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 generalized function projective synchronization of uncertain chaotic systems

This paper mainly investigates adaptive generalized function projective synchronization of two different uncertain chaotic systems, which is a further extension of many existing projection synchronization schemes, such as modified projection synchronization, function projective synchronization and so on. On the basis of Lyapunov stability theory, an adaptive controller for the synchronization of two different chaotic systems is designed, and some parameter update laws for estimating the unknown parameters of the systems are also gained. This technique is applied to achieve synchronization between Lorenz and Rössler chaotic systems. The numerical simulations demonstrate the validity and feasibility of the proposed method.     

受扰不确定混沌系统的双路组合函数投影同步

研究了具有未知参数和外界扰动的多个混沌系统之间的双路组合函数投影同步问题.首先给出了由四个混沌驱动系统和两个混沌响应系统组成的双路组合函数投影同步系统的定义,然后以Lyapunov稳定性理论和不等式变换方法为分析依据,设计了鲁棒自适应控制器和参数自适应律,使得两路同步系统中的响应系统和驱动系统按照相应的函数比例因子矩阵实现同步,并有效克服未知有界干扰和未知参数的影响.相应的理论分析和数值仿真证明了该同步方案的可行性和有效性.     

Adaptive impulsive synchronization of uncertain delayed chaotic system with full unknown parameters via discrete‐time drive signals

This article investigates the adaptive impulsive synchronization of delayed chaotic system with full unknown parameters. Aiming at this problem, we propose a new adaptive strategy, in which both the adaptive–impulsive controller and the parameters adaptive laws are designed via the discrete‐time signals from the drive system. The corresponding theoretical proof is given to guarantee the effectiveness of the proposed strategy. Moreover, the concrete adaptive strategies are achieved for delayed Hopfield neural network, optical Ikeda system and the well‐known delayed Lü chaotic system. As expected, numerical simulations show the effectiveness of the proposed strategy. This method has potential applications in parameters estimation, secure communication, and cryptanalysis when only discrete signals are transmitted in communication channel. © 2014 Wiley Periodicals, Inc. Complexity 21: 43–51, 2016     

Adaptive synchronization of T–S fuzzy chaotic systems with unknown parameters

This paper presents a fuzzy model-based adaptive approach for synchronization of chaotic systems which consist of the drive and response systems. Takagi–Sugeno (T–S) fuzzy model is employed to represent the chaotic drive and response systems. Since the parameters of the drive system are assumed unknown, we design the response system that estimates the parameters of the drive system by adaptive strategy. The adaptive law is derived to estimate the unknown parameters and its stability is guaranteed by Lyapunov stability theory. In addition, the controller in the response system contains two parts: one part that can stabilize the synchronization error dynamics and the other part that estimates the unknown parameters. Numerical examples, including Duffing oscillator and Lorenz attractor, are given to demonstrate the validity of the proposed adaptive synchronization approach.     

受扰不确定混沌系统的降阶修正函数投影同步

研究了具有不同阶数的受扰不确定混沌系统的降阶修正函数投影同步问题.基于Lyapunov稳定性理论和自适应控制方法,设计了统一的非线性状态反馈控制器和参数更新规则,使得混沌响应系统按照相应的函数尺度因子矩阵和混沌驱动系统的部分状态变量实现同步.方法考虑了实际系统中的模型不确定性和外界扰动,具有较强的实用性和鲁棒性.数值仿真证明了控制方法的有效性.     

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.     

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.     

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.     

Chaos projective synchronization of the chaotic finance system with parameter switching perturbation and input time‐varying delay

This paper discusses some basic dynamical properties of the chaotic finance system with parameter switching perturbation, and investigates chaos projective synchronization of the chaotic finance system with the time‐varying delayed feedback controller, which are not fully considered in the existing research. Different from the previous methods, in this paper, the delayed feedback controller is not only time‐varying, but also the time‐varying delay is adaptive. Finally, an illustrate example is provided to show the effectiveness of this method. Copyright © 2014 John Wiley & Sons, Ltd.     

