Synchronization of chaotic systems with unknown parameters using adaptive passivity-based control

This paper investigates the problem of complete synchronization of chaotic systems with unknown parameters. An adaptive control scheme based on a feedback passivity approach is proposed. The convergence of the synchronization error is guaranteed. The unified chaotic and hyperchaotic Lü systems are taken as illustrative examples. The feasibility and effectiveness of the proposed scheme are demonstrated through numerical simulations.     

Adaptive single input sliding mode control for hybrid-synchronization of uncertain hyperchaotic Lu systems

This paper addresses the problem of hybrid synchronization for hyperchaotic Lu systems without and with uncertain parameters via a single input sliding mode controller (SMC). Based on the SMC approach, the proposed controller not only minimizes the influence of uncertainty but also enhances the robustness of the system. The uncertain parameters are estimated by using new adaptation laws which ensure the uncertain parameters convergence to their original value. A hybrid synchronization scheme is useful to maintain the vastly secured and secrecy in the area of secure communication by using the control theory approach. The proposed hybrid synchronization results are providing a superiority of forming a chaotic secure communication scheme. Finally, a numerical example is provided to demonstrate the validity of the theoretical analysis.     

Hybrid modified function projective synchronization of two different dimensional complex nonlinear systems with parameters identification

In this article, a novel synchronization scheme is proposed to achieve hybrid modified function projective synchronization (HMFPS) in two different dimensional complex nonlinear systems with fully unknown parameters. In the complex space, the response system are asymptotically synchronized up to the different order’s drive system by the state transformation with a scaling function matrix, and all of unknown parameters in both drive and response systems are achieved to be identified. Based on the Lyapunov stability theory, an adaptive controller and updated laws of parameters are developed. Respectively on the ways of increased order and reduced order, the corresponding numerical simulations demonstrate the effectiveness and feasibility of the proposed scheme.     

Propagation of projective synchronization in a series connection of chaotic systems

Projective synchronization is a type of chaos synchronization where the response system states are scaled replicas of the drive system states. This paper deals with the propagation of projective synchronization in a series connection of N chaotic discrete-time drive systems and N response systems. By exploiting an observer-based approach, the paper demonstrates that dead-beat projective synchronization (i.e., exact synchronization in finite time for any scaling factor) is achieved between the nth drive and nth response systems. In particular, it is shown that projective synchronization starts from the innermost (Nth) drive-response system pair and propagates toward the outermost (first) drive-response system pair. Only a single scalar synchronizing signal connects the cascaded drive and response systems. Finally, an example illustrates the propagation of different types of chaos synchronization in a series connection consisting of a Gingerbreadman map, a third order hyperchaotic Henon map and a Lozi map.     

Design of adaptive sliding mode tracking controllers for chaotic synchronization and application to secure communications

Synchronization of two identical chaotic systems with matched and mismatched perturbations by utilizing adaptive sliding mode control (ASMC) technique is presented in this paper. The sliding surface function is specially designed based on the Lyapunov stability theorem and linear matrix inequality (LMI) optimization technique. The designed tracking controller can not only suppress the mismatched perturbations when the controlled dynamics (master–slave) are in the sliding mode, but also drive the trajectories of synchronization errors into a small bounded region whose size can be adjusted through the designed parameters. Adaptive mechanisms are employed in the proposed control scheme for adapting the unknown upper bounds of the perturbations, and the stability of overall controlled synchronization systems is guaranteed. The comparison of the proposed chaotic synchronization technique with an existing generalized chaotic synchronization (GCS) method as well as application of the proposed control method to secure communications is also demonstrated in this paper.     

Synchronization of second-order chaotic systems via adaptive terminal sliding mode control with input nonlinearity

In the presence of system uncertainties, external disturbances and input nonlinearity, this paper is concerned with the adaptive terminal sliding mode controller to achieve synchronization between two identical attractors which belong to a class of second-order chaotic system. The proposed controller with adaptive feedback gains can compensate nonlinear dynamics of the synchronous error system without calculating the magnitudes of them. Meanwhile, these feedback gains are updated by the novel adaptive rules without required that the bounds of system uncertainties and external disturbances have to be known in advance. Some sufficient conditions for stability are provided based on the Lyapunov theorem and numerical studies are performed to verify the effectiveness of presented scheme.     

General synchronization criteria for nonlinear Markovian systems with random delays

It is well known that control of Markovian systems is a difficult problem. This paper considers synchronization control of Markovian coupled nonlinear systems with random delays. A new control scheme is proposed. Sufficient conditions in terms of linear matrix inequalities (LMIs) are obtained such that the coupled system can be asymptotically synchronized onto an isolated system. The synchronization criteria include classical mode-dependent and mode-independent results as special cases. The design method of the control gains is also given. Compared with mode-dependent and mode-independent control methods, our results are more practical and have lower conservatism, respectively. Numerical simulations are given to verify the effectiveness of the theoretical results.     

