Adaptive generalized function matrix projective lag synchronization of uncertain complex dynamical networks with different dimensions

Generalized function matrix projective lag synchronization of uncertain complex dynamical networks with different dimension of nodes via adaptive control method is investigated in this paper. Based on Lyapunov stability theory, adaptive controller is obtained and unknown parameters of both the drive network and the response network are estimated by adaptive laws. In addition, the three-dimension chaotic system and the four-dimension hyperchaotic system, respectively, as the nodes of the drive and response network are analyzed in detail, and numerical simulation results are presented to illustrate the effectiveness of the theoretical results.     

Complex projective synchronization in drive-response networks coupled with complex-variable chaotic systems

In drive-response complex-variable systems, projective synchronization with respect to a real number, real matrix, or even real function means that drive-response systems evolve simultaneously along the same or inverse direction in a complex plane. However, in many practical situations, the drive-response systems may evolve in different directions with a constant intersection angle. Therefore, this paper investigates projective synchronization in drive-response networks of coupled complex-variable chaotic systems with respect to complex numbers, called complex projective synchronization (CPS). The adaptive feedback control method is adopted first to achieve CPS in a general drive-response network. For a special class of drive-response networks, the CPS is achieved via pinning control. Furthermore, a universal pinning control scheme is proposed via the adaptive coupling strength method, several simple and useful criteria for CPS are obtained, and all results are illustrated by numerical examples.     

Adaptive-impulsive projective synchronization of drive-response delayed complex dynamical networks with time-varying coupling

This paper investigates the adaptive-impulsive projective synchronization of drive-response delayed complex dynamical networks with time-varying coupling, in which the weights of links between two connected nodes are time varying. By the stability analysis of the impulsive functional differential equation, the sufficient conditions for achieving projective synchronization are obtained, and a hybrid controller, that is, an adaptive feedback controller with impulsive control effects is designed. The numerical examples are presented to illustrate the effectiveness and advantage of the proposed synchronization criteria.     

Contraction theory based synchronization analysis of impulsively coupled oscillators

Impulsively coupled oscillators which are assumed to interact with each other only at discrete times have many applications in practice. In this paper, we introduce the concept of partial contraction theory of impulsive systems, which is used to investigate the synchronization problem of impulsively coupled oscillators. Contraction analysis of two impulsively coupled oscillators and networked impulsively coupled oscillators is provided, respectively. Very simple but very general results for synchronization of impulsively coupled oscillators are derived. Numerical simulations show the effectiveness of our theoretical results.     

Mixed outer synchronization of dynamical networks with nonidentical nodes and output coupling

Outer synchronization between the drive network and the response network has attracted much more attention in various fields of science and engineering. In this paper, mixed outer synchronization between two complex dynamical networks with nonidentical nodes and output coupling is investigated via impulsive hybrid control, that is, an adaptive feedback controller with impulsive control effects. Moreover, both the cases of complex networks without and with coupling delay are considered. According to the stability analysis of the impulsive functional differential equation, several sufficient conditions for the networks to achieve mixed outer synchronization are derived. Numerical examples are presented finally to illustrate the effectiveness and advantage of the proposed synchronization criteria.     

Projective synchronization of neural networks with mixed time-varying delays and parameter mismatch

In this paper, the projective synchronization of neural networks with mixed time-varying delays and parameter mismatch is discussed. Due to parameter mismatch and projective factor, complete projective synchronization cannot be achieved. Therefore, a new weak projective synchronization scheme is proposed to ensure that coupled neural networks are in a state of synchronization with an error level. Several criteria are derived and the error level is estimated by applying a generalized Halanay inequality and matrix measure. Finally, a numerical example is given to verify the efficiencies of theoretical results.     

