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.     

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.     

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.     

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.     

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.     

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.     

Cascades-based synchronization of hyperchaotic systems: Application to Chen systems

We discuss the cascaded-based controlled synchronization method for hyperchaotic systems. The control approach is based on analysis tools for cascaded time-varying systems. That is, the closed-loop system takes the form of two subsystems which are interconnected in a manner that the state of one system enters into another but without feedback loop. The advantage of such construction is that the controller is largely simplified relative to other design methods such as backstepping. We apply the method to Chen’s hyperchaotic system and show that global synchronization is achieved via linear control. Also, we assume that only three instead of four control inputs are available. The method is tested in numerical simulations.     

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.     

A new hyper-chaotic system and its synchronization

This paper presents a new hyper-chaotic system obtained by adding a nonlinear controller to the third equation of the three-dimensional autonomous Chen–Lee chaotic system. Computer simulations demonstrated the hyper-chaotic dynamic behaviors of the system. Numerical results revealed that the new hyper-chaotic system possesses two positive exponents. It was also found that the structure of the hyper-chaotic attractors is more complex than those of the Chen–Lee chaotic system. Furthermore, the hybrid projective synchronization (HPS) of the new hyper-chaotic systems was studied using a nonlinear feedback control. The nonlinear controller was designed according to Lyapunov’s direct method to guarantee HPS, which includes synchronization, anti-synchronization, and projective synchronization. Numerical examples are presented in order to illustrate HPS.     

Robust adaptive modified function projective synchronization of different hyperchaotic systems subject to external disturbance

Robust adaptive modified function projective synchronization between two different hyperchaotic systems is investigated, where the external uncertainties are considered and the scale factors are different from each other. The synchronization criterion is presented, which can be realized by adaptive feedback controller with compensator to eliminate the influence of uncertainties effectively. The update laws of the unknown parameters are given and the sufficient conditions are deduced based on stability theory and adaptive control. And some mistakes in the previous works are pointed out and revised. Finally, the hyperchaotic Lü and new hyperchaotic Lorenz systems are taken for example and the numerical simulations are presented to verify the effectiveness and robustness of the proposed control scheme.     

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.     

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.     

The parametric synchronization scheme of chaotic system

By constructing the parametric error vectors between drive system and response system, a parametric synchronization scheme of chaotic system which is different from all other schemes is proposed in this paper. Controller of the scheme is designed. The proposed scheme and controller not only realize the synchronization of the state vectors, but also synchronize the unknown response parameters to the given drive parameter as time goes to infinity. That is to say, to achieving the synchronization, we have no need to know the parameters of response system when the parameters of drive system are given. The scheme and controller are successfully applied to the Rössler and the hyperchaotic Rössler systems, corresponding numerical simulations are presented to show the validity of the proposed synchronization scheme and effectiveness of the controller.     

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.     

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.     

Tracking the state of the delay hyperchaotic Lü system using the coullet chaotic system via a single controller

In this article, a partial synchronization scheme is proposed based on Lyapunov stability theory to track the signal of the delay hyperchaotic Lü system using the Coullet system based on only one single controller. The proposed tracking control design has two advantages: only one controller is adopted in our approach and it can allow us to drive the hyperchaotic system to a simple chaotic system even with uncertain parameters. Numerical simulation results are given to demonstrate the effectiveness and robustness of the proposed partial synchronization scheme. © 2014 Wiley Periodicals, Inc. Complexity 21: 125–130, 2016     

Lag synchronization of hyperchaos with application to secure communications

In this paper, hyperchaotic lag synchronization is restated as a nonlinear and lag-in-time observer design issue. This approach leads to a systematic tool, which guarantees the lag synchronization of a wide class of chaotic or hyperchaotic systems via a scalar signal. By exploiting this result, we propose a hyperchaos-based cryptosystem scheme that combines the conventional cryptographic methods and the lag synchronization of chaotic circuits. The computer simulation results show that the lag synchronization scheme and the cryptosystem proposed in this paper are both feasible.     

Synchronization of mechanical horizontal platform systems in finite time

The horizontal platform system (HPS) is a mechanical device that exhibits rich and chaotic dynamics. In this paper, the problem of finite-time synchronization of two non-autonomous chaotic HPSs is investigated. It is assumed that both drive and response systems are disturbed by model uncertainties, external disturbances and fully unknown parameters. Appropriate update laws are proposed to undertake the unknown parameters. Using the update laws and finite-time control theory, a robust adaptive controller is derived to synchronize the two uncertain HPSs in a given finite time. Subsequently, the effects of input nonlinearities are taken into account and a robust adaptive controller is introduced to synchronize the two uncertain HPSs within a finite time. The finite-time stability and convergence of the proposed schemes are analytically proved. Two illustrative examples are presented to show the robustness and applicability of the proposed adaptive finite-time control techniques.     

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.     

不同超混沌系统的最优同步

在参数未知的情况下,通过设计最优控制器和参数自适应律实现了新的四维混沌系统与超混沌吕系统的同步.接着根据Lyapunov稳定性原理和Hamilton-Jacobi-Bellman方程,选取Lyapunov函数和合适的性能指标函数从理论上证明这种方法的有效性.理论证明结果表明所设计的控制器能使性能指标函数取得最小值,是最优的.最后又通过matlab软件对同步系统进行数值仿真,仿真结果显示驱动系统与响应系统能够很好地达到了同步,表明方法是可行有效的.