in Pattern Formation with Traveling-Wave Forcing

2007

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We study dynamics of domain walls in pattern forming systems that are externally forced by a moving space-periodic modulation close to 2:1 spatial resonance. The motion of the forcing induces nongradient dynamics, while the wave number mismatch breaks explicitly the chiral symmetry of the domain walls. The combination of both effects yields an imperfect nonequilibrium Ising-Bloch bifurcation, where all kinks (including the Ising-like one) drift. Kink velocities and interactions are studied within the generic amplitude equation. For nonzero mismatch, a transition to traveling bound kink-antikink pairs and chaotic wave trains occurs. DOI: 10.1103/PhysRevLett.99.028302 PACS numbers: 82.40.Ck, 05.45.Yv, 47.54.ÿr The effects of external forcing on pattern forming systems exhibit fascinating nonlinear behavior from a fundamental point of view and at the same time provide a valuable tool for the control of pattern forming systems. While the cases of purely temporal [1][2][3] and spatial forcing [4 -6] have been considered for many years, the spatiotemporal modulation of control parameters has been introduced only recently [7][8][9][10][11][12][13]. Simultaneously, both fundamental questions [14] and interesting applications of pattern control have arisen for possible information processing devices based on nonequilibrium patterns [15,16].In the simplest case of a spatiotemporal forcing, using the form of a traveling wave, one allows for a periodic dependence on both space and time. The consequences of spatial resonance of such a forcing with a Turing-like mode were studied in terms of the frequency, or velocity, !, of the traveling-wave forcing and the deviation q from exact spatial resonance. A central finding was the occurrence of kinks or domain walls [7], which, similar to the case of purely spatial forcing [5], mediate the competition between the inherent and imposed wavelength.In this Letter we study how a mismatch q and the motion of the forcing affect domain walls. We show that a combination of both effects, studied in the exemplary case of 2:1 resonance, provides a new scenario of complex spatiotemporal dynamics. Beginning with exact spatial resonance, q 0, an illuminating analogy with the resonance of a subharmonic temporal forcing of an extended oscillatory system appears. For the latter system it is well known that the resonance generates stable walls and a transition between Ising and Bloch walls. The motion of the forcing pattern -playing the role of the frequency detuning in oscillatory systems-endows the system with nongradient dynamics. This transition is called nonequilibrium IsingBloch (NIB) transition [17][18][19][20][21].The effects of a spatial resonance mismatch are more difficult to analyze. While for large bifurcation parameter (or, equivalently, for small forcing) the effect of the mismatch becomes negligible and a phase approximation can be used [11], we here focus on intermediate values of . We employ numerical continuation methods to calculate kink solutions and show that the mismatch q...

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Traveling-Stripe Forcing Generates Hexagonal Patterns

Cited by 48 publications

References 14 publications

Theory of pattern forming systems under traveling-wave forcing

Theory of pattern forming systems under traveling-wave forcing

Interaction of chemical patterns in coupled layers

Dynamics of Domain Walls in Pattern Formation with Traveling-Wave Forcing

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