Cluster synchronization of two-layer nonlinearly coupled multiplex networks with multi-links and time-delays

The cluster synchronization problem of two-layer nonlinearly coupled multiplex networks with multi-links and time-delays is investigated in this paper. The main feature of this literature can be summarized as follows: (i) introduce a double-layer multiplex networks composed of many pairs of matching leaders' subnets and followers' subnets with multi-links, which gives much more chances for these leaders to transmit information to their matching followers; (ii) configure the multi-links with nonlinear coupling and time-varying delays; (iii) give out the designed pinning scheme that the nodes with low degrees are in need of applying controllers first and some appropriate adaptive update laws, which are used for processing the unknown feedback factors between the pinned nodes and their leaders; (iv) free from the general constraints on the node sizes in each subnet pairs. In view of Lyapunov stability theory of delayed dynamical systems and with the design of an adaptive pinning strategy, some cluster synchronization criteria of two-layer multi-linked multiplex network are proposed. Both theoretical analysis and numerical simulations demonstrate that the proposed adaptive pinning scheme is feasible and effective. (C) 2019 Elsevier B.V. All rights reserved.