Adaptive synchronization of identical chaotic and hyper-chaotic systems with uncertain parameters

This paper addresses a unified mathematical expression describing a class of chaotic systems, for which the problem of adaptive synchronization between two nearly identical chaotic and hyper-chaotic systems with uncertain parameters is studied. Based on Lyapunov stability theory, a novel adaptive synchronization controller is designed, and the analytic expression of the controller and the adaptive laws of parameters are developed. The controller is simple and systemic, no parameters of the slave system are included in the controller, and, for some specific error systems, the controller can be simplified ulteriorly. New chaotic and a new hyper-chaotic systems with uncertain parameters are taken as the examples to show the effectiveness of the proposed adaptive synchronization method.     

Adaptive synchronization of Cohen-Grossberg neural networks with unknown parameters and mixed time-varying delays

In this paper, we investigate the synchronization problem of chaotic Cohen-Grossberg neural networks with unknown parameters and mixed time-varying delays. An adaptive linear feedback controller is designed to guarantee that the response system can be synchronized with a drive system by utilizing Lyapunov stability theory and parameter identification. Our synchronization criteria are easily verified and do not need to solve any linear matrix inequality. These results generalize a few previous known results and remove some restrictions on amplification function and time delay. This research also demonstrates the effectiveness of application in secure communication. Numerical simulations are carried out to illustrate the main results.     

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.     

Synchronization of Genesio chaotic system via backstepping approach

Backstepping design is proposed for synchronization of Genesio chaotic system. Firstly, the control problem for the chaos synchronization of nominal Genesio systems without unknown parameters is considered. Next, an adaptive backstepping control law is derived to make the error signals between drive Genesio system and response Genesio system with an uncertain parameter asymptotically synchronized. Finally, the approach is extended to the synchronization problem for the system with three unknown parameters. The stability analysis in this article is proved by using a well-known Lyapunov stability theorem. Note that the approach provided here needs only a single controller to realize the synchronization. Two numerical simulations are presented to show the effectiveness of the proposed chaos synchronization scheme.     

Adaptive fuzzy control‐based projective synchronization of uncertain nonaffine chaotic systems

This article aims to introduce a projective synchronization approach based on adaptive fuzzy control for a class of perturbed uncertain multivariable nonaffine chaotic systems. The fuzzy‐logic systems are employed to approximate online the uncertain functions. A Lyapunov approach is used to design the parameter adaptation laws and to demonstrate the boundedness of all signals of the closed‐loop system as well as the convergence of the synchronization errors to bounded residual sets. Finally, numerical simulation results are presented to verify the feasibility and effectiveness of the proposed synchronization system based on fuzzy adaptive controller. © 2014 Wiley Periodicals, Inc. Complexity 21: 180–192, 2015     

Adaptive Q–S synchronization between coupled chaotic systems with stochastic perturbation and delay

This work investigates the adaptive Q–S synchronization of coupled chaotic (or hyper-chaotic) systems with stochastic perturbation, delay and unknown parameters. The sufficient conditions for achieving Q–S synchronization of two stochastic chaotic systems are derived based on the invariance principle of stochastic differential equation. By the adaptive control technique, the control laws and the corresponding parameter update laws are proposed such that the stochastic Q–S synchronization of non-identical chaotic (or hyper-chaotic) systems is to be obtained. Finally, two illustrative numerical simulations are also given to demonstrate the effectiveness of the proposed scheme.     

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.     

Further results on the impulsive synchronization of uncertain complex‐variable chaotic delayed systems

Synchronization and control of nonlinear dynamical systems with complex variables has attracted much more attention in various fields of science and engineering. In this article, we investigate the problem of impulsive synchronization for the complex‐variable delayed chaotic systems with parameters perturbation and unknown parameters in which the time delay is also included in the impulsive moment. Based on the theories of adaptive control and impulsive control, synchronization schemes are designed to make a class of complex‐variable chaotic delayed systems asymptotically synchronized, and unknown parameters are identified simultaneously in the process of synchronization. Sufficient conditions are derived to synchronize the complex‐variable chaotic systems include delayed impulses. To illustrate the effectiveness of the proposed schemes, several numerical examples are given. © 2014 Wiley Periodicals, Inc. Complexity 21: 131–142, 2016     

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.