Locking of Turing patterns in the chlorine dioxide-iodine-malonic acid reaction with one-dimensional spatial periodic forcing

We use the photosensitive chlorine dioxide-iodine-malonic acid reaction-diffusion system to study wavenumber locking of Turing patterns with spatial periodic forcing. Wavenumber-locked stripe patterns are the typical resonant structures that labyrinthine patterns exhibit in response to one-dimensional forcing by illumination when images of stripes are projected on a working medium. Our experimental results reveal that segmented oblique, hexagonal and rectangular patterns can also be obtained. However, these two-dimensional resonant structures only develop in a relatively narrow range of forcing parameters, where the unforced stripe pattern is in close proximity to the domain of hexagonal patterns. Numerical simulations based on a model that incorporates the forcing by illumination using an additive term reproduce well the experimental observations. These findings confirm that additive one-dimensional forcing can generate a two-dimensional resonant response. However, such a response is considerably less robust than the effect of multiplicative forcing.     

Superlattice patterns and spatial instability induced by delay feedback

Various oscillatory superlattice patterns in a reaction-diffusion system are observed by means of delay feedback (DF) in the parametric domain where the system without DF displays uniform bulk oscillation. By varying DF parameters within an appropriate range, the system undergoes transitions to oscillatory hexagons, stripes and squares, and square superlattices with different wavenumbers are also obtained. Linear stability analysis reveals that the patterns do not result from the Turing instability and a possible mechanism of pattern formation is suggested and proved analytically: DF induces instability of a homogeneous limit cycle with respect to spatial perturbations even if the Turing instability does not occur, so that oscillatory patterns possessing the corresponding spatial modes are produced. The different behavior of the dominant characteristic multiplier seems to be connected to the pattern selection. Here it is clearly demonstrated that DF can play a destabilizing role in spatially extended system instead of stabilizing the periodic orbits or turbulent states, which most earlier works have usually focused on.     

Turing patterns in the chlorine dioxide-iodine-malonic acid reaction with square spatial periodic forcing

We use the photosensitive chlorine dioxide-iodine-malonic acid reaction-diffusion system to study wavenumber locking of Turing patterns to two-dimensional "square" spatial forcing, implemented as orthogonal sets of bright bands projected onto the reaction medium. Various resonant structures emerge in a broad range of forcing wavelengths and amplitudes, including square lattices and superlattices, one-dimensional stripe patterns and oblique rectangular patterns. Numerical simulations using a model that incorporates additive two-dimensional spatially periodic forcing reproduce well the experimental observations.     

Spontaneous Formation of Microgroove Arrays on the Surface of p‐Type Porous Silicon Induced by a Turing Instability in Electrochemical Dissolution

Self‐organization plays an imperative role in recent materials science. Highly tunable, periodic structures based on dynamic self‐organization at micrometer scales have proven difficult to design, but are desired for the further development of micropatterning. In the present study, we report a microgroove array that spontaneously forms on a p‐type silicon surface during its electrodissolution. Our detailed experimental results suggest that the instability can be classified as Turing instability. The characteristic scale of the Turing‐type pattern is small compared to self‐organized patterns caused by the Turing instabilities reported so far. The mechanism for the miniaturization of self‐organized patterns is strongly related to the semiconducting property of silicon electrodes as well as the dynamics of their surface chemistry.     

Alteration in Cross Diffusivities Governs the Nature and Dynamics of Spatiotemporal Pattern Formation

Realizing spatiotemporal patterns out of a chemical reaction diffusion system remains an experimental challenge owing to the difficulty in overcoming the stringent condition of diffusion driven instability. Herein, by considering the spatially extended Gray-Scott model system, we have investigated how the cross diffusivities of the reactants involved influence the nature and dynamics of spatiotemporal patterns. Our study unravels that in absence of diffusion driven instability, spatially inhomogeneous patterns can be obtained for the Gray-Scott model system, and unstable time dependent patterns can be stabilized just by adjusting cross diffusivities of the reactants. Interestingly, the effect of cross diffusion in presence of the diffusion driven instability can differentially alter the speed of pattern formation, and potentially modify the nature of the spatiotemporal patterns obtained under different parametric conditions. Experimental verification of our findings may allow us to observe spatiotemporal patterns beyond the regime of classical Turing instability.     

Formation and control of Turing patterns and phase fronts in photonics and chemistry

We review the main mechanisms for the formation of regular spatial structures (Turing patterns) and phase fronts in photonics and chemistry driven by either diffraction or diffusion. We first demonstrate that the so-called ‘off-resonance’ mechanism leading to regular patterns in photonics is a Turing instability. We then show that negative feedback techniques for the control of photonic patterns based on Fourier transforms can be extended and applied to chemical experiments. The dynamics of phase fronts leading to locked lines and spots are also presented to outline analogies and differences in the study of complex systems in these two scientific disciplines.     

