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This article proposes two control strategies to realize matrix projective synchronization (MPS) for a class of discrete-time complex dynamical networks (DTCDNs). It is worth pointing out that our network model is composed of nodes with different state dimensions and its outer coupling configuration matrix can be asymmetric. In addition, since a small percentage of nodes that play more important role than others usually exist in a network, thus, one important node, which is named as crucial node in this paper, is taken into account in our network model. Besides, the commonality between each node and the crucial node is also precisely expressed. Further, based on the crucial node and the commonality and associated with the Lyapunov stability theory, two control strategies are addressed to realize the exponential matrix projective synchronization (EMPS) and asymptotic matrix projective synchronization (AMPS), respectively. It should be stressed that only the crucial node is controlled and only the state of the crucial node is applied to design the EMPS and AMPS controllers. This is promising to reduce the control cost as much as possible and to improve the feasibility of our MPS strategies. The effectiveness and feasibility of our MPS schemes are rigorously proved in theory as well as verified by two numerical examples.

This paper investigates the matrix projective synchronization (MPS) problem for time-varying disturbed complex dynamical networks (CDNs) with uncertain nonlinear structures and different dimensional nodes. In order to well describe the practical networks, the unavoidable unknown nonlinear structure and the inevitable uncertain disturbance of each node are considered into our network model, which is different from the previous works. Besides, it is worth pointing out that the outer coupling configuration matrix, which represents the coupling strength and the topological structure, is not restricted by the dissipatively coupled condition in this paper. The definition of MPS is also introduced for the networks with different dimensional nodes. Moreover, several MPS schemes are respectively put forward for our network model according to the norm bound of the uncertain nonlinear structures and disturbances being unknown or not. Finally, two proper examples associated with numerical simulations are given to verify the effectiveness of our theoretical results.

This paper investigates the generalized outer synchronization (GOS) between two non-dissipatively coupled complex dynamical networks (CDNs) with different time-varying coupling delays. Our drive-response networks also possess nonlinear inner coupling functions and time-varying outer coupling configuration matrices. Besides, in our network models, the nodes in the same network are nonidentical and the nodes in different networks have different state dimensions. Asymptotic generalized outer synchronization (AGOS) and exponential generalized outer synchronization (EGOS) are defined for our CDNs. Our main objective in this paper is to design AGOS and EGOS controllers for our drive-response networks via the open-plus-closed-loop control technique. Distinguished from most existing literatures, it is the partial intrinsic dynamics of each node in response network that is restricted by the QUAD condition, which is easy to be satisfied. Representative simulation examples are given to verify the effectiveness and feasibility of our GOS theoretical results in this paper.

We consider a nonlinear Neumann problem driven by a nonhomogeneous differential operator and an indefinite potential. Using variational methods together with flow invariance arguments, we show that the problem has at least one nodal solution. The result presented in this paper gives an answer to the open question raised by Papageorgiou and Radulescu (2016).

This paper investigates the generalized matrix projective outer synchronization (GMPOS) between two non-dissipatively coupled time-varying complex dynamical networks via open-plus- closed-loop dynamical compensation controllers. To be more consistent with the real-world networks, besides non-dissipatively couplings, the drive and response networks in our paper can possess different nodes, different time-varying outer coupling configuration matrices and different nonlinear inner coupling functions. Thus, our network models are more general and extensive than almost all of those in the existing literatures about outer synchronization of networks. In order to make our drive and response networks realize the GMPOS, the open-plus-closed-loop dynamical compensation controllers are designed based on the Lyapunov stability theory and Barbalat's lemma. Moreover, the speed of achieving the GMPOS between two networks can be improved by adjusting the parameters in our dynamical compensation systems. A simulation example with different hyperchaotic nodes is given to verify the effectiveness and feasibility of our theoretical results.