双层耦合非对称反应扩散系统中的超点阵斑图

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

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.

The coupling mechanism is one of most important approaches to generating multiple-scaled spatialtemporal patterns.In this paper,the mode interaction between two different Turing modes and the pattern forming mechanisms in the non-symmetric reaction diffusion system are numerically investigated by using a two-layered coupled model.This model is comprised of two different reaction diffusion models:the Brusselator model and the Lengyel-Epstein model.It is shown that the system gives rise to superlattice patterns if these two Turing modes satisfy the spatial resonance condition,otherwise the system yields simple patterns or superposition patterns.A suitable wave number ratio and the same symmetry are two necessary conditions for the spatial resonance of Turing modes.The eigenvalues of these two Turing modes can only vary in a certain range in order to make the two sub-system patterns have the same symmetry.Only when the long wave mode becomes the unstable mode,can it modulate the other Turing mode and result in the formation of spatiotemporal patterns with multiple scale.As the wave number ratio increases,the higher-order harmonics of the unstable mode appear,and the sub-system with short wave mode undergoes a transition from the black-eye pattern to the white-eye pattern,and finally to a temporally oscillatory hexagon pattern.It is demonstrated that the resonance between the Turing mode and its higher-order harmonics located in the wave instability region is the dominant mechanism of the formation of this oscillatory hexagon pattern.Moreover,it is found that the coupling strength not only determines the amplitudes of these patterns,but also affects their spatial structures.Two different types of white-eye patterns and a new super-hexagon pattern are obtained as the coupling strength increases.These results can conduce to understanding the complex spatial-temporal behaviors in the coupled reaction diffusion systems.