Finite-time synchronization of complex networks with nonidentical discontinuous nodes

In this paper, we study the finite-time synchronization problem for linearly coupled complex networks with discontinuous nonidentical nodes. Firstly, new conditions for general discontinuous chaotic systems is proposed, which is easy to be verified. Secondly, a set of new controllers are designed such that the considered model can be finite-timely synchronized onto any target node with discontinuous functions. Based on a finite-time stability theorem for equations with discontinuous right-hand and inequality techniques, several sufficient conditions are obtained to ensure the synchronization goal. Results of this paper are general, and they extend and improve existing results on both continuous and discontinuous complex networks. Finally, numerical example, in which a BA scale-free network with discontinuous Sprott and Chua circuits is finite-timely synchronized onto discontinuous Chen system, is given to show the effectiveness of our new results.     

Improving synchronous ability between complex networks

We investigate the synchronization ability between complex networks and propose a near optimal connection strategy based on one connection. Numerical simulations on scale-free, small-world and random network are presented to prove the effectiveness of the proposed strategy. Furthermore, we find that the synchronization ability of the networks can be improved more largely by enhancing inter-network coupling strength than by enhancing intra-network coupling strength. We find that there is an upper limit for the synchronization ability of the complex networks, and we analyze the corresponding reason.     

Modified projective and modified function projective synchronization of a class of real nonlinear systems and a class of complex nonlinear systems

The modified projective and modified function projective synchronization of a class of chaotic real nonlinear systems, or a class of chaotic complex nonlinear systems, have been widely reported in the previous studies, respectively. In the paper, the modified projective and modified function projective synchronization between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems are first investigated. Based on the Lyapunov stability theory, the drive real system and response complex system can be synchronized up to the desired scaling constants and functions, respectively. The corresponding numerical simulations are performed to verify and demonstrate the validity and feasibility of the presented idea.     

Generalized synchronization of the fractional-order chaos in weighted complex dynamical networks with nonidentical nodes

A fractional-order weighted complex network consists of a number of nodes, which are the fractional-order chaotic systems, and weighted connections between the nodes. In this paper, we investigate generalized chaotic synchronization of the general fractional-order weighted complex dynamical networks with nonidentical nodes. The well-studied integer-order complex networks are the special cases of the fractional-order ones. Based on the stability theory of linear fraction-order systems, the nonlinear controllers are designed to make the fractional-order complex dynamical networks with distinct nodes asymptotically synchronize onto any smooth goal dynamics. Numerical simulations are provided to verify the theoretical results. It is worth noting that the synchronization effect sensitively depends on both the fractional order ?? and the feedback gain k _i . Moreover, generalized synchronization of the fractional-order weighted networks can still be achieved effectively with the existence of noise perturbation.     

Generalized heterochronous synchronization in coupled time-delayed chaotic systems

A new kind of generalized heterochronous synchronization phenomenon is reported. Different kinds of generalized synchronous states (including generalized anticipated, isochronous and lag projective synchronization) coexist among different state variables between two unidirectionally coupled time-delayed chaotic systems. The analytical conditions for generalized heterochronous synchronization are obtained. We also find that the synchronization conditions are independent of the delay times in the original time-delayed system. The theoretical results are well confirmed by the numerical simulations and electronic circuit experiments.     

Projective synchronization in a driven-response dynamical network with coupling time-varying delay

This paper studies the projective synchronization in a driven-response dynamical network with coupling time-varying delay model via impulsive control, in which the weights of links are time varying. Based on the stability analysis of the impulsive functional differential equations, some sufficient conditions for the projective synchronization are derived. Numerical simulations are provided to verify the correctness and effectiveness of the proposed method and results.     

On pinning synchronization of general coupled networks

The synchronization in general coupled networks subjected to pinning control is investigated. Some generic stability criteria based on the Lyapunov approach are derived for such general controlled networks, which guarantee that the whole network can be pinned to a synchronization state by placing feedback control on only a small fraction of nodes. A real network of television audience flows across 28 satellite channels in China and a representative BA scale-free network composed of chaotic systems are shown, respectively, for illustration and verification. It is found that pinning stability can be improved via increasing pinning density and/or pinning strength for complete diagonal inner coupling.     

