SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 366-411, January 2020.
Center manifold reduction is a standard technique in bifurcation theory, reducing the essential features of local bifurcations to equations in a small number of variables corresponding to critical eigenvalues. This method can be applied to admissible differential equations for a network, but it bears no obvious relation to the network structure. A fully inhomogeneous network is one in which all nodes and couplings can be different. For this class of networks, there are general circumstances in which the center manifold reduced equations inherit a network structure of their own. This structure arises by decomposing the network into path components, which connect to each other in a feedforward manner. Critical eigenvalues can then be associated with specific components, and the network structure on the center manifold depends on how these critical components connect within the network. This observation is used to analyze codimension-1 and codimension-2 local bifurcations. For codimension-1, only one critical component is involved, and generic local bifurcations are saddle-node and standard Hopf. For codimension-2, we focus on the case when one component is downstream from the other in the feedforward structure. This gives rise to four cases: steady or Hopf upstream combined with steady or Hopf downstream. Here the generic bifurcations, within the realm of network-admissible equations, differ significantly from generic codimension-2 bifurcations in a general dynamical system. In each case, we derive singularity-theoretic normal forms and unfoldings, present bifurcation diagrams, and tabulate the bifurcating states and their stabilities.

中文翻译：

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完全不均匀网络上的分歧

SIAM应用动力系统杂志，第19卷，第1期，第366-411页，2020年1月。
中心流形约简是分叉理论中的一种标准技术，它将局部分叉的基本特征简化为方程式中的少量与临界特征值相对应的变量。该方法可以应用于网络的容许微分方程，但与网络结构没有明显的关系。完全不均匀的网络是其中所有节点和耦合都可以不同的网络。对于此类网络，在一般情况下，中心流形归约方程继承其自身的网络结构。这种结构是通过将网络分解为路径组件而形成的，这些路径组件以前馈方式相互连接。然后可以将关键特征值与特定组件相关联，中央歧管上的网络结构取决于这些关键组件在网络中的连接方式。该观察结果用于分析codimension-1和codimension-2局部分支。对于codimension-1，仅涉及一个关键组件，而通用的局部分支是鞍形节点和标准Hopf。对于codimension-2，我们关注前馈结构中一个组件位于另一组件下游的情况。这产生了四种情况：稳态或Hopf上游与稳态或Hopf下游相结合。在此，在网络允许方程范围内，一般分叉与一般动力系统中的一般codimension-2分叉明显不同。在每种情况下，我们导出奇异理论的法线形式和展开，当前的分叉图，