Super-lattice patterns in two-layered coupled non-symmetric reaction diffusion systems

Liu, Fucheng; Liu, Yahui; Zhou, Zhixiang; Guo, Xue; Mengfei, Dong

doi:10.7498/aps.69.20191353

Cited by 3 publications

(1 citation statement)

References 27 publications

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“…它由放电层和电介质层组成, 放电产生的体电荷部分积累到介质表面形成表面电荷, 反过来表面电荷也会直接影响体放电, 两者相互耦合, 从而可以产生丰富多彩的时空斑图, 例如, 四边形斑图、六边形斑图、超点阵斑图、螺旋波斑图等 [27−29] . 研究表明, 气体放电系统一种特殊的反应扩散系统, 可以唯象地用反应扩散模型来描述 [25,26,30] . 为了进一步弄清介质阻挡放电系统中各类时空斑图形成的机理, 尤其是体电荷和表面电荷之间的耦合, 最近本小组线性耦合了两种不同的反应扩散模型, 获得了许多与实验相符合的模拟结果 [31] .…”

Section: 近年来时空振荡斑图由于其丰富的动力学行unclassified

Oscillatory Turing patterns in two-layered coupled non-symmetric reaction diffusion systems

河北大学物理科学与技术学院¹,

河北大学²

2021

Acta Phys. Sin.

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Pattern formation and self-organization are ubiquitous in nature and commonly observed in spatially extended non-equilibrium systems. As is well known, the origin of spatio-temporal patterns can be traced to the instability of the system, and is always accompanied by a symmetry breaking phenomenon. In reality, most of non-equilibrium systems are constructed by interactions among several different units, each of which has its unique symmetry breaking mechanism. The interaction among different units described by coupled pattern forming system gives rise to a variety of self-organized patterns including stationary and/or oscillatory patterns. In this paper, the dynamics of oscillatory Turing patterns in two-layered coupled non-symmetric reaction diffusion systems are numerically investigated by linearly coupling the Brusselator model and the Lengyel-Epstein model. The interaction among the Turing modes, higher-order harmonics and Hopf mode, and their effects on oscillatory Turing pattern are also analyzed. It is shown that the supercritical Turing mode <inline-formula><tex-math id="M5">\begin{document}${k_1}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20201710_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20201710_M5.png"/></alternatives></inline-formula> in the Lengyel-Epstein model is excited and interacts with the higher-order harmonics <inline-formula><tex-math id="M6">\begin{document}$\sqrt 3 {k_1}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20201710_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20201710_M6.png"/></alternatives></inline-formula> located in the Hopf region in the Brusselator model, and thus giving rise to the synchronous oscillatory hexagon pattern. The harmonic <inline-formula><tex-math id="M7">\begin{document}$\sqrt 2 {k_1}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20201710_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20201710_M7.png"/></alternatives></inline-formula> that can also be excited initially is some parameter domain, but it is unstable and vanishes finally. As the parameter <i>b</i> is increased, this oscillatory hexagon pattern first undergoes period-doubling bifurcation and transits into two-period oscillation, and then into multiple-period oscillation. When the Hopf mode participates in the interaction, the pattern will eventually transit into chaos. The synchronous oscillatory hexagon pattern can only be obtained when the subcritical Turing mode <inline-formula><tex-math id="M8">\begin{document}${k_2}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20201710_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20201710_M8.png"/></alternatives></inline-formula> in the Brusselator model is weaker than the higher-order harmonics <inline-formula><tex-math id="M9">\begin{document}$\sqrt 3 {k_1}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20201710_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20201710_M9.png"/></alternatives></inline-formula> located in the Hopf region and neither of the two Turing modes satisfies the spatial resonance condition. The system favorites the spatial resonance and selects the super-lattice patterns when these modes interact with each other. The interaction between Hopf mode and Turing mode can only give rise to non-synchronous oscillatory patterns. Moreover, the coupling strength also has an important effect on the oscillatory Turing pattern. These results not only provide a new pattern forming mechanism which can be extended to other nonlinear systems, but also gives an opportunity for more in-depth understanding the nature and their relevance to technological applications.

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Section: 近年来时空振荡斑图由于其丰富的动力学行unclassified

Oscillatory Turing patterns in two-layered coupled non-symmetric reaction diffusion systems

河北大学物理科学与技术学院¹,

河北大学²

2021

Acta Phys. Sin.

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Numerical analysis on formation and transition of white-eye square superlattice patterns in dielectric barrier discharge system

Bai

et al. 2020

Mod. Phys. Lett. B

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The mechanism of formation and transformation of white-eye square patterns in dielectric barrier discharge system is investigated numerically, using the two-layer Lengyel–Epstein model with asymmetric and symmetric coupling. When the scale of the simulation system [Formula: see text] is two to three times of pattern wavelength [Formula: see text], it is found that an obvious intermediate state with square distribution appears by adjusting the ratio of diffusion coefficients [Formula: see text]/[Formula: see text]. When it is coupled with a suitable short-wavelength Turing mode in the range of [Formula: see text] to [Formula: see text], a new spatial resonance structure can be formed in the short-wavelength mode subsystem, and the pattern evolves from a simple square pattern to a white-eye square pattern. Although the two coupling methods achieve the same results, the duration time of the white-eye square pattern in the symmetric coupling method is significantly longer than that in the asymmetric coupling method. Because the quadratic coefficient of the amplitude equation in the reaction–diffusion system is not zero, the simple square pattern of the long wavelength mode subsystem gradually transits into a stable hexagon pattern gradually. As a result, the white-eye pattern transits from a square to a hexagon.

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Effects of spatial periodic forcing on Turing patterns in two-layer coupled reaction diffusion system

刘倩¹,

田淼²,

范伟丽³

et al. 2022

Acta Phys. Sin.

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Periodic forcing of pattern-forming systems is always a hot point in the field of pattern formation since it is one of the most effective methods for controlling patterns. In reality, most of the pattern-forming systems are the multilayered systems, in which each layer is a reaction-diffusion system coupled to adjacent layers. However, few works on this issue has been done in the multilayered systems and their response to the periodic forcing has yet been well understood. In this paper, the influence of the spatial periodic forcing on the Turing patterns in a two linearly coupled layers described by the Brusselator（Bru）model and the Lengyel-Epstein（LE）model respectively has been investigated by introducing a spatial periodic forcing into the LE layer. It is found that the subcritical Turing mode in the LE layer can be excited either the spatial forcing or the supercritical Turing mode (refers to as internal forcing mode) in the Bru layer is served as the longer wave mode. These three modes interact together and give rise to complex patterns with three different spatial scales. If both the spatial forcing mode and the internal forcing mode are the short wave mode, the subcritical Turing mode in the LE layer cannot be excited. Only the spatial forcing mode and the internal forcing mode interact each other in this case, giving rise to superlattice patterns when they are satisfied the spatial resonance condition. When the eigenmode of the LE layer is supercritical, a simple and robust hexagon pattern with its characteristic wavelength appears and response to the spatial forcing only when the forcing intensity is large enough. It is found that the wave number of forcing has a powerful influence on the spatial symmetry of patterns.

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Super-lattice patterns in two-layered coupled non-symmetric reaction diffusion systems

Cited by 3 publications

References 27 publications

Oscillatory Turing patterns in two-layered coupled non-symmetric reaction diffusion systems

Oscillatory Turing patterns in two-layered coupled non-symmetric reaction diffusion systems

Numerical analysis on formation and transition of white-eye square superlattice patterns in dielectric barrier discharge system

Effects of spatial periodic forcing on Turing patterns in two-layer coupled reaction diffusion system

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