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.     

Post-Double Hopf Bifurcation Dynamics and Adaptive Synchronization of a Hyperchaotic System

In this paper a four-dimensional hyperchaotic system with only one equilibrium is considered and its double Hopf bifurcations are investigated. The general post-bifurcation and stability analysis are carried out using the normal form of the system obtained via the method of multiple scales. The dynamics of the orbits predicted through the normal form comprises possible regimes of periodic solutions, two-period tori, and three-period tori in parameter space. Moreover, we show how the hyperchaotic synchronization of this system can be realized via an adaptive control scheme. Numerical simulations are included to show the effectiveness of the designed control.     

不确定混沌系统的混合投影同步

本文研究了一类不确定混沌(超混沌)系统的混合投影问题.利用自适应方法和Lyapunov稳定性理论,获得了两个恒同或不同混沌系统实现混沌投影同步的一般方法.最后,数值仿真的结果验证了方法的有效性和鲁棒性.     

Adaptive anti-synchronization of two identical and different hyperchaotic systems with uncertain parameters

This paper brings attention to hyperchaos anti-synchronization between two identical and different hyperchaotic systems by using adaptive control. The sufficient conditions for achieving the anti-synchronization of two hyperchaotic systems are derived based on Lyapunov stability theory. An adaptive control law and a parameter update rule for unknown parameters are introduced such that the hyperchaotic Chen system is controlled to be the hyperchaotic Lü system. Theoretical analysis and numerical simulations are shown to verify the results.     

Achieving synchronization between the fractional-order hyperchaotic Novel and Chen systems via a new nonlinear control technique

In this work, we discuss the stability conditions for a nonlinear fractional-order hyperchaotic system. The fractional-order hyperchaotic Novel and Chen systems are introduced. The existence and uniqueness of solutions for two classes of fractional-order hyperchaotic Novel and Chen systems are investigated. On the basis of the stability conditions for nonlinear fractional-order hyperchaotic systems, we study synchronization between the proposed systems by using a new nonlinear control technique. The states of the fractional-order hyperchaotic Novel system are used to control the states of the fractional-order hyperchaotic Chen system. Numerical simulations are used to show the effectiveness of the proposed synchronization scheme.     

Adaptive generalized function projective lag synchronization of different chaotic systems with fully uncertain parameters

In this paper, a novel projective synchronization scheme called adaptive generalized function projective lag synchronization (AGFPLS) is proposed. In the AGFPLS method, the states of two different chaotic systems with fully uncertain parameters are asymptotically lag synchronized up to a desired scaling function matrix. By means of the Lyapunov stability theory, an adaptive controller with corresponding parameter update rule is designed for achieving AGFPLS between two diverse chaotic systems and estimating the unknown parameters. This technique is employed to realize AGFPLS between uncertain Lü chaotic system and uncertain Liu chaotic system, and between Chen hyperchaotic system and Lorenz hyperchaotic system with fully uncertain parameters, respectively. Furthermore, AGFPLS between two different uncertain chaotic systems can still be achieved effectively with the existence of noise perturbation. The corresponding numerical simulations are performed to demonstrate the validity and robustness of the presented synchronization method.     

Generation and suppression of a new hyperchaotic nonlinear model with complex variables

In this paper, we introduce a new hyperchaotic complex Chen model. This hyperchaotic complex system is constructed by adding a complex nonlinear term to the third equation of the chaotic complex Chen system with consideration it’s all variables are complex. The new system is a 6-dimensional continuous real autonomous hyperchaotic system. The properties of this system including invariance, dissipation, equilibria and their stability, Lyapunov exponents, Lyapunov dimension, bifurcation diagrams and hyperchaotic behavior are studied. Different forms of hyperchaotic complex Chen systems are constructed. We suppress the hyperchaotic behavior of our system via passive control method by using one complex controller. The hyperchaotic attractors of the new system are converted to its unstable trivial fixed point and tracked to its unstable non trivial fixed points and periodic orbits. Block diagrams of our system are designed by using Matlab/Simulink after and before the suppression process to ensure the validity of the analytical results.     

