The lattice of balanced equivalence relations of a coupled cell network
The lattice of balanced equivalence relations of a coupled cell network
复制标题
耦合单元网络的平衡等价关系格
DOI：
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发表时间：
2007
影响因子：
0.8
通讯作者：
I. Stewart
中科院分区：
文献类型：
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作者：
I. Stewart
摘要 耦合单元系统是耦合在一起的动力系统或“单元”的集合。关联的耦合单元网络是一个带标签的有向图，指示单元如何耦合以及哪些单元是等效的。 Golubitsky、Stewart、Pivato 和 Török 提出了耦合单元系统的框架，该框架允许根据“平衡等价关系”的概念对稳健同步进行分类，该概念仅取决于网络架构。在他们的方法中，假设网络是有限的。我们证明网络上所有平衡等价关系的集合形成一个格，在偏序集合的意义上，其中任何两个元素都有相交和连接。偏序是通过细化来定义的。该理论的某些方面利用了无限网络，因此我们在“有限类型”网络类别中工作，这一类包括所有局部有限网络。这种情况需要对标准框架进行一些修改。作为部分补偿，平衡等价关系格可以被证明是完整的。然而，正如我们通过一个简单的例子所示，两个平衡的等价关系的交集不需要平衡，因此该格不是所有等价关系的格及其通常的相遇和连接操作的子格。我们讨论该晶格的结构以及与之相关的计算问题。特别地，我们描述了如何确定格是否包含除等式关系之外的内容。作为一个例子，我们推导出具有反馈的相同细胞线性链的晶格形式。
Abstract A coupled cell system is a collection of dynamical systems, or ‘cells’, that are coupled together. The associated coupled cell network is a labelled directed graph that indicates how the cells are coupled, and which cells are equivalent. Golubitsky, Stewart, Pivato and Török have presented a framework for coupled cell systems that permits a classification of robust synchrony in terms of the concept of a ‘balanced equivalence relation’, which depends solely on the network architecture. In their approach the network is assumed to be finite. We prove that the set of all balanced equivalence relations on a network forms a lattice, in the sense of a partially ordered set in which any two elements have a meet and a join. The partial order is defined by refinement. Some aspects of the theory make use of infinite networks, so we work in the category of networks of ‘finite type’, a class that includes all locally finite networks. This context requires some modifications to the standard framework. As partial compensation, the lattice of balanced equivalence relations can then be proved complete. However, the intersection of two balanced equivalence relations need not be balanced, as we show by a simple example, so this lattice is not a sublattice of the lattice of all equivalence relations with its usual operations of meet and join. We discuss the structure of this lattice and computational issues associated with it. In particular, we describe how to determine whether the lattice contains more than the equality relation. As an example, we derive the form of the lattice for a linear chain of identical cells with feedback.