含有不确定项的混沌系统自适应广义同步

针对一类含有不确定项的混沌系统,设计了广义同步响应系统,利用系统稳定性理论设计了自适应广义同步控制器及自适应率,实现了驱动系统和所设计的响应系统广义同步,所设计的控制策略对外界干扰有较强的鲁棒性,而且通过引入加速因子,可任意配置同步响应速度,具有较高的应用价值,理论分析及仿真结果验证了该方法的有效性。     

Adaptive synchronization to a general non-autonomous chaotic system and its applications

In this paper, new adaptive synchronous criteria for a general class of n-dimensional non-autonomous chaotic systems with linear and nonlinear feedback controllers are derived. By suitable separation between linear and nonlinear terms of the chaotic system, the phenomenon of stable chaotic synchronization can be achieved using an appropriate adaptive controller of feedback signals. This method can also be generalized to a form for chaotic synchronization or hyper-chaotic synchronization. Based on stability theory on non-autonomous chaotic systems, some simple yet less conservative criteria for global asymptotic synchronization of the autonomous and non-autonomous chaotic systems are derived analytically. Furthermore, the results are applied to some typical chaotic systems such as the Duffing oscillators and the unified chaotic systems, and the numerical simulations are given to verify and also visualize the theoretical 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.     

Global finite-time synchronization of different dimensional chaotic systems

This paper addresses the problem of global finite-time synchronization of two different dimensional chaotic systems. Firstly, the definition of global finite-time synchronization of different dimensional chaotic systems are introduced. Based on the finite-time stability methods, the controller is designed such that the chaotic systems are globally synchronized in a finite time. Then, some uncertain parameters are adopted in the chaotic systems, new control law and dynamical parameter estimation are proposed to guarantee that the global finite-time synchronization can be obtained. By considering a dynamical parameter designed in the controller, the adaptive updated controller is also designed to achieve the desired results. At last, the results of two different dimensional chaotic systems are also extended to two different dimensional networked chaotic systems. Finally, three numerical examples are given to verify the validity of the proposed methods.     

An adaptive feedback control of linearizable chaotic systems

This paper proposes an adaptive feedback controller for a class of chaotic systems. This controller can be used for tracking a smooth orbit that can be a limit cycle or a chaotic orbit of another system. Based on Lyapunov approach, the adaptation law is determined to tune the controller gain vector in order to track a predetermined linearizing feedback control. To demonstrate the efficiency of the proposed scheme, two well-known chaotic systems namely Chua’s circuit and a Lur’e-like system are considered as illustrative examples.     

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 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.     

Adaptive synchronization for two identical generalized Lorenz chaotic systems via a single controller

This paper presents a systematic design procedure to synchronize two identical generalized Lorenz chaotic systems based on a sliding mode control. In contrast to the previous works, this approach only needs a single controller to realize synchronization, which has considerable significance in reducing the cost and complexity for controller implementation. A switching surface only including partial states is adopted to ensure the stability of the error dynamics in the sliding mode. Then an adaptive sliding mode controller (ASMC) is derived to guarantee the occurrence of the sliding motion even when the parameters of the drive and response generalized Lorenz systems are unknown. Last, an example is included to illustrate the results developed in this paper.     

Generalized lag-synchronization of chaotic mix-delayed systems with uncertain parameters and unknown perturbations

Chaotic systems in practice are always influenced by some unknown factors, which may make the chaotic behavior completely different from that of unaffected system. In this paper, generalized lag-synchronization for a general class of coupled chaotic systems with mixed delays, uncertain parameters, as well as external perturbations is investigated. A simple but all-powerful robust adaptive controller is designed to achieve this goal. Based on Lyapunov stability theory, integral inequality and Barbalat lemma, rigorous proofs are given for the asymptotic stability of the error systems of the coupled systems with or without external perturbations. Sufficient conditions for inaccuracy or accuracy estimation of unknown parameters are also given. Moreover, the designed adaptive controller has better anti-interference capacity than those of references. Numerical simulations verify the effectiveness of the theoretical results.     

Adaptive projective synchronization for a class of switched chaotic systems

This paper addresses the problem of projective synchronization of chaotic systems and switched chaotic systems by adaptive control methods. First, a necessary and sufficient condition is proposed to show how many state variables can realize projective synchronization under a linear feedback controller for the chaotic systems. Then, accordingly, a new algorithm is given to select all state variables that can realize projective synchronization. Furthermore, according to the results of the projective synchronization of chaotic systems, the problem of projective synchronization of the switched chaotic systems comprised by the unified chaotic systems is investigated, and an adaptive global linear feedback controller with only one input channel is designed, which can realize the projective synchronization under the arbitrary switching law. It is worth mentioning that the proposed method can also realize complete synchronization of the switched chaotic systems. Finally, the numerical simulation results verify the correctness and effectiveness of the proposed method.     

Some criteria for the global finite-time synchronization of two Lorenz–Stenflo systems coupled by a new controller

This paper investigates the global finite-time synchronization of two chaotic Lorenz–Stenflo systems coupled by a new controller called the generalized variable substitution controller. First of all, the generalized variable substitution controller is designed to establish the master–slave finite-time synchronization scheme for the Lorenz–Stenflo systems. And then, based on the finite-time stability theory, a sufficient criterion on the finite-time synchronization of this scheme is rigorously verified in the form of matrix and the corresponding estimation for the synchronization time is analytically given. Applying this criterion, some sufficient finite-time synchronization criteria under various generalized variable substitution controllers are further derived in the algebraic form. Finally, some numerical examples are introduced to compare the results proposed in this paper with those proposed in the existing literature, verifying the effectiveness of the criteria obtained.     