Chaos synchronization of a new hyperchaotic system

In this paper, two kinds of synchronization schemes for a new hyperchaotic system are presented. Firstly, on the basis of stability criterion of linear system, synchronization is achieved with the help of the active control theory. Secondly, a nonlinear controller is designed according to Lyapunov stability theory to assure that synchronization can be achieved. Furthermore, an adaptive control approach for synchronization of uncertain hyperchaotic systems is proposed. Finally numerical simulations are provided to show the effectiveness and feasibility of the developed methods.     

Chaos synchronization between two novel different hyperchaotic systems with unknown parameters

This work is devoted to investigating the synchronization between two novel different hyperchaotic systems with fully unknown parameters, i.e., an uncertain hyperchaotic Lorenz system and an uncertain hyperchaotic Lü system. Based on the Lyapunov stability theory, a new adaptive controller with parameter update law is designed to synchronize these two hyperchaotic systems asymptotically and globally. Numerical simulations are presented to verify the effectiveness of the synchronization scheme.     

Switched modified function projective synchronization of hyperchaotic Qi system with uncertain parameters

This work is involved with switched modified function projective synchronization of two identical Qi hyperchaotic systems using adaptive control method. Switched synchronization of chaotic systems in which a state variable of the drive system synchronize with a different state variable of the response system is a promising type of synchronization as it provides greater security in secure communication. Modified function projective synchronization with the unpredictability of scaling functions can enhance security. Recently formulated hyperchaotic Qi system in the hyperchaotic mode has an extremely broad frequency bandwidth of high magnitudes, verifying its unusual random nature and indicating its great potential for some relevant engineering applications such as secure communications. By Lyapunove stability theory, the adaptive control law and the parameter update law are derived to make the state of two chaotic systems modified function projective synchronized. Synchronization under the effect of noise is also considered. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers.     

Adaptive modified function projective synchronization of hyperchaotic systems with unknown parameters

This paper is involved with the adaptive modified function projective synchronization (MFPS) problem of hyperchaotic systems with unknown parameters. Based on the Lyapunov stability theorem and adaptive control method, adaptive controllers and parameters update laws can be presented for the MFPS not only between two identical hyperchaotic systems but particularly also between two different hyperchaotic systems with fully unknown or partially unknown parameters. Moreover, the coupling strength can be automatically adapted to a updated law. Numerical simulations are presented to show the effectiveness of the proposed synchronization schemes.     

Adaptive synchronization between two different hyperchaotic systems

This work presents chaos synchronization between two different hyperchaotic systems using adaptive control. The sufficient conditions for achieving synchronization of two high dimensional chaotic systems are derived based on Lyapunov stability theory, and an adaptive control law and a parameter update rule for unknown parameters are given such that generalized Henon–Heiles system is controlled to be hyperchaotic Chen system. Theoretical analysis and numerical simulations are shown to verify the results.     

Adaptive synchronization of two novel different hyperchaotic systems with partly uncertain parameters

This paper investigates adaptive synchronization between two novel different hyperchaotic systems with partly uncertain parameters. Based on the Lyapunov stability theorem and the adaptive control theory, synchronization between these two hyperchaotic systems is achieved by proposing a new adaptive controller and a parameter estimation update law. Numerical simulations are presented to demonstrate the analytical results.     

Optimize design of adaptive synchronization controllers and parameter observers in different hyperchaotic systems

Based on the Lyapunov stability and adaptive synchronization theory, optimization design of adaptive controllers and parameter observers with controllable gain coefficient are investigated in detail. The linear errors of corresponding variables and parameters are used to construct different appropriate positive Lyapunov functions V and the parameter observers and adaptive controllers are approached analytically by simplifying the differential inequality dV/dt?0. Particularly, an optional gain coefficient is selected in the parameter observers and positive Lyapunov function, which decides the transient period to identify the unknown parameters and reach synchronization. The scheme is used to study the synchronization of two non-identical hyperchaotic Rössler systems. The theoretical and numerical results confirm that the four unknown parameters in the drive system are estimated exactly and the two hyperchaotic systems reach complete synchronization when the controllers and parameter observers work on the driven system. To confirm the model independence of this scheme, an alternative hyperchaotic system is investigated, whereby the results confirm that the five unknown parameters are identified rapidly and exactly, and that the two hyperchaotic systems reach complete synchronization as well.     

