双层非线性耦合反应扩散系统中复杂Turing斑图

Complex Turing patterns in two-layer non-linearly coupling reaction diffusion systems

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.

The influence of Turing modes in two subsystems on pattern formation is investigated by the two-layer non-linearly coupled Brusselator model. It is found that the coupling coefficient and wave number ratio between two Turing modes take an important role in the pattern formation and pattern selection. The kind of pattern changes from simple pattern to complex one with the increase of wave number ratio. When nonlinear coupling coefficient is smaller than 0.1, the short wave mode in system 1 under the action of instability mode in system 2 can form not only simple pattern （such as simple hexagon and quadrilateral and stripe pattern）, but also complex pattern due to the resonance coupling between the two Turing modes （such as honeycomb hexagon and super hexagon and complex black-eye pattern）, and the transformation process of pattern from quadrilateral to superlattice pattern is observed for the first time under the specific parameters. When nonlinear coupling coefficient is more than 0.1, the simple patterns such as simple hexagon and stripe pattern are obtained only in system 1, because there is no resonance coupling between the two Turing modes in system 1.