Travelling-stripe forcing of Turing patterns

doi:10.1016/j.physd.2004.08.017

Physica D: Nonlinear Phenomena

Volume 199, Issues 1–2, 1 December 2004, Pages 235-242

https://doi.org/10.1016/j.physd.2004.08.017 Get rights and content

Abstract

We report on the effect of a spatio-temporal forcing applied on Turing stripe patterns under spatial resonance. Experiments conducted with the photosensitive CDIMA reaction unveil a large variety of dynamical responses of striped patterns, from entrained to waving, when changing the forcing velocity from small to large values. In between, and quite unexpectedly, we also observe a symmetry breaking phenomenon leading to hexagonal lattices, either entrained with the forcing or oscillating. We propose a new set of coupled amplitude equations [1] which provides a theoretical framework where the experimental phenomena can be interpreted.

Introduction

Pattern forming systems have captivated science for decades in a rich variety of natural and laboratory scenarios [2]. In particular, the response of this class of systems to external forcing provides a powerful tool to deeply probe their inherently nonlinear behaviour and, eventually, possible mechanisms for their control. Up to very recently, much effort had focused on the resonant or locked responses of oscillatory media to time periodic (see e.g. [3], [4], [5], [6], [7], [8], [9], and references therein) and spatially periodic forcing [10]. Also the forcing of spatially periodic patterns with, either oscillatory [11], [12] or stationary [13], [14], [15], [16], [17], [18], [19], [20] forcing has been considered. Here we further report [21], [1] on our ongoing research devoted to the response of Turing patterns [22], [23], forming stripes, to the simplest possible type of spatiotemporal forcing (a less generic forcing, in the form of a single travelling stripe, has also been recently considered in [24]) at spatial resonance. Experiments refer to the photosensitive CDIMA reaction [25], [26] (CDIMA stands for the chlorine dioxide–iodine–malonic acid system) conducted in a gel reactor, which is subjected to a passing, spatially periodic, patterned illumination. A striking variety of dynamical regimes are found when varying the velocity of the sweeping forcing. In particular, a remarkable symmetry breaking phenomenon, giving rise to the emergence of hexagonal lattices, intervenes between either entrained or modulated stripes, at small velocities, and seemingly waving bands at large velocities. We interpret these experimental scenarios using as a theoretical framework an amplitude equation based description that we believe is generic enough to reveal the essentials of the response of such a striped pattern to the external travelling-wave forcing.

This paper is divided in two sections plus the conclusions. In the first section we will give a detailed description of the experimental setup and summarize the results of the experiment. In the second section we will present our theoretical analysis of the symmetry breaking phenomenon seen in the experiment.

Section snippets

Experimental setup

The chemical reactants and respective concentrations in the CDIMA reaction as used here are: iodine (0.45 mM), malonic acid (1.2 mM), chlorine dioxide (0.1 mM), poly(vynil) alcohol (80% hydrolized; 10 g/l)and sulfuric acid (10 mM). The reaction occurs in a one feeding chamber chemical reactor, continuously stirred (CSTR), and the residence time of the reactants inside the reactor was kept at 250 s. Structures are formed in a agarose gel (Fluka, gelling temperature 40–43 ^∘C) made from a 2%

Amplitude equations

In order to interpret the experimental observations reported and to understand them within a more general theoretical framework, we adopt an amplitude equation based description.

Conclusions

In summary, we have demonstrated using a prototype chemical system, that simple spatio-temporal modulations (with a single wavelength and a single frequency) of a control parameter may induce a rich variety of oscillatory modes for otherwise steady patterns, provided the respective wavelengths are in resonance. Most of the time-dependent patterns have not been found before and do not reflect in a simple way, neither the structure of the forcing nor the patterns intrinsic to the unforced system.

Acknowledgements

This work has been supported by the European Commission under network HPRN-CT-2002-00312, DGI (Spain) under projects BQU2003-05042 and BFM2000-0348, and Xunta de Galicia (Spain) under project PGIDT00PX120610PR. We would like to thank S. Rüdiger and W. Pesch for useful discussions and comments. Part of the simulations were done with the XDim Interactive Simulation Package developed by P. Coullet and M. Monticelli, who L. K. thanks for their support. All experimental results here reported were

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Travelling-stripe forcing of Turing patterns

Abstract

Introduction

Section snippets

Experimental setup

Amplitude equations

Conclusions

Acknowledgements

Physica D

Phys. Rev. Lett.

Travelling-stripe forcing generates hexagonal patterns

Phys. Rev. Lett.

Pattern formation out of equilibrium

Rev. Mod. Phys.

Breaking chiriality in nonequilibrium systems

Phys. Rev. Lett.

A phase front instability in periodically forced oscillatory systems

Phys. Rev. Lett.

Resonant pattern formation in a chemical system

Nature

Resonant phase patterns in a reaction–diffusion system

Phys. Rev. Lett.

Frequency locking in spatially extended systems

Phys. Rev. Lett.

Siam J. Appl. Dyn. Sys.

Resonant spatio-temporal forcing of oscillatory media

Europhys. Lett.

Control of Turing structures by periodic illumination

Phys. Rev. Lett.

Resonant suppression of Turing patterns by periodic illumination

Phys. Rev. E

Pattern formation in a temporally modulated chemical instability

J. Chem. Phys.