Global impulsive exponential synchronization of time-delayed coupled chaotic systems

In this paper, the impulsive exponential synchronization problem for time-delayed coupled chaotic systems is investigated. By establishing an impulsive differential delay inequality and using the property of P-cone, some simple conditions of impulsive exponential synchronization of two coupled chaotic systems are derived. To illustrate the effectiveness of the new scheme, some numerical examples are given.     

Linearly coupled synchronization of the unified chaotic systems and the Lorenz systems

This paper investigates the synchronization of two linearly coupled unified chaotic systems and two linearly coupled Lorenz systems. Some sufficient conditions for synchronization are attained through rigorous mathematical theory. Compared with the results in the reference [Chaos, Solitons & Fractals 2002;14:529], the sufficient condition for the synchronization of two linearly coupled Lorenz systems is simpler and less conservative. Numerical simulations are provided for illustration and verification.     

Synchronization and antisynchronization of N‐coupled fractional‐order complex chaotic systems with ring connection

This paper is devoted to investigate synchronization and antisynchronization of N‐coupled general fractional‐order complex chaotic systems described by a unified mathematical expression with ring connection. By means of the direct design method, the appropriate controllers are designed to transform the fractional‐order error dynamical system into a nonlinear system with antisymmetric structure. Thus, by using the recently established result for the Caputo fractional derivative of a quadratic function and a fractional‐order extension of the Lyapunov direct method, several stability criteria are derived to ensure the occurrence of synchronization and antisynchronization among N‐coupled fractional‐order complex chaotic systems. Moreover, numerical simulations are performed to illustrate the effectiveness of the proposed design.     

Linearly and nonlinearly bidirectionally coupled synchronization of hyperchaotic systems

To date, there have been many results about unidirectionally coupled synchronization of chaotic systems. However, much less work is reported on bidirectionally-coupled synchronization. In this paper, we investigate the synchronization of two bidirectionally coupled Chen hyperchaotic systems, which are coupled linearly and nonlinearly respectively. Firstly, linearly coupled synchronization of two hyperchaotic Chen systems is investigated, and a theorem on how to choose the coupling coefficients are developed to guarantee the global asymptotical synchronization of two coupled hyperchaotic systems. Analysis shows that the choice of the coupling coefficients relies on the bound of the chaotic system. Secondly, the nonlinearly coupled synchronization is studied; a sufficient condition for the locally asymptotical synchronization is derived, which is independent of the bound of the hyperchaotic system. Finally, numerical simulations are included to verify the effectiveness and feasibility of the developed theorems.     

Some simple global synchronization criterions for coupled time-varying chaotic systems

Based on the Lyapunov stabilization theory and matrix measure, this paper proposes some simple generic criterions of global chaos synchronization between two coupled time-varying chaotic systems from a unidirectional linear error feedback coupling approach. These simple criterions are applicable to some typical chaotic systems with different types of nonlinearity, such as the original Chua’s circuit and the Rössler chaotic system. The coupling parameters are determined according to the new criterion so as to ensure the coupled systems’ global chaos synchronization.     

The control of chaotic systems with unknown parameters and external disturbance via backstepping‐like scheme

This article addresses the adaptive control of chaotic systems with unknown parameters, model uncertainties, and external disturbance. We first investigate the control of a class of chaotic systems and then discuss the control of general chaotic systems. Based on the backstepping‐like procedure, some novel criteria are proposed via adaptive control scheme. As an example to illustrate the application of the proposed method, the control and synchronization of the modified Chua's chaotic system is also investigated via a single input. Some numerical simulations are given to demonstrate the robustness and efficiency of the proposed approach. © 2016 Wiley Periodicals, Inc. Complexity 21: 573–583, 2016     

Synchronization of N-coupled fractional-order chaotic systems with ring connection

In this paper, the synchronization of N-coupled fractional-order chaotic systems with ring connection is firstly investigated in detail. Based on stability criteria of fractional-order system, the synchronization of N-coupled fractional-order chaotic systems with unidirectional coupling and bidirectional coupling is achieved. Moreover, some appropriate comparisons are made to contrast to some of existing results. Finally, some numerical examples are provided to illustrate and verify the effectiveness of the proposed schemes.     

