Patterns of Synchrony in Coupled Cell Networks with Multiple Arrows

动力系统理论 等价类(音乐) 等价关系 数学 多稳态 Lift(数据挖掘) 拓扑(电路) 计算机科学 纯数学 非线性系统 物理 组合数学 量子力学 数据挖掘
作者
Martin Golubitsky,Ian Stewart,Andrei Török
出处
期刊:Siam Journal on Applied Dynamical Systems [Society for Industrial and Applied Mathematics]
卷期号:4 (1): 78-100 被引量:264
标识
DOI:10.1137/040612634
摘要

A coupled cell system is a network of dynamical systems, or "cells," coupled together. The architecture of a coupled cell network is a graph that indicates how cells are coupled and which cells are equivalent. Stewart, Golubitsky, and Pivato presented a framework for coupled cell systems that permits a classification of robust synchrony in terms of network architecture. They also studied the existence of other robust dynamical patterns using a concept of quotient network. There are two difficulties with their approach. First, there are examples of networks with robust patterns of synchrony that are not included in their class of networks; and second, vector fields on the quotient do not in general lift to vector fields on the original network, thus complicating genericity arguments. We enlarge the class of coupled systems under consideration by allowing two cells to be coupled in more than one way, and we show that this approach resolves both difficulties. The theory that we develop, the "multiarrow formalism," parallels that of Stewart, Golubitsky, and Pivato. In addition, we prove that the pattern of synchrony generated by a hyperbolic equilibrium is rigid (the pattern does not change under small admissible perturbations) if and only if the pattern corresponds to a balanced equivalence relation. Finally, we use quotient networks to discuss Hopf bifurcation in homogeneous cell systems with two-color balanced equivalence relations.
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