Finite/fixed-time cluster synchronization for directed and multiplex coupled dynamic networks

This article deals with finite/fixed-time cluster synchronization (FnTCS/FdTCS) issue for directed and multiplex coupled dynamic networks (MCDNs). Finite/fixed-time synchronization of multiplex networks has become a popular research topic, which indicates that, state information is utilized for synchronization, and the settling time depends on original condition for a range of finite time, while the time is bounded with arbitrary original value for fixed-time one. Through constructing appropriate feedback control protocol, FnTCS and FdTCS matters for MCDNs are settled under re-arranging variables' order technique. The present model contains the case that outer matrices (OMs) can be directed, with competitive elements, and not even connected, while previous and related researches assumed that OMs were undirected, cooperative and connected, so existing results can be improved for more general and broader regulations. This strategy above is presented to solve directed CDNs with multiweights from angle of inner and outer matrices, and we prove that if the weighted group of added OMs for each dimension is strongly connected, then cluster synchronization rules are established in finite/fixed time. Moreover, we also earn finite/fixedtime synchronization criteria when the cluster is single. In addition, the issue of FnTCS/FdTCS is developed for directed and multiplex coupled reaction-diffusion networks (MCRDNs) as an application. The validity of these gained results is verified by simulated examples.