Analyzing projective synchronization on different scaling factors in a drive-response coupling dynamical network with time-varying delays

In this paper, projective synchronization of drive-response coupled dynamical network with delayed system nodes and coupling time-varying delay is investigated via impulsive control, where the scaling factors are different from each other. Different controllers are designed to achieve the projective synchronization: only impulsive control is used when the scaling factors need extra limitation, while an extra controller, that is, a simple linear feedback controller, is added when the scaling factors don??t need extra limitation. Based on the stability analysis of the impulsive functional differential equation, the sufficient conditions for achieving projective synchronization of such coupled network are established, and an estimate of the upper bound of impulsive intervals ensuring global exponential synchronization of drive-response coupled dynamical network is also given. Numerical examples on the time-delay Lorenz chaotic systems are presented finally to illustrate the effectiveness and advantage of the proposed synchronization criteria.     

Projective synchronization of nonlinear-coupled spatiotemporal chaotic systems

The projective synchronization of one-dimensional discrete spatiotemporal chaotic systems is discussed in this paper. The coupling equation is determined by suitably separating the linear term and the nonlinear term of the dynamic function, and two coupled map lattices reach projective synchronization by the nonlinear-coupling method. Besides, this method is expanded to the projective synchronization of the complex network composed by coupled map lattices. Numerical simulations show the effectiveness of the scheme.     

Projective synchronization of the small-world delayed network with uncertainty

Spatiotemporal chaotic systems are taken as nodes to constitute the small-world delayed network with uncertainty, and projective synchronization of the network is researched. The control input of the network and the adaptive law of adjustment parameters are designed based on Lyapunov theorem. Concretely, the Burgers systems with spatiotemporal chaotic behavior in physics domain are taken as nodes to constitute the complex network, and the Fisher–Kolmogorov system is taken as the target system. The simulation results show that the synchronization performance of the network is very stable.     

Adaptive synchronization of T-S fuzzy complex networks with time-varying delays via the pinning control method

In this paper, the synchronization of Takagi–Sugeno (T-S) fuzzy complex networks with time-varying delays and adaptive coupling weights is studied. Using the pinning control and adaptive feedback strategy, a new general class of complex networks with fuzzy logic is proposed and its synchronization is investigated in terms of linear matrix inequalities (LMIs). The adaptive update law of coupling weight is only related to the dynamical behaviors of directly connected nodes. Based on the Lyapunov stability theory, it is proven that the synchronization of the addressed network can be achieved under those control strategies. Finally, two numerical examples are given to verify the effectiveness of our theoretical results.     

Spatiotemporal chaos synchronization between uncertain complex networks with diverse structures

Spatiotemporal chaos synchronization between uncertain complex networks with diverse structures is investigated. The identification law of unknown parameters and the adaptive law of the configuration matrix element in state equations of network nodes are determined based on stability theory, and the conditions of realizing spatiotemporal chaos synchronization between uncertain complex networks with different structures are discussed and obtained. Further, the Fisher–Kolmogorov system with spatiotemporal chaotic behavior is taken as the nodes of drive and response networks to imitate the experiment. It is found that the synchronization performance between two networks is very stable.     

Function projective synchronization of impulsive neural networks with mixed time-varying delays

In this paper, the exponential function projective synchronization of impulsive neural networks with mixed time-varying delays is investigated. Based on the contradiction method and analysis technique, some novel criteria are obtained to guarantee the function projective synchronization of considered networks via combining open-loop control and linear feedback control. As some special cases, several control strategies are given to ensure the realization of complete synchronization, anti-synchronization, and the stabilization of the addressed neural networks. Finally, two examples and their numerical simulations are given to show the effectiveness and feasibility of the proposed synchronization schemes.     

Synchronization in complex dynamical networks with interval time-varying coupling delays

This paper deals with the synchronization problem of complex dynamical networks with interval time-varying coupling delays. A simple local linear feedback controller is introduced to guarantee the synchronizability of the networks. Some delay-dependent synchronization conditions for the controlled complex dynamical networks are presented by using the Lyapunov–Krasovskii functional method and the reciprocally convex combination approach. Theoretical analysis and numerical examples show that the obtained conditions have less computational complexity and less conservatism than some recently reported ones.