Superlattice Turing structures in a photosensitive reaction-diffusion system

Families of complex superlattice structures, consisting of combinations of basic hexagonal or square patterns, are found in a photosensitive reaction-diffusion system. The structures are induced by simple illumination patterns whose wavelengths are appropriately related to that of the system's intrinsic Turing pattern. Computer simulations agree with the structures and their stability. The technique offers a general approach to generating superlattices for use in information storage and other applications.     

Traveling-stripe forcing generates hexagonal patterns

We study the response of Turing stripe patterns to a simple spatiotemporal forcing. This forcing has the form of a traveling wave and is spatially resonant with the characteristic Turing wavelength. Experiments conducted with the photosensitive chlorine dioxide-iodine-malonic acid reaction reveal a striking symmetry-breaking phenomenon of the intrinsic striped patterns giving rise to hexagonal lattices for intermediate values of the forcing velocity. The phenomenon is understood in the framework of the corresponding amplitude equations, which unveils a complex scenario of dynamical behaviors.     

双层耦合介质中四边形图灵斑图的数值研究

采用双层线性耦合Lengyel-Epstein模型,在二维空间对简单正四边和超点阵四边形进行了数值分析.结果表明:当两子系统波数比N1时,随耦合强度的增大,基模的波矢空间共振形式发生改变,系统由简单六边形自发演化为结构复杂的新型斑图,除已报道的超六边形外,还获得了简单正四边和多种超点阵四边形,包括大小点、点线、白眼和环状超四边等斑图.当耦合系数α和β在一定范围内同步增大时,两子系统形成相同波长的Ⅰ型简单正四边;当α和β不同步增大时,由于两图灵模在短波子系统形成共振,系统斑图经相变发生Ⅰ型正四边→Ⅱ型正四边→超点阵四边形的转变;当系统失去耦合作用时,短波子系统波长为λ的Ⅰ型正四边斑图迅速失稳并形成波长为λ/N的Ⅰ型正四边,随模拟时间的延长,两子系统中不同波长的正四边均会经相变发生Ⅰ型正四边→Ⅱ型正四边→六边形的转变.     

Numerical simulation and analysis of complex patterns in a two-layer coupled reaction diffusion system

The resonance interaction between two modes is investigated using a two-layer coupled Brusselator model. When two different wavelength modes satisfy resonance conditions, new modes will appear, and a variety of superlattice patterns can be obtained in a short wavelength mode subsystem. We find that even though the wavenumbers of two Turing modes are fixed, the parameter changes have influences on wave intensity and pattern selection. When a hexagon pattern occurs in the short wavelength mode layer and a stripe pattern appears in the long wavelength mode layer, the Hopf instability may happen in a nonlinearly coupled model, and twinkling-eye hexagon and travelling hexagon patterns will be obtained. The symmetries of patterns resulting from the coupled modes may be different from those of their parents, such as the cluster hexagon pattern and square pattern. With the increase of perturbation and coupling intensity, the nonlinear system will convert between a static pattern and a dynamic pattern when the Turing instability and Hopf instability happen in the nonlinear system. Besides the wavenumber ratio and intensity ratio of the two different wavelength Turing modes, perturbation and coupling intensity play an important role in the pattern formation and selection. According to the simulation results, we find that two modes with different symmetries can also be in the spatial resonance under certain conditions, and complex patterns appear in the two-layer coupled reaction diffusion systems.     

Hexagonal Standing-Wave Patterns in Periodically Forced Reaction-Diffusion Systems

The periodically forced spatially extended Brusselator is investigated in the oscillating regime. The temporal response and pattern formation within the 2：1 frequency-locking band where the system oscillates at one half of the forcing frequency are examined. An hexagonal standing-wave pattern and other resonant patterns are observed. The detailed phase diagram of resonance structure in the forcing frequency and forcing amplitude parameter space is calculated. The transitions between the resonant standing-wave patterns are of hysteresis when control parameters are varied, and the presence of multiplicity is demonstrated. Analysis in the framework of amplitude equation reveals that the spatial patterns of the standing waves come out as a result of Turing bifurcation in the amplitude equation.     

双层耦合非对称反应扩散系统中的超点阵斑图

通过线性耦合Brusselator模型和Lengyel-Epstein模型,数值研究了双层耦合非对称反应扩散系统中图灵模之间的相互作用以及斑图的形成机理.模拟结果表明,合适的波数比以及相同的对称性是两个图灵模之间达到空间共振的必要条件,而耦合强度则直接影响了图灵斑图的振幅大小.为了保证对称性相同,两个图灵模的本征值高度要位于一定的范围内.只有失稳模为长波模时,才能对另一个图灵模产生调制作用,并形成多尺度时空斑图.随着波数比的增加,短波模子系统依次经历黑眼斑图、白眼斑图以及时序振荡六边形斑图的转变.研究表明失稳图灵模与处于短波不稳定区域的高阶谐波模之间的共振是产生时序振荡六边形的主要原因.     

