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.     

Function vector synchronization based on fuzzy control for uncertain chaotic systems with dead‐zone nonlinearities

In this article, a fuzzy adaptive control scheme is designed to achieve a function vector synchronization behavior between two identical or different chaotic (or hyperchaotic) systems in the presence of unknown dynamic disturbances and input nonlinearities (dead‐zone and sector nonlinearities). This proposed synchronization scheme can be considered as a generalization of many existing projective synchronization schemes (namely the function projective synchronization, the modified projective synchronization, generalized projective synchronization, and so forth) in the sense that the master and slave outputs are assumed to be some general function vectors. To practically deal with the input nonlinearities, the adaptive fuzzy control system is designed in a variable‐structure framework. The fuzzy systems are used to appropriately approximate the uncertain nonlinear functions. A Lyapunov approach is used to prove the boundedness of all signals of the closed‐loop control system as well as the exponential convergence of the corresponding synchronization errors to an adjustable region. The synchronization between two identical systems (chaotic satellite systems) and two different systems (chaotic Chen and Lü systems) are taken as two illustrative examples to show the effectiveness of the proposed method. © 2015 Wiley Periodicals, Inc. Complexity 21: 234–249, 2016     

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.     

A new hyperchaotic system and its synchronization

In this paper, a four-dimensional (4D) continuous autonomous hyperchaotic system is introduced and analyzed. This hyperchaotic system is constructed by adding a linear controller to the 3D autonomous chaotic system with a reverse butterfly-shape attractor. Some of its basic dynamical properties, such as Lyapunov exponents, Poincare section, bifurcation diagram and the periodic orbits evolving into chaotic, hyperchaotic dynamical behavior by varying parameter d are studied. Furthermore, the full state hybrid projective synchronization (FSHPS) of new hyperchaotic system with unknown parameters including the unknown coefficients of nonlinear terms is studied by using adaptive control. Numerical simulations are presented to show the effective of the proposed chaos 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.     

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.     

Generalized projective synchronization of a class of hyperchaotic systems based on state observer

In this paper, the generalized projective synchronization of a class of hyperchaotic systems is studied. On the basis of the state observer, it is not necessary to calculate the Lyapunov exponents, which makes this scheme simpler. Hyperchaotic Lü system and hyperchaotic Rössler systems are used as examples to validate the effectiveness of the proposed method.     

Projective and lag synchronization of a novel hyperchaotic system via impulsive control

Recently, Yang, Zhang, and Chen (2009) have proposed a novel hyperchaotic system. This paper studies the projective and lag synchronization of this novel hyperchaotic system using an impulsive control technique. Some sufficient conditions of projective and lag synchronization of such new system are derived from strict mathematical theories. Numerical examples are worked through for illustrating the main results.     

Modified generalized projective synchronization of a new fractional-order hyperchaotic system and its application to secure communication

This paper presents a new fractional-order hyperchaotic system. The chaotic behaviors of this system in phase portraits are analyzed by the fractional calculus theory and computer simulations. Numerical results have revealed that hyperchaos does exist in the new fractional-order four-dimensional system with order less than 4 and the lowest order to have hyperchaos in this system is 3.664. The existence of two positive Lyapunov exponents further verifies our results. Furthermore, a novel modified generalized projective synchronization (MGPS) for the fractional-order chaotic systems is proposed based on the stability theory of the fractional-order system, where the states of the drive and response systems are asymptotically synchronized up to a desired scaling matrix. The unpredictability of the scaling factors in projective synchronization can additionally enhance the security of communication. Thus MGPS of the new fractional-order hyperchaotic system is applied to secure communication. Computer simulations are done to verify the proposed methods and the numerical results show that the obtained theoretic results are feasible and efficient.     