Hyperchaotic secure communication via generalized function projective synchronization

This paper presents two different hyperchaotic secure communication schemes by using generalized function projective synchronization (GFPS), where the drive and response systems could be synchronized up to a desired scaling function matrix. The unpredictability of the scaling functions can additionally enhance the security of communication. First, a hyperchaotic secure communication scheme applying GFPS of the uncertain Chen hyperchaotic system is proposed. The transmitted information signal is modulated into the parameter of the Chen hyperchaotic system in the transmitter and it is assumed that the parameter of the receiver system is unknown. Based on the Lyapunov stability theory and the adaptive control technique, the controllers are designed to make two identical Chen hyperchaotic systems with unknown parameter asymptotically synchronized; thus, the uncertain parameter of the receiver system is identified. The information signal can be recovered accurately by the estimated parameter. Secondly, another secure communication scheme by the coupled GFPS of the Chen hyperchaotic system is introduced. The information signal transmitted can be extracted exactly through simple operation in the receiver. The corresponding theoretical proofs and numerical simulations demonstrate the validity and feasibility of the proposed hyperchaotic secure communication schemes.     

Adaptive synchronization of a hyperchaotic system with uncertain parameter

This paper addresses the synchronization problem of two Lü hyperchaotic dynamical systems in the presence of unknown system parameters. Based on Lyapunov stability theory an adaptive control law is derived to make the states of two identical Lü hyperchaotic systems with unknown system parameters asymptotically synchronized. Numerical simulations are presented to show the effectiveness of the proposed chaos synchronization schemes.     

Decentralized Adaptive Robust State Feedback for Uncertain Large-Scale Interconnected Systems with Time Delays

The problem of the decentralized robust control is considered for a class of large-scale time-varying systems withdelayed state perturbations and external disturbances in the interconnections. Here, the upper bounds of the delayed stateperturbations and external disturbances in the interconnections are assumed to be unknown. Adaptation laws areproposed to estimate such unknown bounds; by making use of the updated values of the unknown bounds, decentralized linear and nonlinear memoryless robust state feedback controllers are constructed. Based on Lyapunov stability theoryand Lyapunov–Krasovskii functionals, as well as employing the proposed decentralized nonlinear robust state feedback controllers, it is shown that the solutions of the resulting adaptive closed-loop large-scale time-delay system can be guaranteed to be uniformly bounded and that the states converge uniformly and asymptotically to zero. It is also shown that the proposed decentralized linear robust state feedback controllers can guarantee the uniform ultimate boundedness of the resulting adaptive closed-loop large-scale time-delay system. Finally, a numerical example is given to demonstrate the validity of the results.     

Tracking control and generalized projective synchronization of a class of hyperchaotic system with unknown parameter and disturbance

In this paper, the tracking control and generalized projective synchronization of a class of hyperchaotic system with unknown parameter and disturbance are investigated. Based on the LaSalle’s invariant set theorem, a robust adaptive controller is contrived to acquire tracking control and generalized projective synchronization and parameter identification simultaneously. It is proved theoretically that the proposed scheme can allow us to drive the hyperchaotic system to any desired reference signals, including hyperchaotic signals, chaotic signals, periodic orbits or fixed value by the given scaling factor. The presented simulation results further demonstrate that the proposed method is effective and robust.     

Modified projective synchronization of uncertain fractional order hyperchaotic systems

Base on the stability theory of fractional order system, this work mainly investigates modified projective synchronization of two fractional order hyperchaotic systems with unknown parameters. A controller is designed for synchronization of two different fractional order hyperchaotic systems. The method is successfully applied to modified projective synchronization between fractional order Rössler hyperchaotic system and fractional order Chen hyperchaotic system, and numerical simulations illustrate the effectiveness of the obtained results.     

基于神经网络的一类非线性时滞系统自适应控制器设计

针对一类带有扰动和未知时滞的非线性系统,通过反步方法设计一种鲁棒自适应控制器.提出了一种新的Lyapunov-Krasovskii泛函,补偿了未知时滞项的不确定性.引入一种合适的偶函数,避免了控制器的奇异性问题.通过Lyapunov直接方法,证明了所设计的控制器能保证闭环系统所有信号全局一致最终有界.