Hidden symmetries have been previously explored in the context of coupled cell networks and coupled cell systems. These include interior symmetry, quotient symmetry and quotient interior symmetry. We introduce here an equivariant degree theory that incorporates these different forms of hidden symmetry based on lattice structures of synchrony subspaces. The result is a unified theory capable of treating synchrony-breaking Hopf bifurcation problems in coupled cell systems with various forms of hidden symmetries, which leads to full topological classifications of bifurcating branches arising from single bifurcation points under mutual influence of these hidden symmetries.

中文翻译：

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隐藏的对称性、耦合网络和等变度

先前已经在耦合细胞网络和耦合细胞系统的背景下探索了隐藏的对称性。这些包括内部对称性、商对称性和商内部对称性。我们在这里介绍一种等变度理论，该理论基于同步子空间的晶格结构结合了这些不同形式的隐藏对称性。结果是一个统一的理论能够处理具有各种形式的隐藏对称性的耦合单元系统中的同步破坏 Hopf 分岔问题，这导致在这些隐藏对称性的相互影响下由单个分岔点产生的分岔分支的完整拓扑分类。