Symmetric patterns in linear arrays of coupled cells
Symmetric patterns in linear arrays of coupled cells
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DOI：
10.1063/1.165974
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发表时间：
1993-01-01
期刊：
影响因子：
2.9
通讯作者：
Golubitsky, Martin
中科院分区：
文献类型：
--
作者：
Epstein, Irving R.;Golubitsky, Martin
在这篇文章中，我们将展示如何在耦合单元的线性阵列中找到模式化的解决方案。通过将系统嵌入到具有两倍单元数的圆形阵列中来找到解决方案。单个细胞具有独特的稳定状态，因此图案化溶液代表连续介质中图灵结构的离散模拟。然后，我们使用的对称性的圆形阵列（和分歧从一个不变的平衡），以确定对称的圆形阵列的解决方案，限制到原来的线性阵列的解决方案。我们将这些抽象的结果耦合到一个系统，证明模式化的解决方案的存在。此外，我们表明，在某些情况下，这些图案的解决方案可以通过数值积分，因此可能是渐近稳定的。
In this note we show how to find patterned solutions in linear arrays of coupled cells. The solutions are found by embedding the system in a circular array with twice the number of cells. The individual cells have a unique steady state, so that the patterned solutions represent a discrete analog of Turing structures in continuous media. We then use the symmetry of the circular array (and bifurcation from an invariant equilibrium) to identify symmetric solutions of the circular array that restrict to solutions of the original linear array. We apply these abstract results to a system of coupled Brusselators to prove that patterned solutions exist. In addition, we show, in certain instances, that these patterned solutions can be found by numerical integration and hence are presumably asymptotically stable.