Projective Synchronization Of Chaotic Systems Research Articles

A fixed-time robust controller based on zeroing neural network for generalized projective synchronization of chaotic systems

Generalized projective synchronization (GPS) as a deeply influential chaos synchronization has always attracted lots of attention. However, plenty of traditional control methods do not predict its synchronization time or have no regard for the interference of noise in practical applications. Inspired by the fact that zeroing neural network (ZNN) can solve the time-varying problems well, this paper adopts the design method of the ZNN to construct a fixed-time robust controller (FXTRC), realizing the GPS of a class of chaotic systems. The fixed-time synchronization and robustness of chaotic systems under the FXTRC are clearly demonstrated by detailed theoretical analyses. Moreover, the upper bound of the synchronization time can be calculated by introducing the Beta function when the FXTRC is applied to control the GPS of chaotic systems. Numerical simulations prove the correctness of the theoretical analyses and the superiority of the FXTRC over the previous control methods.

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.

Improved Control Schemes for Projective Synchronization of Delayed Neural Networks with Unmatched Coefficients

Projective synchronization (PS) is a type of chaos synchronization where the states of slave system are scaled replicas of the states of master system. This paper studies the asymptotic projective synchronization (APS) between master–slave chaotic neural networks (NNs) with mixed time-delays and unmatched coefficients. Based on useful inequality techniques and constructing a suitable Lyapunov functional, some simple criteria are derived to ensure the APS of considered networks via designing a novel adaptive feedback controller. In addition, a numerical example and its MATLAB simulations are provided to check the feasibility of the obtained results. The main innovation of our work is that we dealt with the APS problem between two different chaotic NNs, while most of the existing works only concerned with the PS of chaotic systems with the same topologies. In addition, compared with the controllers introduced in the existing papers, the designed controller in this paper does not require any knowledge about the activation functions, which can be seen as another novelty of the paper.

A Novel Adaptive Active Control Projective Synchronization of Chaotic Systems

This paper investigates adaptive active control projective synchronization scheme. A general synchronization controller and parameter update laws are proposed to stabilize the error system for the identical structural chaotic systems. It is the first time that the active synchronization, the projective synchronization, and the adaptive synchronization are combined to achieve the synchronization of chaotic systems, which extend the control capability of achieving chaotic synchronization. By using a constant diagonal matrix, the active control is developed. Especially, when designing the controller, we just need to ensure that the diagonal elements of the diagonal matrix are less than or equal 0. So, the synchronization of chaotic systems can be realized more easily. Furthermore, by proposing an active controller, in combination with several different control schemes, we lower the complexity of the design process of the controller. More importantly, the larger the absolute value of product of the diagonal elements of diagonal matrix is, the smoother the curve of chaotic synchronization is and the shorter the time of chaotic synchronization is. In our paper, we take Lorenz system as an example to verify the effectiveness of the proposed synchronization scheme. Theoretical analysis and numerical simulations demonstrate the feasibility of this control method.

Modified Projective Synchronization of Chaotic Systems with Noise Disturbance, an Active Nonlinear Control Method

<p>The synchronization problem of chaotic systems using active modified projective nonlinear control method is rarely addressed. Thus the concentration of this study is to derive a modified projective controller to synchronize the two chaotic systems. Since, the parameter of the master and follower systems are considered known, so active methods are employed instead of adaptive methods. The validity of the proposed controller is studied by means of the Lyapunov stability theorem. Furthermore, some numerical simulations are shown to verify the validity of the theoretical discussions. The results demonstrate the effectiveness of the proposed method in both speed and accuracy points of views.</p>

Bounded Scaling Function Projective Synchronization of Chaotic Systems with Adaptive Finite-Time Control

This paper investigates the bounded scaling function projective synchronization of uncertain chaotic systems using adaptive finite-time control. Based on finite-time control and inequality principle, the new adaptive finite-time controller is designed to achieve two chaotic systems scaling function projective synchronized, and uncertain parameters of chaotic systems are also identified. Moreover, in comparison with those of the existing scaling function synchronization, the given scaling function can be more complex bounded functions. Some numerical are also given to show the effectiveness of the proposed method.

Hybrid function projective synchronization of chaotic systems via adaptive control

In this manuscript, hybrid function projective synchronization of Bhalekar–Gejji and Pehlivan chaotic systems is established by applying adaptive control technique where the system parameters are unknown. In this manuscript both the master and slave system are chosen in such a way that none of them can be derived from the member of the unified chaotic system. We construct an adaptive controller in such a manner that master and slave system attain global chaos synchronization. The results derived for the synchronization have been established using adaptive control theory and Lyapunov stability theory. Fundamental dynamical properties of both the chaotic systems are also described. The results are validated by numerical simulation which are performed by using Matlab.

