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.     

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.     

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.     

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     

Dead-beat full state hybrid projective synchronization for chaotic maps using a scalar synchronizing signal

This paper provides a contribution to the topic of full state hybrid projective synchronization (FSHPS) by introducing an observer-based approach that enables synchronization to be achieved via a scalar synchronizing signal. The method is based on a theorem that assures dead-beat synchronization (i.e., exact synchronization in finite time) to a wide class of discrete-time chaotic (hyperchaotic) systems. Two examples, involving the hyperchaotic Grassi-Miller map and the hyperchaotic double scroll map, show that FSHPS can be effectively achieved in finite time using a scalar synchronizing signal only.     

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.     

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     

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.     

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 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.     

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.     

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 new less conservative criteria for impulsive synchronization of a hyperchaotic Lorenz system based on small impulsive signals

In this Letter the issue of impulsive Synchronization of a hyperchaotic Lorenz system is developed. We propose an impulsive synchronization scheme of the hyperchaotic Lorenz system including chaotic systems. Some new and sufficient conditions on varying impulsive distances are established in order to guarantee the synchronizability of the systems using the synchronization method. In particular, some simple conditions are derived for synchronizing the systems by equal impulsive distances. The boundaries of the stable regions are also estimated. Simulation results show the proposed synchronization method to be effective.     

Simplified method and synchronization for a class of complex chaotic systems

This paper is devoted to investigate a class of complex chaotic systems and a linear correlation between the real and imaginary component of complex variables in these systems is found. Based on this linear relationship, a simplified law is proposed. First, complex Lorenz system is given to show the linear correlation, then it is simplified. Second, a simplified law is proposed to determine whether the complex system can be simplified, and the complex Lü system and hyperchaotic complex Lü system are used to verify the simplified law. Finally, a new synchronization control is proposed to synchronize complex Lorenz system and real Lorenz system. The theoretical analysis and numerical simulation prove the feasibility and better performance of this method.     

A special full-state hybrid projective synchronization in symmetrical chaotic systems

A special full-state hybrid projective synchronization type is proposed in this paper. The anti-synchronization and complete synchronization can be achieved simultaneously in this new synchronization phenomenon. We point out how to realize this synchronization in chaotic systems: anti-synchronization in symmetrical coordinate subspace and complete synchronization in its normal coordinate subspace. Two illustrative examples, multi-scroll chaotic system by the partial Lyapunov stability theory, and a four-dimensional chaotic system by the invariance principle of differential equation are presented to exhibit this new synchronization.     

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.     

Complex complete synchronization of two nonidentical hyperchaotic complex nonlinear systems

In this paper, we introduce the definition of complex complete synchronization (CCS) of hyperchaotic complex nonlinear systems that have not been introduced recently in the literature. This type of synchronization can study only for complex nonlinear systems. On the basis of Lyapunov function, a scheme is designed to achieve the CCS of two nonidentical hyperchaotic attractors of these systems. The effectiveness of the obtained results is illustrated by a simulation example. Numerical results are plotted to show state variables, modules errors, and phases errors of these hyperchaotic attractors after synchronization to prove that CCS is achieved. 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.     

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.     

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.