Pinning‐controlled synchronization of hybrid‐coupled complex dynamical networks with mixed time‐delays

This paper investigates the pinning‐controlled synchronization of a general complex dynamical network with hybrid coupling, which includes constant coupling, discrete‐delay coupling and distributed‐delay coupling. Furthermore, the network can be composed of coupled identical nonlinear oscillators with or without internal time‐delay. To be more consistent with the realistic networks, the internal time‐delay, the discrete time‐delay and the distributed time‐delay can be different from each other, the inner coupling matrices are not necessarily to be diagonal, and the coupling configuration matrices are not required to be symmetric or irreducible. Under some sufficient conditions, it is shown that a hybrid‐coupled complex network with or without internal time‐delay can be asymptotically pinned to a homogenous trajectory by applying adaptive control actions to a small fraction of network nodes. In particular, the paper addresses what kind of nodes should be pinned and how many nodes are needed to be pinned to achieve synchronization in a hybrid‐coupled network with fixed coupling matrices and strengths. Numerical examples are given to verify the theoretical analysis. Copyright © 2011 John Wiley & Sons, Ltd.