Turing pattern amplitude equation for a model glycolytic reaction-diffusion system

For a reaction-diffusion system of glycolytic oscillations containing analytical steady state solution in complicated algebraic form, Turing instability condition and the critical wavenumber at the Turing bifurcation point, have been derived by a linear stability analysis. In the framework of a weakly nonlinear theory, these relations have been subsequently used to derive an amplitude equation, which interprets the structural transitions and stability of various forms of Turing structures. Amplitude equation also conforms to the expectation that time-invariant amplitudes are independent of complexing reaction with the activator species.     

低浓度三分子双曲型反应-扩散方程的非线性理论

建立了低浓度三分子模型双曲型反应-扩散的波动方程，研究了定态的稳定性，重点研究了Turing不稳定问题，指出双曲型方程的Turing不稳定不受扩散系数不相等（Dx≠Dy）这一条件的约束，进而对方程作近似的分支分析，讨论了出现极限环的条件，最后对极限环和定态不稳定作了数值研究．     

Oxalate ester addition from a solid reagent bed for chemiluminescence detection in HPLC

Summary The combination of solid-state peroxyoxalate chemiluminescence detection and post-column chemical reaction systems in liquid chromatography is investigated. Bis-2,4,6-trichlorophenyloxalate (TCPO) is added from a solid reagent bed with the fluorophore, 3-aminofluoranthene, immobilized on glass beads. Hydrogen peroxide generated photochemically by quinone analytes is measured. Flow splitting is shown to be a simple means of reagent addition with a negligible band broadening effect. This system is compared to a more flexible dualpump design. To minimize band broadening in both systems, the reagent bed is located out of the path of the chromatographic effluent. Detection limits are in the sub-picomole range, an imporvement relative to those previously reported with TCPO added in the liquid phase. These systems can be easily be adapted for detection of peroxide generated by other post-column reactions.     

A graph-theoretic method for detecting potential Turing bifurcations

The conditions for diffusion-driven (Turing) instabilities in systems with two reactive species are well known. General methods for detecting potential Turing bifurcations in larger reaction schemes are, on the other hand, not well developed. We prove a theorem for a graph-theoretic condition originally given by Volpert and Ivanova [Mathematical Modeling (Nauka, Moscow, 1987) (in Russian), p. 57] for Turing instabilities in a mass-action reaction-diffusion system involving n substances. The method is based on the representation of a reaction mechanism as a bipartite graph with two types of nodes representing chemical species and reactions, respectively. The condition for diffusion-driven instability is related to the existence of a structure in the graph known as a critical fragment. The technique is illustrated using a substrate-inhibited bifunctional enzyme mechanism which involves seven chemical species.     

Nanoscale changes induce microscale effects in Turing patterns

A water-in-oil microemulsion loaded with a reaction-diffusion chemical system (Belousov-Zhabotinsky reaction) is able to exhibit Turing patterns that are believed to be responsible for differentiation processes in Nature. Using polymers, such as polyethylene oxide, longer than the droplet size changes the distribution of droplets due to cluster formation. This difference in the nanoscale has relevant consequences in the observed the Turing pattern's wavelength, which is three orders of magnitude larger than the droplet size.     

Analysis of a model scheme incorporating reactant inhibition: Instability of homogeneous solution and formation of spatio–temporal structures

A model scheme incorporating reactant inhibition in the rate process has been analyzed with a view to study the instability of homogeneous solution due to diffusion. Conditions for the occurrence of Turing as well as phase instability are derived and show the existence of multiplicity in the parameter space. The Ginzburg-Landau equation for the system is developed and solved numerically in various regions of the parameter space. The simple model system shows the existence of very rich behavior including normal and inverted bifurcations in the super and subcritical regimes. The various results are analyzed and discussed. © 1993 John Wiley & Sons, Inc.     

Resonant Excitation of Boundary Layer Instability of DC Arc Plasma Jet by Current Modulation

Instabilities of thermal plasma jets were studied on the basis of analysis of plasma radiation fluctuations recorded by an array of high frequency photodiodes. Characteristic frequencies of jet oscillations were found and spatial distribution of amplitude of plasma fluctuations was determined. The influence of arc current ripple on plasma instabilities was investigated for two types of power supply—classical thyristor controlled unit with the frequency of the current ripple 300 Hz and the rectifier with the high frequency converter and frequency of the current modulation 30 kHz. Generation of boundary layer instability with the current modulation frequency and its harmonics was proved using fast Fourier transform, contour plots and phase portraits. It was found that the character of fluctuations of plasma jet was substantially influenced by current ripple with the frequency or its harmonics close to the frequency of oscillations generated by boundary layer instability.     

