Multiswitching dual combination synchronization of time‐delay chaotic systems

This paper presents a novel synchronization scheme of multiswitching dual combination synchronization which is first of its kind. Multiswitching dual combination synchronization is achieved for 6 time‐delay chaotic systems. Asymptotically stable synchronization states are established by nonlinear control method and Lyapunov Krasovskii functional. To elaborate the proposed scheme, an example of time‐delay Rossler, Chen, and Shimizu Morioka systems is considered, where time‐delay Rossler system and Chen system are considered as drive systems and time‐delay Shimizu Morioka system is considered as response system. Theoretical analysis and computational results are in excellent agreement.     

Multi-synchronization of chaos via linear output feedback strategy

Multi-synchronization of chaotic systems based on the master-slave scheme as an extension of the dual synchronization problem is introduced. It is assumed that the only information available from the master systems is a linear combination of their state vectors. The design procedure for multi-synchronization through output feedback strategy is described and the sufficient condition is given. The performance of the proposed algorithm is numerically examined by applying it to the Chen–Lorenz–Rossler and the Duffing–Van der Pol chaotic systems. Simulation results show the effectiveness of the proposed scheme.     

Adaptive synchronization of two chaotic systems with stochastic unknown parameters

Using the Lyapunov stability theory an adaptive control is proposed for chaos synchronization between two different systems which have stochastically time varying unknown coefficients. The stochastic variations of the coefficients about their unknown mean values are modeled through white Gaussian noise produced by the Weiner process. It is shown that using the proposed adaptive control the mean square of synchronization error converges to an arbitrarily small bound around zero. To demonstrate the effectiveness of the proposed technique, it is applied to the Lorenz–Chen and the Chen–Rossler dynamical systems, as some case studies. Simulation results indicate that the proposed adaptive controller has a high performance in synchronization of chaotic systems in noisy environment.     

超混沌系统的混合同步控制

基于Lyapunov稳定性理论,提出了一种超混沌系统混合同步控制方法,给出并详细证明了Rossler超混沌系统实现自同步的充分条件以及控制律参数的取值范围,并构建了两个不同结构的Rossler超混沌系统的异结构快速同步的数学模型。数值仿真表明了所设控制器的有效性和方法的可操作性．     

Chaos synchronization between two different chaotic dynamical systems

This work presents chaos synchronization between two different chaotic systems by nonlinear control laws. First, synchronization problem between Genesio system and Rossler system has been investigated, and then the similar approach is applied to the synchronization problem between Genesio system and a new chaotic system developed recently in the literature. The control performances are verified by two numerical examples.     

Synchronization and anti-synchronization of chaotic oscillators under input saturation

This paper addresses the design of simple state feedback controllers for synchronization and anti-synchronization of chaotic oscillators under input saturation and disturbance. By employing sector condition, linear matrix inequality (LMI)-based sufficient conditions are derived to design (global or local) controllers for chaos synchronization. The proposed local synchronization strategy guarantees a region of stability in terms of difference between states of the master–slave systems. This region of stability can be enlarged by means of an LMI-based optimization algorithm, through which asymptotic synchronization of chaotic oscillators can be ensured for a large difference in their initial conditions. Further, a novel LMI-based robust control strategy is developed, for local synchronization of input-constrained chaotic oscillators, by providing an upper bound on synchronization error in terms of disturbance and initial conditions of chaotic systems. Moreover, the proposed robust state feedback control methodology is modified to provide an inaugural treatment for robust anti-synchronization of chaotic systems under input saturation and disturbance. The results of the proposed methodologies are verified through numerical simulations for synchronization and anti-synchronization of the master–slave chaotic Chua’s circuits under input saturation.     

Controlling chaotic behaviour for spin generator and Rossler dynamical systems with feedback control

Different methods are proposed to control chaotic behaviour of the Nuclear Spin Generator (NSG) and Rossler continuous dynamical systems. Linear and nonlinear feedback control techniques are used to suppress chaos. The stabilization of unstable fixed point or unstable periodic solution of chaotic behaviour is achieved. The controlled system is stable under some conditions on the parameters of the system. Stability of the controlled system is determined by the Routh–Hurwitz criterion and Lyapunov direct method. Numerical simulation results are included to show the control process.     