One input control for exponential synchronization in generalized Lorenz systems with uncertain parameters

By taking account of uncertain slave system parameters, the main goal of this paper is to investigate exponential master–slave synchronization between two nearly identical generalized Lorenz systems via one control input, which including a single state proportional feedback, associated with system parameter estimated laws, which not including states of the master system. Sufficient conditions are provided for the guaranteed exponential stability of both synchronized errors and system parameter errors. Meanwhile, numerical studies are also performed to verify the effectiveness of presented schemes.     

Synchronization of IT2 stochastic fuzzy complex dynamical networks with time-varying delay via fuzzy pinning control

In this paper, the problem of synchronization on interval type-2 (IT2) stochastic fuzzy complex dynamical networks (CDNs) with time-varying delay via fuzzy pinning control is fully studied. Firstly, a more general complex network model is considered, which involves the time-varying delay, IT2 fuzzy and stochastic effects. More specifically, IT2 fuzzy model, as a meaningful fuzzy scheme, is investigated for the first time in CDNs. Then, with the aid of Lyapunov stability theory and stochastic analysis technique, some new sufficient criteria are established to ensure synchronization of the addressed systems. Moreover, on basis of the parallel-distributed compensation (PDC) scheme, two effective fuzzy pinning control protocols are proposed to achieve the synchronization. Finally, a numerical example is performed to illustrate the effectiveness and superiority of the derived theoretical results.     

Synchronization of coupled heterogeneous complex networks

By only designing the internal coupling, quasi synchronization of heterogeneous complex networks coupled by N nonidentical Duffing-type oscillators without any external controller is investigated in this paper. To achieve quasi synchronization, the average of states of all nodes is designed as the virtual target. Heterogeneous complex networks with two kinds of nonlinear node dynamics are analyzed firstly. Some sufficient conditions on quasi synchronization are obtained without designing any external controller. Quasi synchronization means that the states of all nonidentical nodes will keep a bounded error with the virtual target. Then the heterogeneous complex network with impulsive coupling which means the network only has coupling at some discrete impulsive instants, is further discussed. Some sufficient conditions on heterogeneous complex network with impulsive coupling are derived. Based on these results, heterogeneous complex network can still reach quasi synchronization even if its nodes are only coupled at discrete impulsive instants. Finally, two examples are provided to verify the theoretical results.     

Synchronization for chaotic systems via mixed-objective dynamic output feedback robust model predictive control

This paper focuses on mixed-objective dynamic output feedback robust model predictive control (OFRMPC) for the synchronization of two identical discrete-time chaotic systems with polytopic uncertainties, energy bounded disturbances, and input constraint. Using active control strategy, the chaos synchronization is transformed into standard dynamic OFRMPC scenarios tractable through receding horizon min–max optimization. Utilizing the notion of quadratic boundedness, the augmented closed-loop stability is further characterized. Then, the concepts of mixed performance criteria are firstly incorporated into the dynamic OFRMPC scheme to guarantee both the robust stability and the disturbance attenuation ability while preserving better dynamical behaviors. Necessary and/or sufficient conditions for desired mixed-objective dynamic OFRMPC are formulated involving linear matrix inequalities (LMIs). Finally, two numerical examples are given to demonstrate the theoretical results.     

Adaptive synchronization of multi-agent systems via variable impulsive control

This paper mainly focuses on the adaptive synchronization problem of multi-agent systems via distributed impulsive control method. Different from the existing investigations of impulsive synchronization with fixed time impulsive inputs, the proposed distributed variable impulsive protocol allows that the impulsive inputs are chosen within a time period (namely impulsive time window) which can be described by the distances of the left (right) endpoints or the centers between two adjacent impulsive time windows. Obviously, this kind of flexible control scheme is more effective in practical systems (especially for the complex environment with physical restrictions). Moreover, the proposed adaptive control technique is helpful to solve the problem with uncertain system parameters. By means of Lyapunov stability theory, impulsive differential equations and adaptive control technique, three sufficient impulsive consensus conditions are given to realize the synchronization of a class of multi-agent nonlinear systems. Finally, two numerical simulations are provided to illustrate the validity of the theoretical analysis.     

Robust function projective synchronization of two different chaotic systems with unknown parameters

This paper deals with the function projective synchronization problem of two different chaotic systems with unknown and perturbed parameters. The parameter perturbations are assumed to appear in both drive and response systems, which perturbed about the nominal parameter values. A new robust function projective synchronization method is proposed, which is able to overcome random uncertainties of all model parameters. Corresponding numerical simulations are performed to verify and illustrate the analytical results.     

