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 investigation on square Turing patterns in medium with two coupled layers

In this paper, the simple and superlattice square patterns in two-dimensional space are investigated numerically by the two-layer coupled Lengyel-Epstein model. When the wave number ratio of Turing modes is greater than one, our results show that the spatial resonance form of the fundamental mode is changed with the increase of coupling strength, and simple hexagon pattern evolves spontaneously into a new pattern with a complicated structure. In addition to the reported superlattice hexagonal pattern, simple square pattern and superlattice square pattern are obtained, such as the complicated big-small spot, spot-line, ring and white-eye square pattern. The characteristics of simple and complicated superlattice square pattern are investigated by the intermediate process of evolution. When the coupling parameters \$\alpha $\ and \$\beta $\ increase synchronously within a certain range, the type I square patterns of the same wavelength are obtained in the two subsystems. When the coupling parameters \$\alpha $\ and \$\beta $\ increase asynchronously, the type I square pattern can evolve into the type Ⅱ square pattern on the same spatial scale through phase transition. Then, the new subharmonic modes are generated, and the complicated superlattice square patterns are obtained due to the resonance between the two Turing modes in a short wavelength mode subsystem. The influence of coupling between two subsystems on the square pattern is investigated. When the type I square pattern of wavelength \$\lambda $\ emerges, the square pattern will quickly lose its stability in the short wavelength mode subsystem, since the coupling coefficient is equal to zero. Finally a new square pattern of wavelength \$\lambda $\/N is formed. The type I square patterns of two subsystems successively evolve into the type Ⅱ square patterns through the phase transition. The spots move relatively with the extension of simulation time, and a new mode is generated and forms three-wave resonance in two subsystems, and then the hexagonal pattern dominates the system. Our results also show that the type Ⅱ square pattern spontaneously transforms into a hexagonal pattern.