Matrix Projective Synchronization of Chaotic Systems and the Application in Secure Communication

First of all, we investigate adaptive matrix projective synchronization of the chaotic system. Finally, this method is applied to secure communication through improved chaotic masking. The information signal is mixed with the chaotic signal before being transmitted, and is recovered without distortion through the synchronized receiver. Simulation results show that the scheme has a good performance.

Generalized Hybrid Dislocated Function Projective Synchronization of Chaotic Systems with Time Delay

In order to improve the security of secure communication, a novel generalized hybrid dislocated function projective synchronization (GHDFPS) was proposed and GHDFPS of time delay chaotic systems with uncertain parameters were researched in this paper. Due to time delay, the chaotic system can produce multiple positive Lyapunov exponential; this characteristic can enhance security in secure communications noticeably. Based on Lyapunove stability theory and modified hybrid feedback control method, the modified hybrid feedback controller and the parameter updating laws were designed for the GHDFPS between the two time delay chaotic systems with uncertain parameters. The feedback gain can be adjusted automatically according to the synchronization error values. Under the controller, generalized hybrid dislocated function projective synchronization of the two chaotic systems is achieved, and the uncertain parameters of response systems are identified. The chaotic item is added in the function scale factor. The chaotic item in the function scaling factor makes function scaling factor more complex and unpredictable. So this can enhance the features of indeterminism in secure communication. The time delay feedback Lorenz system as an example; by numerical simulations the effectiveness of the proposed method is demonstrated.

Projective synchronization of chaotic systems with bidirectional nonlinear coupling

This paper presents a new scheme for constructing bidirectional nonlinear coupled chaotic systems which synchronize projectively. Conditions necessary for projective synchronization (PS) of two bidirectionally coupled chaotic systems are derived using Lyapunov stability theory. The proposed PS scheme is discussed by taking as examples the so-called unified chaotic model, the Lorenz–Stenflo system and the nonautonomous chaotic Van der Pol oscillator. Numerical simulation results are presented to show the efficiency of the proposed synchronization scheme.

Projective synchronization of chaotic systems via backstepping design

Chaos synchronization of discrete dynamical systems are investigated. An algorithm is proposed for projective synchronization of chaotic 2D Duffing map and chaotic Tinkerbell map. The control law was derived from the Lyapunov stability theory. The numerical simulation results are presented to verify the effectiveness of the proposed algorithm.

Modified projective synchronization of unknown chaotic dissipative gyroscope systems via Gaussian radial basis adaptive variable structure control

This paper proposes the modified projective synchronization method for unknown chaotic dissipative gyroscope systems via Gaussian radial basis adaptive variable structure control. Because of the nonlinear terms of the dissipative gyroscope system, the system exhibits chaotic motions. As chaotic signals are usually broadband and noise-like, synchronized chaotic systems can be used as cipher generators for secure communication. Obviously the importance of obtaining these objectives is specified when the dynamics of the gyroscope system are unknown. In this paper, using the neural variable structure control technique, control laws are established, which guarantees the modified projective synchronization of an unknown chaotic dissipative gyroscope system. Switching surfaces are adopted to ensure the stability of the error dynamics in variable structure control. In the neural variable structure control, Gaussian radial basis functions are utilized online to estimate the system dynamic functions. Also, the adaptation laws of the online estimators are derived in the sense of Lyapunov function. Thus, the unknown chaotic gyroscope system can be guaranteed to be asymptotically stable. Also, the synchronization objectives have been achieved. The proposed method allows us to arbitrarily adjust the desired scaling by controlling the slave system. It is not necessary to calculate the Lyapunov exponents and the eigenvalues of the Jacobian matrix, which makes it simple and convenient. Also, it is a systematic procedure for modified projective synchronization of chaotic systems and it can be applied to a variety of chaotic systems no matter whether it contains external excitation or not. The designed control system is robust versus model uncertainty. Numerical simulations are presented to verify the proposed synchronization method.

Adaptive Hybrid Function Projective Synchronization of Chaotic Systems with Time‐Varying Parameters

The adaptive hybrid function projective synchronization (AHFPS) of different chaotic systems with unknown time‐varying parameters is investigated. Based on the Lyapunov stability theory and adaptive bounding technique, the robust adaptive control law and the parameters update law are derived to make the states of two different chaotic systems asymptotically synchronized. In the control strategy, the parameters need not be known throughly if the time‐varying parameters are bounded by the product of a known function of t and an unknown constant. In order to avoid the switching in the control signal, a modified robust adaptive synchronization approach with the leakage‐like adaptation law is also proposed to guarantee the ultimately uni‐formly boundedness (UUB) of synchronization errors. The schemes are successfully applied to the hybrid function projective synchronization between the Chen system and the Lorenz system and between hyperchaotic Chen system and generalized Lorenz system. Moreover, numerical simulation results are presented to verify the effectiveness of the proposed scheme.

