论文标题
同步子空间及其指数的晶格减少
Reduced Lattices of Synchrony Subspaces and their Indices
论文作者
论文摘要
对于常规的耦合单元网络，同步子空间是在网络邻接矩阵下不变的多径元子空间。 $ n $ cell常规网络的同步子空间的完整晶格可以看作是$ n $元素的分区晶格和相关邻接矩阵的不变子空间的相交。我们分配了具有同步子空间的整数元组，并使用它们来识别要合并的等效同步子空间。基于此等效性，可以将同步子空间的初始晶格还原为同步子空间的晶格，这与我们以前的工作中讨论的简单特征值相对应。结果是同步子空间的晶格降低，该子空间提供了一个定义明确的非负整数指数，该指数导致常规耦合细胞网络中的分叉分析。
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.
