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.     

Adaptive lag synchronization in coupled chaotic systems with unidirectional delay feedback

Lag synchronization in unidirectionally coupled chaotic systems is investigated in this paper. Based on the invariance principle of differential equations, a new adaptive delay feedback scheme is proposed to realize the lag synchronization effectively in the coupled chaotic systems. As an example, numerical simulations for the coupled Hindmarsh-Rose (HR) neuron models are conducted, which is in good agreement with the theoretical analysis. More interestingly, it is found that there is a fine U-shaped structure in the lag synchronization curve for the HR neuron model. Furthermore, lag synchronization and the corresponding U-shaped structure are robust against the small mismatch of parameters and noisy disturbances.     

Secure communication using chaotic synchronization in mutually coupled semiconductor lasers

In this paper, we propose a scheme of the secure communication systems, in the frame of the chaotic synchronization of two semiconductor lasers with optical feedback in mutually coupled configuration within the new modified rate equation of lasers by using the numerical simulations. We explore the robustness of this synchronization scheme with parameter mismatches. A high sensitivity of the synchronization of the lasers via the new control parameter mismatch is discussed in the temporal and frequency domains. Moreover, a possible complete synchronization with negative time lag is discussed. As a result, this leads the receiver to be in the future state of the transmitter. To sustain this proposed scheme, we present the encrypted transmission of binary messages at a bit rate of 250?Mbits/s, and we show that the digital messages at the output receiver laser are perfectly recovered using a finite impulse response filter. The error bits, the error rates as well as the effects of parameter mismatches on the recovered messages are evaluated.     

Coupled Chaotic Colpitts Oscillators: Identical and Mismatched Cases

A system consisting of two linearly coupled chaotic Colpitts oscillators is considered. Two different coupling configurations, namely coupled collector nodes (C–C) and coupled emitter nodes (E–E) have been investigated. In addition to identical oscillators the case of mismatched circuits has been studied. Specifically the influence of the transistor parameter mismatch has been analyzed. The relative synchronization error has been estimated for different mismatch levels provided the coupling coefficient is twice larger than the synchronization threshold. Illustrative experimental results, including phase portraits and synchronization error are presented.     

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.     

Synchronization of stochastic reaction–diffusion systems via boundary control

This paper studies the problem of mean square asymptotical synchronization and \(H_\infty \) synchronization for coupled stochastic reaction–diffusion systems (SRDSs) via boundary control. Based on the deduced synchronization error dynamic, we design boundary controllers to achieve mean square asymptotical synchronization. By virtue of Lyapunov functional method and Wirtinger’s inequality, sufficient conditions are obtained for ensuring mean square asymptotical synchronization. When coupled SRDSs are subject to external disturbance, mean square \(H_\infty \) synchronization is investigated and corresponding criterion is presented under a designed boundary controller. In addition to focusing on systems with Neumann boundary conditions, we also briefly study coupled SRDSs with mixed boundary conditions and sufficient conditions are provided to achieve the desired performance. Numerical examples are used to verify the effectiveness of our theoretical results.     

Effect of the coupling matrix with a weight parameter on synchronization pattern in a globally coupled network

In this paper, effect of the coupling matrix with a weight parameter on synchronization pattern in a globally coupled network is investigated. On the basis of matrix theory, the threshold values of the coupling strength and the weight parameter for cluster synchronization have been obtained by utilizing the attractiveness criteria of the invariant synchronization manifold. It shows that cluster synchronization bifurcation comes forth, which concept is first put forward. That is to say, via changing the weight parameter and the coupling strength, the purpose of controlling the number of clusters is achieved, which provides a new idea for control the number of clusters in a network. Numerical simulations are given to demonstrate the theoretical results. In addition, the theoretical results and the numerical simulations also show that full synchronization may not be realized even if the network is globally coupled when there are some negative couplings.     

Practical synchronization of second-order nonautonomous systems with parameter mismatch and its applications

This paper is concerned with robust practical synchronization for general second-order nonautonomous systems with parameter mismatch. Some simple yet general algebraic criteria are derived based on practical stability theory of nonautonomous dynamical systems. A?distinctive feature of this work is that the parameter mismatch cannot only be existed in system parameters, but also in external excitation ones. Furthermore, the obtained results are applied to a typical horizontal platform system and the representative forced Van der Pol oscillator. Subsequently, numerical simulations demonstrate the effectiveness of the criteria and the robustness of the control technique.     

Parameter identification and chaos synchronization for uncertain coupled map lattices

A method of chaos synchronization and parameter identification is proposed in the paper. The synchronization controller and the parameter recognizer are designed. Two coupled map lattices with different structures are taken as examples to verify the effectiveness of the method. Simulation results show that the identification variables in the parameter recognizer can substitute for the unknown parameters in both target and response systems. Then global synchronization of the two uncertain coupled map lattices can be realized after the designed controller is added.     

Lag synchronization between discrete chaotic systems with diverse structure

A lag synchronization controller is designed in studying discrete chaotic systems with diverse structures to realize synchronization between Henon and Ikeda sys- terns. The structure of the lag synchronization controller and the error equations of state variables between discrete chaotic systems are presented based on the stability theory. The designed controller has unique structures for different chaotic systems. Lag synchro- nization between any discrete chaotic systems with diverse structures can be achieved. Simulation results show that this control method is effective and feasible.     