Mobility-induced instability and pattern formation in a reaction-diffusion system

Ions undergoing a reaction-diffusion process are susceptible to electric field. We show that a constant external field may induce a kind of instability on the state stabilized by diffusion in a reaction-diffusion system giving rise to formation of pattern even when the diffusion coefficients of the reactants are equal. The origin of the pattern is due to the difference in mobilities of the two species and is thus markedly different from that of deformed Turing pattern in presence of the field. While this differential flow instability had been shown earlier to result in traveling waves, we realize in the context of stationary pattern formation in a typical reaction-diffusion-advective system. Our analysis is based on a numerical simulation of a generic model on a two-dimensional domain.     

Turing structures and stability for the 1-D Lengyel–Epstein system

This paper continues the analysis on the Lengyel–Epstein reaction- diffusion system of the chlorite-iodide-malonic acid-starch (CIMA) reaction for the rich Turing structures. The steady state structures, especially the double bifurcation one, and their stability and multiplicity are studied by the use of Lyapunov–Schmidt reduction technique and singularity theory. Numerical simulations are presented to support our theoretical studies. The results show that the richer stationary Turing patterns heavily rely both on the size of the reactor and on the effective diffusion rate in the CIMA reaction.     

Control of Turing pattern by weak spatial perturbation

The control of Turing pattern formation by weak spatial perturbation is investigated. The weak spatial perturbation added before Turing pattern stabilization is found to show prominent spatial orientation effect. The control process of perturbation to Turing patterns is tracked. The effect of perturbation factors, such as amplitude and imposing time are also discussed.     

Turing pattern formation in anisotropic medium

Diffusion of reacting species in chemical and biochemical systems in anisotropic medium is markedly different from those occurring in isotropic medium, therefore approximating diffusion coefficients as constants may not be desirable as this has dynamical consequences. This paper is devoted to the analytical and numerical investigation of the development of spatial patterns in such systems. To this end we consider a general reaction–diffusion system with concentration-dependent diffusion and formulate a scheme to derive the general form of envelope equation for such systems. The theory is applied to the chlorite–iodide–malonic acid system, a standard paradigm for activator–inhibitor mechanism, to derive the instability condition in terms of the anisotropy parameters (\(\kappa _{i}, i = u, v\) that impart concentration-dependence to the diffusion coefficients) and identify the supercritical and subcritical Turing regions in the bifurcation diagram. The theoretical predictions are in good agreement with the numerical simulations.     

Chemical waves and pattern formation in the 1,2-dipalmitoyl-sn-glycero-3-phosphocholine/water lamellar system

The present work deals with the spatially extended oscillatory Belousov Zhabotinsky reaction-diffusion system carried out in an anisotropic environment of phosphatidylcholines/water binary system, which presents layered aqueous domains separated by lipid bilayers. We report the occurrence of stable Turing patterns, spiral waves, and other exotic structures in phospholipids bilayers that are generally used as a models for cell plasma membranes.     

Turing patterns, spatial bistability, and front interactions in the [ClO2, I2, I-, CH2(COOH)2] reaction

Recent experiments by Szalai and De Kepper performed in open spatial reactors have shown that the rich variety of dynamic properties of the chlorine dioxide-iodide-chlorite-iodine-malonic acid family of reactions is far from being exhausted: stable inhomogeneous patterns due to front interactions and transient labyrinthine structures are now added to the spatial bistability and Turing patterns as possible spatial behavior. The two latter phenomena, already observed in the chlorine dioxide-iodide (CDI) and the chlorine dioxide-iodide-malonic acid (CDIMA) reactions, respectively, were kept as limiting cases in the new setup. In this paper, we numerically analyze an extension of the most detailed available model of the CDI system (Lengyel et al.) including a reaction between I2 and MA that comes from the presence of the latter into the flow. The resulting nine-variable model is simulated in one and two dimensions, taking into account the proper constraints of the boundary-fed system. The nonequilibrium phase diagram closely follows the results of the experiments of ref 1. In particular, the model reproduces observations on spatial bistability, stationary front interactions, and Turing patterns. In addition, it predicts a new region of spatial bistability.     

Friction-induced pattern formation and Turing systems

We investigate the possibility of Turing-type pattern formation during friction. Turing or reaction-diffusion systems describe variations of spatial concentrations of chemical components with time due to local chemical reactions coupled with diffusion. Turing systems can lead to a variety of complex spatial patterns evolving with time. During friction, the patterns can form at the sliding interface due to the mass transfer (diffusion), heat transfer, various tribochemical reactions, and wear. We present simulation data showing the possibility of such pattern formation. On the other hand, existing experimental data suggest that in situ tribofilms can form at the frictional interface due to a variety of friction-induced chemical reactions (oxidation, the selective transfer of Cu ions, etc.). These tribofilms as well as other frictional "secondary structures" can form various patterns (islands or honeycomb domains). This mechanism of pattern formation can be attributed to the Turing systems.