论文标题
非正常，最优性和同步
Non-normality, optimality and synchronization
论文作者
论文摘要
人们已经认识到，对于某些矩阵，光谱不足以讲述系统动力学的完整故事，即使是线性odes。虽然特征值确实控制系统的渐近行为，但如果代表系统的矩阵是非正态的，则短期瞬变可能会出现在线性系统中。最近，人们已经认识到，由于代表定向网络的矩阵是非正常的，因此仅基于光谱的分析可能会产生误导。根据主稳定性范式，正常系统和非正常系统都可以稳定，但是非正常系统可能具有对同步状态的任意吸引盆地，而等效的正常系统可能具有明显较大的同步盆地。这表明需要更仔细地研究非正态网络中的同步。在这项工作中，将利用各种工具来检查定向网络中的同步，包括Pseudospectra，这是我们称为Laplacian Pseudospectra的Pseudospectra的改编。我们定义了一个被称为拉普拉斯伪谱弹性（LPR）的概念。可以证明，LPR优于其他标量措施，用于估计同步状态在称为最佳网络的一类网络中的有限扰动的稳定性。最后，我们发现，最佳网络的理想选择，旨在同步，是使LPR最小化的理想选择
It has been recognized for quite some time that for some matrices the spectra are not enough to tell the complete story of the dynamics of the system, even for linear ODEs. While it is true that the eigenvalues control the asymptotic behavior of the system, if the matrix representing the system is non-normal, short term transients may appear in the linear system. Recently it has been recognized that since the matrices representing directed networks are non-normal, analysis based on spectra alone may be misleading. Both a normal and a non-normal system may be stable according to the master stability paradigm, but the non-normal system may have an arbitrarily small attraction basin to the synchronous state whereas an equivalent normal system may have a significantly larger sync basin. This points to the need to study synchronization in non-normal networks more closely. In this work, various tools will be utilized to examine synchronization in directed networks, including pseudospectra, an adaption of pseudospectra that we will call Laplacian pseudospectra. We define a resulting concept that we call Laplacian pseudospectral resilience (LPR). It will be shown that LPR outperforms other scalar measures for estimating the stability of the synchronous state to finite perturbations in a class of networks known as optimal networks. Finally we find that the ideal choice of optimal network, with an eye toward synchronization, is the one which minimizes LPR
