Su Libo, Wei Yanling, Michiels Wim, Steur Erik, Nijmeijer Henk
Department of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001 Heverlee, Belgium.
Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.
Chaos. 2020 Jan;30(1):013126. doi: 10.1063/1.5111745.
Networks of coupled systems may exhibit a form of incomplete synchronization called partial synchronization or cluster synchronization, which refers to the situation where only some, but not all, systems exhibit synchronous behavior. Moreover, due to perturbations or uncertainties in the network, exact partial synchronization in the sense that the states of the systems within each cluster become identical, cannot be achieved. Instead, an approximate synchronization may be observed, where the states of the systems within each cluster converge up to some bound, and this bound tends to zero if (the size of) the perturbations tends to zero. In order to derive sufficient conditions for this robustified notion of synchronization, which we refer to as practical partial synchronization, first, we separate the synchronization error dynamics from the network dynamics and interpret them in terms of a nonautonomous system of delay differential equations with a bounded additive perturbation. Second, by assessing the practical stability of this error system, conditions for practical partial synchronization are derived and formulated in terms of linear matrix inequalities. In addition, an explicit relation between the size of perturbation and the bound of the synchronization error is provided.
耦合系统网络可能会呈现出一种不完全同步的形式，称为部分同步或集群同步，它指的是只有部分而非所有系统表现出同步行为的情况。此外，由于网络中的扰动或不确定性，无法实现每个集群内系统状态完全相同意义上的精确部分同步。相反，可能会观察到一种近似同步，其中每个集群内系统的状态收敛到某个界限，并且如果扰动（的大小）趋于零，这个界限也趋于零。为了推导这种强化同步概念（我们称之为实际部分同步）的充分条件，首先，我们将同步误差动态与网络动态分离，并将它们解释为具有有界加性扰动的延迟微分方程的非自治系统。其次，通过评估这个误差系统的实际稳定性，根据线性矩阵不等式推导并制定实际部分同步的条件。此外，还提供了扰动大小与同步误差界限之间的明确关系。