超混沌系统的函数投影同步与参数辨识

研究了一参数未知超混沌系统的函数投影同步问题．基于李雅谱诺夫稳定性理论,设计了实现混沌系统函数投影同步的有效非线性控制器,可以快速实现超混沌系统的加速函数投影同步,同时设计了参数控制律,有效的辨识了系统的未知参数,数值仿真验证了理论分析和数值计算的正确性．     

Control and synchronize a novel hyperchaotic system

The control and hybrid projective synchronization (HPS) strategies for a novel hyperchaotic system are investigated. Firstly, the novel hyperchaotic system is controlled to the unsteady equilibrium point or limit cycle via only one scalar controller which includes two state variables. Secondly, based on Lyapunov’s direct method HPS between two novel hyperchaotic systems is studied. A new nonlinear feedback vector controller is designed to guarantee HPS, which can be simplified ulteriorly into a single scalar controller to achieve complete synchronization between two novel hyperchaotic systems. Finally, numerical simulations are given to verify the effectiveness of these strategies. The proposed methods have certain significances for reducing the cost and complexity for controller implementation.     

A new scheme of general hybrid projective complete dislocated synchronization

Based on the Lyapunov stability theorem, a new type of chaos synchronization, general hybrid projective complete dislocated synchronization (GHPCDS), is proposed under the framework of drive-response systems. The difference between the GHPCDS and complete synchronization is that every state variable of drive system does not equal the corresponding state variable, but equal other ones of response system while evolving in time. The GHPCDS includes complete dislocated synchronization, dislocated anti-synchronization and projective dislocated synchronization as its special item. As examples, the Lorenz chaotic system, Rössler chaotic system, hyperchaotic Chen system and hyperchaotic Lü system are discussed. Numerical simulations are given to show the effectiveness of these methods.     

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.     

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.     

超混沌Lorenz系统的脉冲控制与修正投影同步

考虑超混沌Lorenz系统的脉冲控制与修正投影同步,基于脉冲控制系统的稳定性理论,给出了脉冲控制与修正投影同步的充分条件,并通过数值仿真验证了所给充分条件的有效性.由定理4易知当同步因子α_1,α_2,α_3,α_4满足α_1~2=1,α_2=α_1α_3=α_4时所给同步方法无需控制器,因此方法可以看做是脉冲完全同步的推广.     

带有不确定参数的一类混沌金融系统的广义投影同步

针对带有不确定参数的一类混沌金融系统,提出了实现驱动系统和响应系统广义投影同步的自适应控制策略,并基于Lyapunov稳定性理论给出和验证了广义投影同步稳定性判据.数值仿真验证了控制策略和理论分析的有效性.     

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     

Adaptive control and synchronization of a new modified hyperchaotic Lü system with uncertain parameters

This paper addresses problems of control and synchronization for a new modified hyperchaotic Lü system with uncertain parameters. This new modified uncertain hyperchaotic Lü system is stabilized to its unique unstable equilibrium by using adaptive control. Furthermore, an adaptive control law and a parameter estimation update law are derived to synchronize two identical modified hyperchaotic Lü systems with uncertain parameters. Numerical examples are proposed to demonstrate and verify the theoretical analysis.     

Generalized projective synchronization for a class of chaotic systems with parameter mismatching,unknown external excitation,and uncertain input nonlinearity

In this paper, the concept of the generalized projective synchronization in practical type (GPSPT) is introduced and the GPSPT of uncertain horizontal platform systems with parameter mismatching, unknown external excitation, and uncertain input nonlinearity is investigated. Based on the differential inequality methodology, a tracking control is proposed to realize the GPSPT for the uncertain horizontal platform systems with parameter mismatching, unknown external excitation, and uncertain input nonlinearity. Not only the desired scaling factor, the guaranteed exponential convergence rate, and convergence radius can be pre-specified, but also the parameter mismatching and unknown external excitation can be simultaneously conquered by the proposed control. Finally, numerical simulations are provided to illustrate the feasibility and effectiveness of the obtained result.     

Ultimate bound sets of a hyperchaotic system and its application in chaos synchronization

In this article, we investigate globally exponentially attractive sets and chaos synchronization for a hyperchaotic system, namely, Lorenz–Stenflo system. For this system, two ellipsoidal globally exponentially attractive sets are derived based on generalized Lyapunov function theory and the extremum principle of function. Furthermore, we propose linear feedback control with a one, two, three, and four inputs to realize globally exponential synchronization of two four‐dimesional hyperchaotic systems using inequality techniques. Numerical simulations are presented to show the effectiveness of the proposed synchronization scheme. © 2014 Wiley Periodicals, Inc. Complexity 20: 30–44, 2015