由图设计异宿网络
On designing heteroclinic networks from graphs
摘要：
鲁棒异宿网络是可以在对称耦合或其他约束动力系统中作为吸引子出现的不变集。这些网络可能有一个非常复杂的结构，人们对它的了解很少，在很大程度上是由系统的约束和维度决定的。由于这些网络作为生物和认知过程的动力学模型非常有趣，因此了解特定的图如何实现为吸引人的强大异宿网络是有用的。本文提出了两种方法实现任意复杂的有向图的鲁棒异宿网络的ODE生成的流-我们说的ODE { \em实现}的图作为异宿网络之间的平衡，代表的顶点。假设我们有一个在$n_v$顶点上的有向图，它的边是$n_e$。“单纯形实现”将图嵌入为流在$(n_v-1)$ -单纯形上的不变集。只要图形没有一个周期和两个周期，该方法就会实现该图形。“柱面实现”将一个图作为流在$(n_e+1)$维空间上的不变集嵌入。这种方法实现的图形，只要它是一个周期自由。在这两种情况下，我们发现图作为一个不变集内的吸引子，并讨论了一些说明性的例子，包括噪声和参数的动态的影响。特别是，我们表明，由此产生的异宿网络可能会或可能不会显示“内存”的顶点访问。
英文摘要：
Robust heteroclinic networks are invariant sets that can appear as attractors in symmetrically coupled or otherwise constrained dynamical systems. These networks may have a very complicated structure that is poorly understood and determined to a large extent by the constraints and dimension of the system. As these networks are of great interest as dynamical models of biological and cognitive processes, it is useful to understand how particular graphs can be realised as robust heteroclinic networks that are attracting. This paper presents two methods of realizing arbitrarily complex directed graphs as robust heteroclinic networks for flows generated by ODEs---we say the ODEs {\em realise} the graphs as heteroclinic networks between equilibria that represent the vertices. Suppose we have a directed graph on $n_v$ vertices with $n_e$ edges. The "simplex realisation" embeds the graph as an invariant set of a flow on an $(n_v-1)$-simplex. This method realises the graph as long as it is one- and two-cycle free. The "cylinder realisation" embeds a graph as an invariant set of a flow on a $(n_e+1)$-dimensional space. This method realises the graph as long as it is one-cycle free. In both cases we find the graph as an invariant set within an attractor, and discuss some illustrative examples, including the influence of noise and parameters on the dynamics. In particular we show that the resulting heteroclinic network may or may not display "memory" of the vertices visited.
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