Nonlinear dynamics of networks: The groupoid formalism
Nonlinear dynamics of networks: The groupoid formalism
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DOI：
10.1090/s0273-0979-06-01108-6
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发表时间：
2006-01-01
影响因子：
1.3
通讯作者：
Stewart, Ian
中科院分区：
文献类型：
--
作者：
Golubitsky, Martin;Stewart, Ian
一种关于耦合动力系统网络对称性的形式化理论，用保持网络拓扑结构的节点排列组来表述，已经存在了一段时间。全局网络对称性对相应的动力系统施加了很强的约束，影响平衡状态、周期状态、异斜循环甚至混沌状态。特别是，网络的对称性会导致同步、相位关系、共振以及同步或循环混沌。对称性是一个相当严格的假设，而网络的一般理论应该更加灵活。最近对群论对称概念的推广，用网络有向边的某些子集之间的对射取代了整体对称，即“输入集”。现在对称群变成了群样，它是一种类似于群的代数结构，只是两个元素的乘积可以不被定义。类群形式化使得将群论方法扩展到更一般的网络成为可能，特别是它根据网络的组合结构对同步的“健壮”模式进行了完整的分类。许多在任意动力系统中是非一般的现象在受到特定网络拓扑结构的约束时可以变成一般的现象。动力系统网络不仅仅是一个具有高维相空间的动力系统。它还配备了一组规范的可观察对象——网络中各个节点的状态。此外，底层ODE的形式受到网络拓扑的约束——哪些变量出现在哪些组件方程中，以及这些方程如何相互关联。其结果是产生了一系列丰富而新颖的现象，其中只有少数尚未得到适当的理解。
A formal theory of symmetries of networks of coupled dynamical systems, stated in terms of the group of permutations of the nodes that preserve the network topology, has existed for some time. Global network symmetries impose strong constraints on the corresponding dynamical systems, which affect equilibria, periodic states, heteroclinic cycles, and even chaotic states. In particular, the symmetries of the network can lead to synchrony, phase relations, resonances, and synchronous or cycling chaos. Symmetry is a rather restrictive assumption, and a general theory of networks should be more flexible. A recent generalization of the group-theoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the directed edges of the network, the 'input sets'. Now the symmetry group becomes a groupoid, which is an algebraic structure that resembles a group, except that the product of two elements may not be defined. The groupoid formalism makes it possible to extend group-theoretic methods to more general networks, and in particular it leads to a complete classification of 'robust' patterns of synchrony in terms of the combinatorial structure of the network. Many phenomena that would be nongeneric in an arbitrary dynamical system can become generic when constrained by a particular network topology. A network of dynamical systems is not just a dynamical system with a high-dimensional phase space. It is also equipped with a canonical set of observables - the states of the individual nodes of the network. Moreover, the form of the underlying ODE is constrained by the network topology - which variables occur in which component equations, and how those equations relate to each other. The result is a rich and new range of phenomena, only a few of which are yet properly understood.