Pinning adaptive synchronization of general time-varying delayed and multi-linked networks with variable structures

This paper is concerned with the issue of pinning synchronization of a general model of complex dynamical networks, which can well describe practical architectures of many realistic complex systems. Compared with some existing works, the distinctive feature of the considered model in this paper includes (i) the network topologies with variable structures corresponding directed graph; (ii) the multi-linked configuration with nonlinear coupling and time-varying delays. Some generic criteria for pinning global synchronization of such dynamical network are presented based on Lyapunov stability theory on delayed dynamical systems. It is shown that these criteria can provide a novel and effective adaptive pinning strategy, which is very convenient to implement in practice since the design of the adaptive control law is independent of time-varying delays. Furthermore, it is interesting to find that when the nodes with low in-degrees are pinned firstly, the pinning control scheme is more efficient. Subsequently, the theoretic results are applied to a general two-linked network consisting of Hopfield neuron oscillator. Finally, numerical simulations demonstrate that the proposed pinning adaptive synchronization criteria are practical and effective to pin two-linked Hopfield neural networks to an equilibrium, periodic orbit and chaotic attractor.