Hybrid combinatorial synchronization on multiple sub-networks of complex network with unknown boundaries of uncertainties

In this paper, hybrid combinatorial synchronization on multiple sub-networks of complex network with unknown boundaries of uncertainties, stochastic disturbances and different dimensions of nodes is investigated. The complex network can be divided into multiple sub-networks, such that the nodes belonging to the same sub-network are identical, while the ones belonging to different sub-networks are non-identical. Each sub-network is composed of two types of nodes: general node and preset-node, and only the nodes with the same type can communicate with each other. In this new setting, the aim is to design some suitable controllers on the preset-nodes, which are taken as the being synchronized nodes in each sub-network, so as to reach not only the combinatorial inner synchronization within a sub-network, but also the combinatorial outer synchronization between multiple different sub-networks. Furthermore, the nodes may be influenced by some stochastic disturbances and uncertainties, and the boundaries of these uncertainties are not always known in advance, appropriate adaptive update laws are used to deal with them. Based on the Lyapunov stability theory and adaptive control method, some synchronization criteria on multiple sub-networks of complex network are established. Numerical simulations are given to demonstrate the feasibility and validity of the proposed scheme.