Finite-time inter-layer projective synchronization of Caputo fractional-order two-layer networks by sliding mode control

Finite-time inter-layer projective synchronization (FIPS) of Caputo fractional-order two-layer networks (FTN) based on sliding mode control (SMC) technique is investigated in this article. Firstly, in order to realize the FIPS of FTN, a fractional-order integral sliding mode surface (SMS) is established. Then, through the theory of SMC, we design a sliding mode controller (SMCr) to ensure the appearance of sliding mode motion. According to the fractional Lyapunov direct method, the trajectories of the system are driven to the proposed SMS, and some novel sufficient conditions for FIPS of FTN are derived. Furthermore, as two special cases of FIPS, finite-time inter-layer synchronization and finite-time inter-layer anti-synchronization for the FTN are studied. Finally, this paper takes the fractional-order chaotic Lü’s system and the fractional-order chaotic Chen’s system as the isolated node of the first layer system and the second layer system, respectively. And the numerical simulations are given to demonstrate the feasibility and validity of the proposed theoretical results.     

Cluster synchronization for directed coupled inertial reaction-diffusion neural networks with nonidentical nodes via non-reduced order method

The cluster synchronization issues are investigated for directed coupled inertial reaction-diffusion neural networks (CIRDNNs) with nonidentical nodes by imposing two effective pinning control. A novel Lyapunov-Krasovskii functional (LKF) is established to directly analyze the dynamic behavior of CIRDNNs and deal with reaction-diffusion term, inertia term and coupling term. Moreover, based on different desired cluster synchronization states including a set of un-decoupled trajectories and the particular solutions of the decoupled node systems, two class of synchronization criteria in view of algebraic inequalities are derived under two different communication topologies, respectively. Finally, two typical examples are given to verify the theoretical results.     

The characteristics and self-time-delay synchronization of two-time-delay complex Lorenz system

Time-delay is frequently encountered in a variety of practical chaotic systems, such as chaos-communication. The behaviours of time-delay chaotic system are greatly different from those of the original system. Self-time-delay synchronization (STDS) implies that the synchronization between the time-delay system and the original system while maintaining the structure and parameters of systems unchanged, thus these various problems produced by time-delay in practice are avoided. Firstly, we investigate the characteristics of two-time-delay complex Lorenz system. Then we take one-time-delay and two-time-delay complex Lorenz system as examples, and design their controllers to realize STDS. One-time-delay complex Lorenz system is a special case of two-time-delay. Numerical simulations verify the validity of the STDS controllers. The controllers only involve error, and it is easy to realize in practice. Moreover, the time-delay chaotic system exhibits highly stochastic behaviors and unpredictable properties, which can be applied to chaos-communication and enhance the security of communication.     

Generalized lag synchronization of multiple weighted complex networks with and without time delay

The generalized lag synchronization of multiple weighted complex dynamical networks with fixed and adaptive couplings is investigated in this paper, respectively. By designing appropriate controller, several synchronization criteria are presented for multiple weighted complex dynamical networks with and without time delay based on the selected Lyapunov functional and inequality techniques. Moreover, an adaptive scheme to update the coupling weights is also developed for ensuring the generalized lag synchronization of multiple weighted complex dynamical networks with and without time delay. Finally, two numerical examples are provided in order to validate effectiveness of the proposed generalized lag synchronization criteria.     

Boundary control for synchronization of fractional-order complex spatiotemporal networks based on PDEs with multiple delays and its application in image encryption

This paper addresses the synchronization problem of fractional-order complex spatiotemporal networks (CSNs) based on partial differential equations with delays via boundary control. First, fractional-order CSNs with time-invariant and time-varying delays are studied separately due to the widespread existence of time delays in complex networks. Moreover, two boundary controllers are proposed to solve the synchronization issue of fractional-order CSNs, in which nodes communicate with each other only on the spatial boundary. Furthermore, according to the fractional-order inequality, the synchronization criteria of fractional-order CNSs with multiple delays are obtained. Finally, the numerical simulations are given to verify the feasibility of the presented results. A case provides the application of CSNs in image encryption.     

Finite-time synchronization of delayed complex dynamic networks via aperiodically intermittent control

In this paper, the finite-time synchronization problem of complex dynamic networks with time delay is studied via aperiodically intermittent control. By compared with the existed results concerning aperiodically intermittent control, some new results are obtained to guarantee the synchronization of networks in a finite time. Especially, a new lemma is proposed to reduce the convergence time. In addition, based on aperiodically intermittent control scheme, the essential condition ensuring finite-time synchronization of dynamic networks is also obtained, and the convergence time is closely related to the topological structure of networks and the maximum ratio of the rest width to the aperiodic time span. Finally, a numerical example is provided to verify the validness of the proposed theoretical results.     

Asynchronous synchronization of complex networks with switched adjacent matrices

This note analyzes the synchronization problem for a class of complex networks. The considered network topology, symbolized by adjacent matrices, is assumed to be time-varying and subjected to a Markov process. When designing controllers to synchronize the network, we consider the asynchronous phenomenon occurred between the mode of the adjacent matrices and that of the controllers. A properly defined stochastic process, which is related to the Markov process by a conditional probability is then introduced to indicate the controller mode. Base on this connection, a sufficient condition is derived by using Lyapunov–Krasoviskii theorem to ensure the global synchronization of the considered complex network with a predefined noise attenuation level. The design scheme is presented by solving a series of LMIs. A numerical example is finally provided to verify the effectiveness of the proposed method.