Projective synchronization of different fractional-order chaotic systems with non-identical orders

This paper investigates the projective synchronization (PS) of different fractional order chaotic systems while the derivative orders of the states in drive and response systems are unequal. Based on some essential properties on fractional calculus and the stability theorems of fractional-order systems, we propose a general method to achieve the PS in such cases. The fractional operators are introduced into the controller to transform the problem into synchronization problem between chaotic systems with identical orders, and the nonlinear feedback controller is proposed based on the concept of active control technique. The method is both theoretically rigorous and practically feasible. We present two examples that illustrate the effectiveness and applications of the method, which include the PS between two 3-D commensurate fractional-order chaotic systems and the PS between two 4-D fractional-order hyperchaotic systems with incommensurate and commensurate orders, respectively. Abundant numerical simulations are given which agree well with the analytical results. Our investigations show that PS can also be achieved between different chaotic systems with non-identical orders. We have further reviewed and compared some relevant methods on this topic reported in several recent papers. A discussion on the physical implementation of the proposed method is also presented in this paper.     

Estimating parameters of chaotic systems under noise-induced synchronization

Kim et al. introduced in 2002 [Kim CM, Rim S, Kye WH. Sequential synchronization of chaotic systems with an application to communication. Phys Rev Lett 2002;88:014103] a hierarchically structured communication scheme based on sequential synchronization, a modification of noise-induced synchronization (NIS). We propose in this paper an approach that can estimate the parameters of chaotic systems under NIS. In this approach, a dimensionally-expanded parameter estimating system is first constructed according to the original chaotic system. By feeding chaotic transmitted signal and external driving signal, the parameter estimating system can be synchronized with the original chaotic system. Consequently, parameters would be estimated. Numerical simulation shows that this approach can estimate all the parameters of chaotic systems under two feeding modes, which implies the potential weakness of the chaotic communication scheme under NIS or sequential synchronization.     

Finite-time synchronization of uncertain unified chaotic systems based on CLF

This paper studies the problem of finite-time synchronization for the unified chaotic systems. We prove that global finite-time synchronization can be achieved for unified chaotic systems which have uncertain parameters. Simulation results for Lorenz, Lü and Chen chaotic systems are provided to illustrate the effectiveness of the proposed scheme.     

A note on phase synchronization in coupled chaotic fractional order systems

The dynamic behaviors of fractional order systems have received increasing attention in recent years. This paper addresses the reliable phase synchronization problem between two coupled chaotic fractional order systems. An active nonlinear feedback control scheme is constructed to achieve phase synchronization between two coupled chaotic fractional order systems. We investigated the necessary conditions for fractional order Lorenz, Lü and Rössler systems to exhibit chaotic attractor similar to their integer order counterpart. Then, based on the stability results of fractional order systems, sufficient conditions for phase synchronization of the fractional models of Lorenz, Lü and Rössler systems are derived. The synchronization scheme that is simple and global enables synchronization of fractional order chaotic systems to be achieved without the computation of the conditional Lyapunov exponents. Numerical simulations are performed to assess the performance of the presented analysis.     

Synchronization of non-autonomous chaotic systems with time-varying delay via delayed feedback control

In this paper, we investigate the synchronization of non-autonomous chaotic systems with time-varying delay via delayed feedback control. Using a combination of Riccati differential equation approach, Lyapunov-Krasovskii functional, inequality techniques, some sufficient conditions for exponentially stability of the error system are formulated in form of a solution to the standard Riccati differential equation. The designed controller ensures that the synchronization of non-autonomous chaotic systems are proposed via delayed feedback control and intermittent linear state delayed feedback control. Numerical simulations are presented to illustrate the effectiveness of these synchronization criteria.     

Some impulsive synchronization criterions for coupled chaotic systems via unidirectional linear error feedback approach

Based on stability theory of impulsive differential equation and new comparison theory of impulsive differential system, we study the chaos impulsive synchronization of two coupled chaotic systems using the unidirectional linear error feedback scheme. Some generic conditions of chaos impulsive synchronization of two coupled chaotic systems are derived, and to apply the conditions to typical chaotic system––the original Chua’s circuit. The example illustrates the effectiveness of the proposed result.     

Synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal

In this paper, a drive-response synchronization method with linear output error feedback is presented for synchronizing a class of fractional-order chaotic systems via a scalar transmitted signal. Based on stability theory of fractional-order systems and linear system theory, a necessary and sufficient condition for the existence of the feedback gain vector such that global synchronization between the fractional-order drive system and response system can be achieved and its design method are given. This synchronization approach that is simple, global and theoretically rigorous enables synchronization of fractional-order chaotic systems be achieved in a systematic way and does not require the computation of the conditional Lyapunov exponents. An example is used to illustrate the effectiveness of the proposed synchronization method.     

Synchronizing chaotic systems up to an arbitrary scaling matrix via a single signal

This paper introduces a novel type of synchronization, where two chaotic systems synchronize up to an arbitrary scaling matrix. In particular, each drive system state synchronizes with a linear combination of response system states by using a single synchronizing signal. The proposed observer-based method exploits a theorem that assures asymptotic synchronization for a wide class of continuous-time chaotic (hyperchaotic) systems. Two examples, involving Rössler’s system and a hyperchaotic oscillator, show that the proposed technique is a general framework to achieve any type of synchronization defined to date.     

Switching among three different kinds of synchronization for delay chaotic systems

We found that the complete synchronization, anticipating synchronization and lag synchronization can be reached by the same kind of one way coupling for a large class of chaotic delay system. By changing the transformation time of the coupling signal we can switch from anticipating synchronization to complete synchronization, and then to lag synchronization. Numerical simulation for three chaotic delay systems were presented, one of them was novel which had two degree of freedoms, and the other two were the well known Ikeda and Mackey–Glass system which are one degree of freedom chaotic delay system. The theoretical analysis and the numerical simulation agreed perfect good.     

Observer-based chaotic synchronization in the presence of unknown inputs

This paper deals with the problem of synchronization of chaotic dynamical systems. We consider a drive-response type of synchronization via a scalar transmitted signal. Unlike most works we consider the presence of some unknown inputs in the drive system and that no knowledge about their nature is available. A reduced-order observer-based response system is designed to synchronize with the missing states. We show that under some assumptions the synchronization is exponentially achieved. The efficiency of our method is confirmed by numerical simulations of two well-known chaotic systems: Chua’s circuit and Lur’e system.     

Fast synchronization of directionally coupled chaotic systems

This paper studies the fast synchronization of directionally coupled chaotic systems under a chained interaction topology. Firstly, by applying finite-time stability theory, it is shown that all chaotic systems can achieve synchronization in finite time as long as the coupling strength is strong enough. Secondly, it is proved that the settling times are determined by the interaction strength, system parameters and initial conditions of the chaotic systems. Furthermore, it is found that the settling times are mainly dependent on the bounded value and dimension of the coupled chaotic systems when the individual chaotic sub-system is bounded. Finally, illustrative examples and numerical simulations are given to show the correctness of theoretical results.     

Universal chaos synchronization control laws for general quadratic discrete systems

This paper addresses the reliable universal synchronization problem between two coupled chaotic quadratic discrete systems. A general nonlinear control method of synchronization for coupled 2D and 3D quadratic dynamical systems in discrete-time is proposed. The proposed synchronization method is based on universal controllers. The synchronization results are derived theoretically using active control method and Lyapunov stability theory. Numerical simulations are performed to assess the performance of the presented analysis.     

Switching synchronized chaotic systems applied to secure communication

The purpose of this paper is to study the behavior of the solutions of two synchronized chaotic systems when the solutions switch from the first to the second system and vice-versa. The initial condition is chosen in the first system and the solutions travels for time $t \in [0, h]$, where $h>0$. The value of the solution at time $h$ is then chosen as the initial condition for the solution of the second system and this solution travels for time $t \in [h, 2h]$. The value of the solution at time $2h$ is then chosen as the initial value for the solution of the first system and so on. The first system is composed of two subsystems, Master and Slave that are synchronized. We present applications using the Lorenz, Chua and Chen systems. Some simulations using Matlab are presented.