Robust partial synchronization of delay-coupled networks

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.