同步子空间的简化格及其索引
Reduced Lattices of Synchrony Subspaces and their Indices
AI总结：
本文提出了一种简化同步子空间格的方法，通过整数元组识别等价子空间并用于分支分析。
摘要：
对于一个正则耦合细胞网络，同步子空间是保持网络邻接矩阵不变的对角线子空间。一个n细胞正则网络的完整同步子空间格可以看作是n元素的划分格和关联邻接矩阵不变子空间格的交集。我们为同步子空间分配整数元组，并利用它们来识别等价的同步子空间以进行合并。基于这种等价性，初始的同步子空间格可以被简化为一个对应的简单特征值情况讨论在我们之前工作的格。结果是一个简化的同步子空间格，它提供了一个明确的非负整数索引，用于正则耦合细胞网络的分支分析。
英文摘要：
For a regular coupled cell network, synchrony subspaces are the polydiagonal subspaces that are invariant under the network adjacency matrix. The complete lattice of synchrony subspaces of an $n$-cell regular network can be seen as an intersection of the partition lattice of $n$ elements and a lattice of invariant subspaces of the associated adjacency matrix. We assign integer tuples with synchrony subspaces, and use them for identifying equivalent synchrony subspaces to be merged. Based on this equivalence, the initial lattice of synchrony subspaces can be reduced to a lattice of synchrony subspaces which corresponds to a simple eigenvalue case discussed in our previous work. The result is a reduced lattice of synchrony subspaces, which affords a well-defined non-negative integer index that leads to bifurcation analysis in regular coupled cell networks.
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