耦合动力系统网络的特征稳定性分析
Signature stability analysis for networks of coupled dynamical systems with Hermitian Jacobian
摘要：
复杂系统研究的中心主题是从微观组成部分的相互作用中理解系统的宏观特性。在这里，我们要问的是，一个微观动力学单元的复杂网络必须满足什么条件才能允许宏观动力学的稳定，比如稳定的平衡或锁相状态。我们提出了一种分析方法，这是基于一个图形符号，允许重写雅可比的签名标准在一个可解释的形式。推导出的条件适用于所有尺度上的拓扑结构，从单个节点到整个交互网络。我们的方法可以适用于许多系统的对称耦合单元。为了说明的目的，我们考虑同步的例子，特别是（异构）仓本模型和自适应变体。此外，我们还讨论了如何利用图记法来研究埃尔米特矩阵的等谱性。结果完成并扩展了Do等人的先前分析[Phys. Rev. Lett. 108，194102（2012）]。
英文摘要：
The central theme of complex systems research is understanding the emergent macroscopic properties of a system from the interplay of its microscopic constituents. Here, we ask what conditions a complex network of microscopic dynamical units has to meet to permit stationary macroscopic dynamics, such as stable equilibria or phase-locked states. We present an analytical approach which is based on a graphical notation that allows rewriting Jacobi's signature criterion in an interpretable form. The derived conditions pertain to topological structures on all scales, ranging from individual nodes to the interaction network as a whole. Our approach can be applied to many systems of symmetrically coupled units. For the purpose of illustration, we consider the example of synchronization, specifically the (heterogeneous) Kuramoto model and an adaptive variant. Moreover, we discuss how the graphical notation can be employed to study isospectrality in Hermitian matrices. The results complete and extend the previous analysis of Do et al. [Phys. Rev. Lett. 108, 194102 (2012)].
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