Anticipatory, complete and lag synchronization of chaos and hyperchaos in a nonlinear delay-coupled time-delayed system

We study the synchronization of chaos and hyperchaos in first-order time-delayed systems that are coupled using the nonlinear time-delay excitatory coupling. We assign two characteristic time delays: the system delay that is same for both the systems, and the coupling delay associated with the coupling path. We show that depending upon the relative values of the system delay and the coupling delay the coupled systems show anticipatory, complete, and lag synchronization. We derive a general stability condition for all the synchronization processes using the Krasovskii–Lyapunov theory. Numerical simulations are carried out to corroborate the analytical results. We compute a quantitative measure to ensure the occurrence of different synchronization phenomena. Finally, we set up an experiment in electronic circuit to verify all the synchronization scenario. It is observed that the experimental results are in good agreement with our analytical results and numerical observations.     

Projective and lag synchronization between general complex networks via impulsive control

This paper mainly investigates the projective and lag synchronization between general complex networks via impulsive control. A general drive complex network and an impulsively controlled slave network are presented in the model. Specially, the coupling matrix in this model is not assumed to be symmetric, diffusive or irreducible. Some criteria and corollaries are, respectively, derived for the projective synchronization and lag synchronization between the presented impulsively controlled complex networks. Finally, the results are illustrated by complex networks composed of the chaotic Lorenz systems. All the numerical simulations verify the correctness of the theoretical results.     

Lag synchronization of hyperchaotic complex nonlinear systems

In this paper, we study the lag synchronization (LS) of n-dimensional hyperchaotic complex nonlinear systems. The idea of the nonlinear control technique based on the complex Lyapunov function with lag in time is used to propose a scheme to investigate LS of hyperchaotic attractors of these systems. Both complex Lyapunov and control functions are introduced. For illustration, the scheme is applied to two hyperchaotic complex Lorenz systems. The real and complex control functions are derived analytically to achieve LS and to show that the complex error dynamical systems are globally stable. Numerical results are calculated to test the validity of the analytical expressions of control functions to achieve LS of two identical hyperchaotic attractors.     

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.     

Hybrid robust modified function projective lag synchronization in two different dimensional chaotic systems

In this article, a novel synchronization scheme, modified function projective lag synchronization (MFPLS) in two different dimensional chaotic systems with parameter perturbations, is proposed. In the proposed method, the states of two nonidentical chaotic systems with different orders are asymptotically lag synchronized up to a desired scaling function matrix by means of reduced order and increased order, respectively. Furthermore, based on the reality situation, the parameter perturbations are involved, which are assumed to appear in both drive and response systems. With the Lyapunov stability theory, an adaptive controller is designed to achieve MFPLS. Theoretical proof and numerical simulations demonstrate the effectiveness and feasibility of the proposed scheme.     

Cluster synchronization of nonlinearly-coupled complex networks with nonidentical nodes and asymmetrical coupling matrix

In this paper, we investigate the cluster synchronization problem for networks with nonlinearly coupled nonidentical dynamical systems and asymmetrical coupling matrix by using pinning control. We derive sufficient conditions for cluster synchronization for any initial values through a feedback scheme and propose an adaptive feedback algorithm that adjusts the coupling strength. Some numerical examples are then given to illustrate the theoretical results.     

A single adaptive controller with one variable for synchronizing two identical time delay hyperchaotic Lorenz systems with mismatched parameters

Time delays are ubiquitous in real world and are often sources of complex behaviors of dynamical systems. This paper addresses the problem of parameter identification and synchronization of uncertain hyperchaotic time-delayed systems. Based on the Lyapunov stability theory and the adaptive control theory, a single adaptive controller with one variable for synchronizing two identical time-delay hyperchaotic Lorenz systems with mismatch parameters is proposed. The parameter update laws and sufficient conditions of the scheme are obtained for both linear feedback and adaptive control approaches. Numerical simulations are also given to show the effectiveness of the proposed method.     

Parameter-dependent synchronization of coupled neurons in cold receptor model

In this paper, synchronization in two coupled neurons with spiking, bursting and chaos firings is investigated as the coupling strength gets increased. Synchronization state can be identified by means of the bifurcation diagram, the correlation coefficient and ISI-distance. It is illustrated that the coupled neurons can exhibit different types of synchronization state when the coupling strength increases. The different synchronization processes appear similar, but their detailed processes are different depending on the parameter values. The synchronization of neuronal network with two different network connectivity patterns is also studied. It is shown that chaotic and high period pattern are more difficult to get complete synchronization than the situation in single spike and low period pattern. It is also demonstrated that the synchronization status of multiple neurons is dependent on the network connectivity patterns. These results may be instructive to understand synchronization in neuronal systems.     

Impulsive control induced effects on dynamics of single and coupled ODE systems

The dynamics of differential system can be changed very obviously after inputting impulse signals. Previous studies show that the single chaotic system can be controlled to periodic motions using impulsive control method. It was well known that the dynamics of hyper-chaotic and coupled systems are very important and more complex than those of a single system. In this paper, particular impulsive control of the hyper-chaotic Lü system was proposed, which is with outer impulsive signals. It can be seen that such impulsive strategy can generate chaos from periodic orbit or control chaos to periodic orbit etc. For the first time, impulsive control induced effects on dynamics of coupled systems are considered in this paper, where the impulse effect has outer input signals. Many interesting and useful results are obtained. The coupled system can realize synchronization and its synchronization manifold can be changed with such impulsive control signals. Strict theories are given, and numerical simulations confirm the correctness of theoretical results.     

Adaptive bidirectionally coupled synchronization of?chaotic?systems with unknown parameters

This paper investigates the chaos synchronization of two bidirectionally coupled chaotic systems. In comparison with previous methods (identical bidirectionally coupled synchronization), the present control scheme is different bidirectionally coupled synchronization, which includes different complete bidirectionally coupled synchronization and different partial bidirectionally coupled synchronization. Based on the Lasalle invariance principle, adaptive schemes are designed to make two different bidirectionally coupled chaotic systems asymptotically synchronized, and unknown parameters are identified simultaneously in the process of synchronization. Theoretical analysis and numerical simulations are shown to verify the results.