Dual synchronization based on two different chaotic systems: Lorenz systems and Rössler systems

In this paper, we improve and extend the works of Liu and Davids [Dual synchronization of chaos, Phys. Rev. E 61 (2000) 2176–2179] which only introduce the dual synchronization of 1-D discrete chaotic systems. The dual synchronization of two different 3-D continuous chaotic systems, Lorenz systems and Rössler systems, is discussed. And a sufficient condition of dual synchronization about the two different chaotic systems is obtained. Theories and numerical simulations show the possibility of dual synchronization and the effectiveness of the method.     

Projective synchronization between different fractional‐order hyperchaotic systems with uncertain parameters using proposed modified adaptive projective synchronization technique

In the present article, the authors have proposed a modified projective adaptive synchronization technique for fractional‐order chaotic systems. The adaptive projective synchronization controller and identification parameters law are developed on the basis of Lyapunov direct stability theory. The proposed method is successfully applied for the projective synchronization between fractional‐order hyperchaotic Lü system as drive system and fractional‐order hyperchaotic Lorenz chaotic system as response system. A comparison between the effects on synchronization time due to the presence of fractional‐order time derivatives for modified projective synchronization method and proposed modified adaptive projective synchronization technique is the key feature of the present article. Numerical simulation results, which are carried out using Adams–Boshforth–Moulton method show that the proposed technique is effective, convenient and also faster for projective synchronization of fractional‐order nonlinear dynamical systems. Copyright © 2013 John Wiley & Sons, Ltd.     

Adaptive fuzzy observer based synchronization design and secure communications of chaotic systems

This paper proposes a synchronization design scheme based on an alternative indirect adaptive fuzzy observer and its application to secure communication of chaotic systems. It is assumed that their states are unmeasurable and their parameters are unknown. Chaotic systems and the structure of the fuzzy observer are represented by the Takagi–Sugeno fuzzy model. Using Lyapunov stability theory, an adaptive law is derived to estimate the unknown parameters and the stability of the proposed system is guaranteed. Through this process, the asymptotic synchronization of chaotic systems is achieved. The proposed observer is applied to secure communications of chaotic systems and some numerical simulation results show the validity of theoretical derivations and the performance of the proposed observer.     

Synchronization and anti-synchronization coexist in Chen–Lee chaotic systems

This study demonstrates that synchronization and anti-synchronization can coexist in Chen–Lee chaotic systems by direct linear coupling. Based on Lyapunov’s direct method, a linear controller was designed to assure that two different types of synchronization can simultaneously be achieved. Further, the hybrid projective synchronization of Chen–Lee chaotic systems was studied using a nonlinear control scheme. The nonlinear controller was designed according to the Lyapunov stability theory to guarantee the hybrid projective synchronization, including synchronization, anti-synchronization, and projective synchronization. Finally, numerical examples are presented in order to illustrate the proposed synchronization approach.     

Synchronization of chaotic systems using feedback controller: An application to Diffie–Hellman key exchange protocol and ElGamal public key cryptosystem

In this paper, designing an appropriate linear and nonlinear feedback control, the two identical integer order chaotic systems are synchronized by analytically and numerically. It has been realizing that, synchronization using linear feedback control method is efficient than nonlinear feedback control method due to the less computational complexity and the synchronization error. ElGamal public key cryptosystem is described through the proposed Diffie–Hellman key exchange protocol based on the synchronized chaotic systems using linear feedback control and their security are analyzed. The numerical simulations are given to validate the correctness of the proposed synchronization of chaotic systems and the ElGamal cryptosystem.     

Adaptive robust synchronization of Rossler systems in the presence of unknown matched time-varying parameters

This paper deals with the problem of adaptive robust synchronization of chaotic systems based on the Lyapunov theory. A controller is designed for a feedback linearizable error system with matched uncertainties. The proposed method shows that the drive and response systems are synchronized and states of the response system track the states of the drive system as time tends to infinity. Since this approach does not require any information about the bound of uncertainties, this information is not needed in advance. In order to prevent the frequent switching phenomenon in the control signal, the method is modified such that the norm of tracking error is bounded. Numerical simulations on two uncertain Rossler systems with matched uncertainties show fast responses of tracking error, while the errors are Uniformly Ultimately Bounded, and the control signal is reasonably smooth.     