Oscillatory Turing patterns in reaction-diffusion systems with two coupled layers

A model reaction-diffusion system with two coupled layers yields oscillatory Turing patterns when oscillation occurs in one layer and the other supports stationary Turing structures. Patterns include "twinkling eyes," where oscillating Turing spots are arranged as a hexagonal lattice, and localized spiral or concentric waves within spot-like or stripe-like Turing structures. A new approach to generating the short-wave instability is proposed.     

Stable squares and other oscillatory turing patterns in a reaction-diffusion model

We study the Brusselator reaction-diffusion model under conditions where the Hopf mode is supercritical and the Turing band is subcritical. Oscillating Turing patterns arise in the system when bulk oscillations lose their stability to spatial perturbations. Spatially uniform external periodic forcing can generate oscillating Turing patterns when both the Turing and Hopf modes are subcritical in the autonomous system. Most of the symmetric patterns show period doubling in both space and time. Patterns observed include squares, rhombi, stripes, and hexagons.     

Turing Patterns in a Reaction-Diffusion System

We have further investigated Turing patterns in a reaction-diffusion system by theoretical analysis and numerical simulations. Simple Turing patterns and complex superlattice structures are observed. We find that the shape and type of Turing patterns depend on dynamical parameters and external periodic forcing, and is independent of effective diffusivity rate σ in the Lengyel-Epstein model. Our numerical results provide additional insight into understanding the mechanism of development of Turing patterns and predicting new pattern formations.     

六边形格子态斑图的数值模拟

采用双层耦合的Lengel-Epstein模型, 通过改变两子系统图灵模的强度比, 获得了四种的六边形格子态和多种非格子态结构. 模拟结果表明: 反应扩散系统的格子态结构由三套子结构叠加而成, 是两图灵模的波数比和强度比共同作用的结果, 两模的强度比决定了三波共振的具体模式; 另外, 系统选择格子态斑图所需的两图灵模的强度比大于非格子态斑图的强度比; 逐步增加两图灵模强度比, 出现的斑图趋于从复杂到简单变化. 深入研究发现: 不同互质数对(a, b)对应的格子态斑图的稳定性不同, 其中(3, 2)对应的格子态结构最为稳定.     

Pattern formation in forced reaction diffusion systems with nearly degenerate bifurcations

The existence and stability of stable standing-wave patterns in an assembly of spatially distributed generic oscillators governed by a couple of complex Ginzburg-Landau equations, subjected to parametric forcing, are reported. The mechanism of a dispersion-induced pattern in dissipative oscillators parametrically forced near the degenerate Turing-Hopf bifurcation is also illustrated. We show that, when excitation occurs just after the Turing bifurcation and before the Hopf instability, the system exhibits a new type of stable standing-wave structures, namely the mixed-mode solutions. The Brussellator-model, parametrically forced below the threshold of oscillations, is analyzed as an example of calculation.     

Turing bifurcation in a reaction-diffusion system with density-dependent dispersal

Motivated by the recent finding [N. Kumar, G.M. Viswanathan, V.M. Kenkre, Physica A 388 (2009) 3687] that the dynamics of particles undergoing density-dependent nonlinear diffusion shows sub-diffusive behaviour, we study the Turing bifurcation in a two-variable system with this kind of dispersal. We perform a linear stability analysis of the uniform steady state to find the conditions for the Turing bifurcation and compare it with the standard Turing condition in a reaction-diffusion system, where dispersal is described by simple Fickian diffusion. While activator-inhibitor kinetics are a necessary condition for the Turing instability as in standard two-variable systems, the instability can occur even if the diffusion constant of the inhibitor is equal to or smaller than that of the activator. We apply these results to two model systems, the Brusselator and the Gierer-Meinhardt model.     

Spatial resonances and superposition patterns in a reaction-diffusion model with interacting Turing modes

Spatial resonances leading to superlattice hexagonal patterns, known as "black-eyes," and superposition patterns combining stripes and/or spots are studied in a reaction-diffusion model of two interacting Turing modes with different wavelengths. A three-phase oscillatory interlacing hexagonal lattice pattern is also found, and its appearance is attributed to resonance between a Turing mode and its subharmonic.     

