Pinning synchronisation in fixed and switching directed networks of Lorenz-type nodes

This study addresses the global pinning synchronisation of directed networks with Lorenz-type node dynamics. By using tools from M-matrix theory and Lyapunov stability theory, the interesting issues of what kind of nodes should be pinned and how large the control strength between neighbouring nodes should be selected for achieving global synchronisation in both fixed and switching networks are clearly addressed. It is theoretically shown that global pinning synchronisation of an arbitrarily given fixed network can be achieved if the network topology contains a directed spanning tree and the coupling strength is larger than a derived critical value depending both on the node dynamics and the network topology. By suitably constructing multiple Lyapunov functions, it is further proved that global pinning synchronisation in switching networks with a suitable coupling strength can be guaranteed if each possible network topology contains a directed spanning tree and the dwell time of switching is less than a positive threshold. Numerical examples are finally given for illustration.