SIAM Review, Volume 62, Issue 4, Page 817-836, January 2020.
The field of network synchronization has seen tremendous growth following the introduction of the master stability function (MSF) formalism, which enables the efficient stability analysis of synchronization in large oscillator networks. However, to make further progress we must overcome the limitations of this celebrated formalism, which focuses on global synchronization and requires both the oscillators and their interaction functions to be identical, while many systems of interest are inherently heterogeneous and exhibit complex synchronization patterns. Here, we establish a generalization of the MSF formalism that can characterize the stability of any cluster synchronization pattern, even when the oscillators and/or their interaction functions are nonidentical. The new framework is based on finding the finest simultaneous block diagonalization of matrices in the variational equation and does not rely on information about network symmetry. This leads to an algorithm that is error-tolerant and orders of magnitude faster than existing symmetry-based algorithms. As an application, we rigorously characterize the stability of chimera states in networks with multiple types of interactions.

中文翻译：

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同步模式的与对称无关的稳定性分析

SIAM评论，第62卷，第4期，第817-836页，2020年1月。
在引入主稳定性函数（MSF）形式主义之后，网络同步领域得到了巨大的发展，该形式可以在大型振荡器网络中进行高效的同步稳定性分析。但是，要取得进一步的进展，我们必须克服这种着名的形式主义的局限性，后者注重全局同步，并且要求振荡器及其交互功能都相同，而许多关注的系统本来就是异构的，并且表现出复杂的同步模式。在这里，我们建立了MSF形式化的概括，可以描述任何群集同步模式的稳定性，即使在振荡器和/或其相互作用函数不相同时也是如此。新框架基于在变分方程中找到矩阵的最佳同时块对角化，并且不依赖于有关网络对称性的信息。与现有的基于对称的算法相比，这导致了一种容错算法，并且速度更快。作为一种应用，我们严格描述了具有多种类型相互作用的网络中嵌合态的稳定性。