Robust Adaptive Generalized Projective Synchronization of Chaotic Systems with Uncertain Disturbances

The generalized projective synchronization (GPS) of chaotic systems with uncertain parameter noise and external disturbance is discussed. Based on the adaptive technique, a response system is constructed, and a novel adaptive controller is designed to guarantee the GPS between the drive‐response systems, and to eliminate the effect of external disturbance and parameters noise on GPS. The conclusion is proved theoretically, and corresponding numerical simulations are provided to verify the effectiveness of the proposed method.

FUNCTION PROJECTIVE SYNCHRONIZATION OF CHAOTIC SYSTEMS VIA NONLINEAR ADAPTIVE–IMPULSIVE CONTROL

In this paper, function projective synchronization of chaotic systems is investigated through nonlinear adaptive–impulsive control. To achieve synchronization, suitable nonlinear continuous and impulsive controllers are designed, according to invariant principle of impulsive dynamical systems. Sufficient conditions are given to ensure the synchronization. Numerical simulation results show the effectiveness of the proposed scheme.

Modified Projective Synchronization of Chaotic Systems with Unknown Parameters

This paper proposes the modified projective synchronization of uncertain chaotic systems with unknown parameters via active adaptive sliding mode control (AASMC). The disturbances are considered both in the drive and the response system. The bounds of the disturbances are unknown. The adaptive updating laws are designed to tackle the unknown parameters. Moreover, the robustness and stability of the proposed AASMC is proved by the Lyapunov stability theory. Some numerical simulations are given to demonstrate the robustness and efficiency of the proposed scheme.

Generalized function projective synchronization of chaotic systems for secure communication

By using the generalized function projective synchronization (GFPS) method, in this paper, a new scheme for secure information transmission is proposed. The Liu system is employed to encrypt the information signal. In the transmitter, the original information signal is modulated into the system parameter of the chaotic systems. In the receiver, we assume that the parameter of receiver system is uncertain. Based on the Lyapunov stability theory, the controllers and corresponding parameter update rule are constructed to achieve GFPS between the transmitter and receiver system with uncertain parameters, and identify unknown parameters. The original information signal can be recovered successfully through some simple operations by the estimated parameter. Furthermore, by means of the proposed method, the original information signal can be extracted accurately in the presence of additional noise in communication channel. Numerical results have verified the effectiveness and feasibility of presented method. Mathematics subject classification (2010) 68M10, 34C28, 93A30, 93C40

Hybrid Function Projective Synchronization of Chaotic Systems with Fully Unknown Parameters

To compensate for projective synchronization (PS) and function projective synchronization (FPS), we propose a hybrid function projective synchronization (HFPS), which applies the different time-varying functions as the synchronization scaling factors. Based on the adaptive control method, we design a simple controller and a set of update laws of unknown parameters to carry out HFPS in identical and different chaotic systems with fully unknown parameters. According to the Lyapunov stability theorem and the Barbalat lemma, we prove the asymptotical stability of the error dynamical system at the origin. Then two numerical examples are given to validate the feasibility and effectiveness of the developed procedure in this paper. Key Words: Hybrid function projective synchronization; Lyapunov stability theorem; Adaptive control; Unknown parameters

Unified projective synchronization of chaotic system

A unified method for the projective synchronization of chaotic system is proposed in this paper. The universal model of chaotic projective synchronization is established by constructing a generalized proportion matrix and an appropriate response system. All kinds of synchronized schemes can be achieved by varying the generalized proportion matrix, including generalized projective synchronization, dislocated projective synchronization and generalized hybrid dislocated projective synchronization and so on. The stability analysis in the paper is proved using Lyapunov stability theory. Numerical simulations of generalized hybrid dislocated projective synchronization for multiscroll chaotic attractors system and hyperchaotic Qi system verify the effectiveness of the proposed method.

Generalized projective synchronization of chaotic systems via modified active control

Based on Lyapunov stability theory, by introducing a special matrix structure, a modified active control is proposed for the generalized projective synchronization of chaotic system. Compared with the traditional active control, the modified active control is independent of the Routh-Hurwitz criterion, which means the complexity of active control is simplified. The method is successfully applied to the energy resource system and nuclear spin generator system. Compared with other method, the proposed method, which could realize the generalized projective synchronization between both identical systems and different systems, is shown to be simple, direct, stable and efficient. Numerical simulation is provided to show the validity of the theoretical analysis and the effectiveness of the method.