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Interior symmetry and local bifurcation in coupled cell networks

机译:耦合细胞网络的内部对称性和局部分叉

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摘要

A coupled cell system is a network of dynamical systems, or 'cells', coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells and edges that preserves all internal dynamics and all couplings. It is well known that symmetry can lead to patterns of synchronized cells, rotating waves, multirhythms, and synchronized chaos. Recently, the introduction of a less stringent form of symmetry, the 'symmetry groupoid', has shown that global group-theoretic symmetry is not the only mechanism that can create such states in a coupled cell system. The symmetry groupoid consists of structure-preserving bisections between certain subsets of the cell network, the input sets. Here, we introduce a concept intermediate between the groupoid symmetries and the global group symmetries of a network: 'interior symmetry'. This concept is closely related to the groupoid structure, but imposes stronger constraints of a group-theoretic nature. We develop the local bifurcation theory of coupled cell systems possessing interior symmetries, by analogy with symmetric bifurcation theory. The main results are analogues for 'synchrony-breaking' bifurcations of the Equivariant Branching Lemma for steady-state bifurcation, and the Equivariant Hopf Theorem for bifurcation to time-periodic states.
机译:耦合单元系统是耦合在一起的动态系统或“单元”的网络。这样的系统可以用有向图来示意性表示,该有向图的节点对应于单元,其边缘代表耦合。耦合单元系统的对称性是单元和边缘的排列,保留所有内部动力学和所有耦合。众所周知,对称可以导致同步单元,旋转波,多重节奏和同步混沌的模式。最近,引入了一种不太严格的对称形式,即“对称类群”,表明整体组理论对称性并不是唯一可以在耦合细胞系统中产生这种状态的机制。对称类群由单元网络的某些子集(输入集)之间的保留结构的二等分组成。在这里,我们介绍了网络的群对称和全局群对称之间的中间概念:“内部对称”。这个概念与类群结构密切相关,但是对群论性质施加了更强的约束。通过类比对称分叉理论,我们发展了具有内部对称性的耦合细胞系统的局部分叉理论。主要结果是稳态分支的等变分支引理的“同步破坏”分支的类似物,以及分支为时间周期状态的等价霍夫定理。

著录项

  • 来源
    《Dynamical Systems》 |2004年第4期|p.389-407|共19页
  • 作者单位
  • 收录信息 美国《科学引文索引》(SCI);美国《工程索引》(EI);美国《生物学医学文摘》(MEDLINE);美国《化学文摘》(CA);
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 工程基础科学;
  • 关键词

  • 入库时间 2022-08-17 13:08:39

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