Generalized projective synchronization for networks with one crucial node and different dimensional nodes via a single controller

This paper investigates the generalized projective synchronization (GPS) among the nodes within a complex dynamical network (CDN). Multiple characteristics of realistic networks are simultaneously considered in our network model. First, one crucial node is introduced into our CDN, which behaves like a leader and exists in many practical networks. Second, the commonalities between the crucial node and every other node are discussed and they are embodied in the network model. Third, our CDN is composed of different dimensional nodes and has an overall time-varying coupling strength. Furthermore, in order to describe more synchronous phenomena, the exponential generalized projective synchronization (EGPS) and asymptotic generalized projective synchronization (AGPS) are defined. The main objective of this paper is to design an effective but simple decentralized controller for the crucial node such that our network realizes GPS. Based on rigorous theoretical derivations, EGPS and AGPS schemes are respectively proposed according to whether the overall coupling strength is certain or adaptive. Few previous studies have focused on this. It is worth pointing out that both the control input and the adaptive law of the coupling strength in this paper only lie in the information of the crucial node, which implies the significance of the crucial node to some extent. Finally, the effectiveness and feasibility of our theoretical results are verified by two numerical simulation examples.