异结构离散型混沌系统的延迟同步

以异结构离散型混沌系统为研究对象,设计了一种延迟同步控制器实现了离散型Henon混沌系统和Ikeda混沌系统之间的同步控制．根据稳定性定理,确定了延迟同步控制器的结构以及系统状态变量之间的误差方程．设计的延迟同步控制器对于不同的离散型混沌系统具有统一的形式,可以实现任意异结构离散型混沌系统之间的延迟同步．数值仿真模拟进一步验证了该控制器的有效性．     

Analysis of generalized projective synchronization for a chaotic gyroscope with a periodic gyroscope

In this paper, a simple nonlinear controller is applied to investigate the generalized projective synchronization for a controlled chaotic gyroscope with a periodic gyroscope dynamical system. The necessary and sufficient conditions for generalized projective synchronization are developed through the theory of discontinuous dynamical systems. The synchronization invariant domain from the synchronization conditions is presented. The parameter maps are explored for a better understanding of the synchronicity of two gyroscopes with different motions. Finally, the partial and full generalized projective synchronizations of two nonlinear coupled gyroscope systems are carried out to verify the effectiveness of the scheme.     

Synchronizing the noise-perturbed Genesio chaotic system by sliding mode control

This paper investigates the chaos synchronization between Genesio chaotic systems with noise perturbation. It is proved theoretically that the synchronization between such noise-perturbed systems can be implemented by choosing a suitable sliding mode surface and designing a sliding mode controller. Numerical simulations show the effectiveness of the theoretical analysis. This proposed method is important because it can be applied to many other chaotic systems.     

分数阶不确定多混沌系统的自适应滑模同步

研究分数阶不确定多混沌系统的自适应滑模同步,通过构造滑模面,设计控制器和适应规则,能够满足滑模面的稳定性与到达性,进而得到分数阶不确定多混沌系统取得自适应滑模同步的充分性条件,研究表明:分数阶不确定多混沌系统满足在一定条件下能够取得自适应滑模同步.     

Generalized projective synchronization in time-delayed chaotic systems

We report on generalized projective synchronization between two identical time delay chaotic systems with single time delays. It overcomes some limitations of the previous work where generalized projective synchronization has been investigated only in finite-dimensional chaotic systems, so we can achieve generalized projective synchronization in infinite-dimensional chaotic systems. This method allows us to arbitrarily direct the scaling factor onto a desired value. Numerical simulations show that this method works very well.     

Global synchronization criteria for a class of third-order non-autonomous chaotic systems via linear state error feedback control

This paper investigates the global synchronization of a class of third-order non-autonomous chaotic systems via the master–slave linear state error feedback control. A sufficient global synchronization criterion of linear matrix inequality (LMI) and several algebraic synchronization criteria for single-variable coupling are proven. These LMI and algebraic synchronization criteria are then applied to two classes of well-known third-order chaotic systems, the generalized Lorenz systems and the gyrostat systems, proving that the local synchronization criteria for the chaotic generalized Lorenz systems developed in the existing literature can actually be extended to describe global synchronization and obtaining some easily implemented synchronization criteria for the gyrostat systems.     

Counteracting the dynamical degradation of digital chaos via hybrid control

Chaotic systems would degrade owing to finite computing precisions, and such degradation often seriously affects the performance of digital chaos-based applications. In this paper, a chaotification method is proposed to solve the dynamical degradation of digital chaotic systems based on a hybrid structure, where a continuous chaotic system is applied to control the digital chaotic system, and a unidirectional coupling controller that combines a linear external state control with a modular function is designed. Moreover, we proof rigorously that a class of digital chaotic systems can be driven to be chaotic in the sense that the system is sensitive to initial conditions. Different from the existing remedies, this method can recover the dynamical properties of system, and even make some properties better than those of the original chaotic system. Thus, this new approach can be applied to the fields of chaotic cryptography and secure communication.     

Synchronization and anti-synchronization coexist in Chen–Lee chaotic systems

This study demonstrates that synchronization and anti-synchronization can coexist in Chen–Lee chaotic systems by direct linear coupling. Based on Lyapunov’s direct method, a linear controller was designed to assure that two different types of synchronization can simultaneously be achieved. Further, the hybrid projective synchronization of Chen–Lee chaotic systems was studied using a nonlinear control scheme. The nonlinear controller was designed according to the Lyapunov stability theory to guarantee the hybrid projective synchronization, including synchronization, anti-synchronization, and projective synchronization. Finally, numerical examples are presented in order to illustrate the proposed synchronization approach.     

Synchronization of generalized Henon map via backstepping design

This paper proposes a backstepping method to resolve the synchronization of discrete-time chaotic systems. The proposed scheme offers systematic design method for the synchronization of a class of discrete-time hyper-chaotic systems, which implies much complicated high-order chaotic systems can be used to improve the security in chaos communications. A well-known chaotic systems: generalized Henon map is considered as illustrative example to demonstrate the general applicability of backstepping design. Numerical simulations verify the effectiveness of the approach.