双层非线性耦合反应扩散系统中复杂Turing斑图

采用双层耦合的Brusselator模型, 研究了两个子系统非线性耦合时Turing 模对斑图的影响, 发现两子系统Turing 模的波数比和耦合系数的大小对斑图的形成起着重要作用. 模拟结果表明: 斑图类型随波数比值的增加, 从简单斑图发展到复杂斑图; 非线性耦合项系数在0–0.1时, 系统1中短波模在系统2失稳模的影响下不仅可形成简单六边形、四边形和条纹斑图, 两模共振耦合还可以形成蜂窝六边形、超六边形和复杂的黑眼斑图等超点阵图形, 首次在一定范围内调整控制参量观察到由简单正四边形向超六边形斑图的转化过程; 耦合系数在0.1–1时, 系统1中短波模与系统2失稳模未发生共振耦合仅观察到与系统2相同形状的简单六边形、四边形和条纹斑图. 关键词： Brusselator模型 非线性耦合 Turing模     

Spontaneous symmetry breaking turing-type pattern formation in a confined Dictyostelium cell mass

We have discovered a new type of patterning which occurs in a two-dimensionally confined cell mass of the cellular slime mold Dictyostelium discoideum. Besides the longitudinal structure reported earlier, we observed a spontaneous symmetry breaking spot pattern whose wavelength shows similar strain dependency to that of the longitudinal pattern. We propose that these structures are due to a reaction-diffusion Turing instability similar to the one which has been exemplified by CIMA (chlorite-iodide-malonic acid) reaction. The present finding may exhibit the first biochemical Turing structure in a developmental system with a controllable boundary condition.     

Control and managing of localized states in two-dimensional systems with periodic forcing

We study the formation of localized structures in two-dimensional systems with periodic forcing, showing that these types of systems provide an adequate framework for the study and control of localized structures. Theoretically, we introduce a dissipative ϕ ⁴ model as a prototype for a bistable spatially forced system, and we show that with different spatial forcings of small amplitudes, such as square or hexagonal grids, this model exhibits a family of localized structures. By changing the forcing parameters, we control the bistability between the various induced patterns. Experimentally, based on an optical feedback with spatially amplitude-modulated beam, we set-up a two-dimensional forced experiment in a nematic liquid crystal cell. By changing the forcing parameters, the system exhibits a family of localized structures that are confirmed by numerical simulations for the average liquid crystal tilt angle.     

Spontaneous motion of cavity solitons in vertical-cavity lasers subject to optical injection and to delayed feedback

We consider a rate equation model describing broad area vertical-cavity surface-emitting lasers subject to injection and to time-delayed optical feedback. We show that the inclusion of an external cavity affects dramatically the space-time behavior of this system by modifying the instability threshold as well as the wavelength of the Turing instability. We show also that the delayed feedback is responsible for the appearance of traveling wave instability. Finally, we demonstrate that a single cavity soliton exhibits a spontaneous motion with a constant velocity. Without delayed feedback transition to the moving cavity soliton does not exist.     

Pattern formation in a predator–prey model with spatial effect

In this paper, spatial dynamics in the Beddington–DeAngelis predator–prey model with self-diffusion and cross-diffusion is investigated. We analyze the linear stability and obtain the condition of Turing instability of this model. Moreover, we deduce the amplitude equations and determine the stability of different patterns. Numerical simulations show that this system exhibits complex dynamical behaviors. In the Turing space, we find three types of typical patterns. One is the coexistence of hexagon patterns and stripe patterns. The other two are hexagon patterns of different types. The obtained results well enrich the finding in predator–prey models with Beddington–DeAngelis functional response.     

Mutual synchronization of two coupled self-oscillators based on GaAs/AlGaAs superlattices

The interaction of self-oscillators based on 30-period weakly coupled GaAs/AlGaAs superlattices is studied. The action of one self-oscillator on the other was observed for a constant bias voltage in the absence of generation of self-sustained oscillations in one of the oscillators. It is shown that induced oscillations in a forced oscillator appear due to excitation of oscillations in the system of coupled oscillators forming the electric-field domain wall at the frequency of one of the higher harmonics of a forcing oscillation.     

用于短波段发光二极管的二维光子晶体禁带研究

运用平面波展开法(PWE),针对光子晶体在短波长段发光二极管(LED)领域上的应用,主要选择高频禁带模式,研究了4种具有较大应用潜力的二维光子晶体结构,包括正方空气柱结构、三角空气柱结构、正方介质柱结构和三角介质柱结构。在不同晶格常数、占空比(AFF)、柱半径和晶格常数比下,分析了TE模式和TM模式光子禁带的变化。数据分析表明,光子禁带中心波长随AFF增加而变小。相比于其他结构,正方介质柱结构更适于短波段光子晶体LED来提高其出光效率,三角介质柱结构和三角空气柱结构适合用于构造短波段偏振光LED。