Projective synchronization of time‒delayed chaotic systems with unknown parameters using adaptive control method

The present article aims to study the projective synchronization between two identical and non?identical time?delayed chaotic systems with fully unknown parameters. Here the asymptotical and global synchronization are achieved by means of adaptive control approach based on Lyapunov–Krasovskii functional theory. The proposed technique is successfully applied to investigate the projective synchronization for the pairs of time?delayed chaotic systems amongst advanced Lorenz system as drive system with multiple delay Rössler system and time?delayed Chua's oscillator as response system. An adaptive controller and parameter update laws for unknown parameters are designed so that the drive system is controlled to be the response system. Numerical simulation results, depicted graphically, are carried out using Runge–Kutta Method for delay?differential equations, showing that the design of controller and the adaptive parameter laws are very effective and reliable and can be applied for synchronization of time?delayed chaotic systems. Copyright © 2014 John Wiley & Sons, Ltd.     

Projective synchronization of chaotic fractional-order energy resources demand–supply systems via linear control

A fractional-order energy resources demand–supply system is proposed. A projective synchronization scheme is proposed as an extension on the synchronization scheme of Odibat et al. (2010). The scheme is applied to achieve projective synchronization of the chaotic fractional-order energy resource demand–supply systems. Numerical simulations are performed to verify the effectiveness of the proposed synchronization scheme.     

Synchronization and anti-synchronization of Lu and Bhalekar–Gejji chaotic systems using nonlinear active control

This paper aims at synchronization and anti-synchronization between Lu chaotic system, a member of unified chaotic system, and recently developed Bhalekar–Gejji chaotic system, a system which cannot be derived from the member of unified chaotic system. These synchronization and anti-synchronization have been achieved by using nonlinear active control since the parameters of both the systems are known. Lyapunov stability theory is used and required condition is derived to ensure the stability of error dynamics. Controller is designed by using the sum of relevant variables in chaotic systems. Simulation results suggest that proposed scheme is working satisfactorily.     

Adaptive projective synchronization for a class of switched chaotic systems

This paper addresses the problem of projective synchronization of chaotic systems and switched chaotic systems by adaptive control methods. First, a necessary and sufficient condition is proposed to show how many state variables can realize projective synchronization under a linear feedback controller for the chaotic systems. Then, accordingly, a new algorithm is given to select all state variables that can realize projective synchronization. Furthermore, according to the results of the projective synchronization of chaotic systems, the problem of projective synchronization of the switched chaotic systems comprised by the unified chaotic systems is investigated, and an adaptive global linear feedback controller with only one input channel is designed, which can realize the projective synchronization under the arbitrary switching law. It is worth mentioning that the proposed method can also realize complete synchronization of the switched chaotic systems. Finally, the numerical simulation results verify the correctness and effectiveness of the proposed method.     

The function cascade synchronization method and applications

In this paper, a new function cascade synchronization method of chaos system is proposed to achieve generalized projective synchronization for chaotic systems. Based on Laypunov stability, the proposed synchronization technique is applied to three famous chaotic systems: the unified chaotic system, Liu system and Rössler system, which can make the states of two identical chaotic systems asymptotically synchronized by choosing different special suitable error functions. Numerical simulations are presented to show the effectiveness.     

Global chaos synchronization of new chaotic system using linear active control

Chaos synchronization is a procedure where one chaotic oscillator is forced to adjust the properties of another chaotic oscillator for all future states. This research paper studies and investigates the global chaos synchronization problem of two identical chaotic systems and two non‐identical chaotic systems using the linear active control technique. Based on the Lyapunov stability theory and using the linear active control technique, the stabilizing controllers are designed for asymptotically global stability of the closed‐loop system for both identical and non‐identical synchronization. Numerical simulations and graphs are imparted to justify the efficiency and effectiveness of the proposed scheme. All simulations have been done by using mathematica 9. © 2014 Wiley Periodicals, Inc. Complexity 21: 379–386, 2015