AI 中文总结
该研究针对给定图结构的微分方程网络，探讨选择拉普拉斯算子以优化同步性的问题，给出三对角及带状拉普拉斯算子满足同步解渐近稳定、最小化特征值归一化 spread、缩短暂态的条件，并通过数值结果验证。
AI 中文摘要
在本研究中，针对具有给定图结构的微分方程网络，我们的目标是展示如何选择网络的拉普拉斯算子以获得最有利的同步性结果，即需满足：（i）保证同步解的渐近稳定性（通过主稳定函数的负值衡量）；（ii）最小化拉普拉斯特征值的归一化 spread；（iii）使暂态尽可能短。在三对角拉普拉斯算子类别内，我们给出满足上述三个准则的充要条件，并扩展至带状拉普拉斯算子。最后，我们给出大量数值结果以阐明理论结果并与现有工作对比。
英文摘要
In this work, for a network of differential equations with a prescribed graph structure, our goal is to show how to select the Laplacian of the network in order to obtain the most favorable outcome insofar as synchronizability. That is, we will want to: (i) guarantee asymptotic stability of a synchronous solution (as measured by a negative value of the master stability function), (ii) minimize the normalized spread of the Laplacian eigenvalues, and (iii) have a transient as short as possible. Within the class of tridiagonal Laplacians, we give both necessary and sufficient conditions for satisfying our three criteria above, and give extension to banded Laplacians as well. Finally, we give extensive numerical results to elucidate our theoretical results and to compare to existing works.
Comments10 pages, 6 figures, v2: Corrected a one-word omission in the statement of Theorem 5; results unchanged
