Matrix projective cluster synchronization for arbitrarily coupled networks with different dimensional nodes via nonlinear control

Recently, many control mechanisms have been put forward to realize the cluster synchronization of networks. However, these results are derived by making very conservative assumptions on the network's nodes and its outer coupling configuration matrix, which greatly limit these results' potential applications. This paper focuses on weakening these restrictive assumptions as much as possible and further proposes nonlinear controllers for our general networks to achieve matrix projective cluster synchronization (MPCS). One outstanding characteristic of our introduced network model is that the state dimensions of the nodes in different clusters are different, which extends the case that the network is composed of identical dimensional nodes. Another distinctive characteristic is that no other restrictions are imposed on our network's outer coupling matrix except for being bounded, namely, both the outer coupling matrix and its block matrices may be time varying and nonzero-row-sum. Besides, the exponential MPCS (EMPCS) and asymptotic MPCS (AMPCS) are defined to make a better description of the cluster synchronization phenomena in this paper. It is worth noted that these two definitions extend the existing ones about cluster synchronization. Furthermore, by means of rigorous theoretical analysis, some nonlinear control strategies are proposed for our network to realize EMPCS or AMPCS. At last, a proper numerical example along with two control patterns is presented to demonstrate the validity and effectivity of our results in this paper.