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.     

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.     

Chaotic communication using generalized synchronization

We present a new technique for the chaotic communication of a signal using the concept of generalized synchronization. We develop a general approach for implementing our technique and illustrate it using a Rössler system driving a Lorenz system. It is demonstrated that the scheme is robust with respect to noise in the communication channel and to small parameter mismatches in the system. Finally, we discuss the advantages of this technique over existing methods and examine ways of improving the scheme.     

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.     

Exponential synchronization of discontinuous chaotic systems via delayed impulsive control and its application to secure communication

This paper investigates drive-response synchronization of chaotic systems with discontinuous right-hand side. Firstly, a general model is proposed to describe most of known discontinuous chaotic system with or without time-varying delay. An uniform impulsive controller with multiple unknown time-varying delays is designed such that the response system can be globally exponentially synchronized with the drive system. By utilizing a new lemma on impulsive differential inequality and the Lyapunov functional method, several synchronization criteria are obtained through rigorous mathematical proofs. Results of this paper are universal and can be applied to continuous chaotic systems. Moreover, numerical examples including discontinuous chaotic Chen system, memristor-based Chua’s circuit, and neural networks with discontinuous activations are given to verify the effectiveness of the theoretical results. Application of the obtained results to secure communication is also demonstrated in this paper.     

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.     

Chaos control and generalized projective synchronization of heavy symmetric chaotic gyroscope systems via Gaussian radial basis adaptive variable structure control

This paper proposes the chaos control and the generalized projective synchronization methods for heavy symmetric gyroscope systems via Gaussian radial basis adaptive variable structure control. Because of the nonlinear terms of the gyroscope system, the system exhibits chaotic motions. Occasionally, the extreme sensitivity to initial states in a system operating in chaotic mode can be very destructive to the system because of unpredictable behavior. In order to improve the performance of a dynamic system or avoid the chaotic phenomena, it is necessary to control a chaotic system with a periodic motion beneficial for working with a particular condition. As chaotic signals are usually broadband and noise like, synchronized chaotic systems can be used as cipher generators for secure communication. This paper presents chaos synchronization of two identical chaotic motions of symmetric gyroscopes. In this paper, the switching surfaces are adopted to ensure the stability of the error dynamics in variable structure control. Using the neural variable structure control technique, control laws are established which guarantees the chaos control and the generalized projective synchronization of unknown gyroscope systems. In the neural variable structure control, Gaussian radial basis functions are utilized to on-line estimate the system dynamic functions. Also, the adaptation laws of the on-line estimator are derived in the sense of Lyapunov function. Thus, the unknown gyro systems can be guaranteed to be asymptotically stable. Also, the proposed method can achieve the control objectives. Numerical simulations are presented to verify the proposed control and synchronization methods. Finally, the effectiveness of the proposed methods is discussed.     

A new adaptive variable structure control for chaotic synchronization and secure communication

A novel adaptive complementary variable structure control is proposed in this paper for chaotic synchronization. The bounded parameters of the model approximation error and the external disturbance are all regarded as unknown constants in this paper. Based on Lyapunov’s stability theory and the Babalat’s lemma the proposed controller has been shown to render the synchronous error to zero. The Duffing–Holmes oscillator was used as an illustrative example. Simulation results validated that the proposed scheme in the application of secure communication.     

Adaptive modified function projective synchronization of hyperchaotic systems with unknown parameters

This paper is involved with the adaptive modified function projective synchronization (MFPS) problem of hyperchaotic systems with unknown parameters. Based on the Lyapunov stability theorem and adaptive control method, adaptive controllers and parameters update laws can be presented for the MFPS not only between two identical hyperchaotic systems but particularly also between two different hyperchaotic systems with fully unknown or partially unknown parameters. Moreover, the coupling strength can be automatically adapted to a updated law. Numerical simulations are presented to show the effectiveness of the proposed synchronization schemes.     

Modified function projective lag synchronization of chaotic systems with disturbance estimations

This paper addresses the modified function projective lag synchronization (MFPLS) for a class of chaotic systems with unknown external disturbances. The disturbances are supposed to be generated by the exogenous systems. By using the disturbance-observer-based control and the linear matrix inequality approach, the disturbance observers are developed to ensure the boundedness of the disturbance error dynamics. Then by employing the sliding mode control (SMC) technique, an active SMC law is established to guarantee the disturbance rejection and realize MFPLS between the master and slave systems. And the corresponding numerical simulation is provided to illustrate the effectiveness of the proposed method.     

Adaptive modified function projective synchronization of unknown chaotic systems with different order

This paper investigates the modified function projective synchronization (MFPS) between two different dimensional chaotic systems with fully unknown or partially unknown parameters via increased order. Based on the Lyapunov stability theorem and adaptive control method, a unified adaptive controller and parameters update law can be designed for achieving the MFPS of the two different chaotic systems with different orders. Numerical simulations are presented to show the effectiveness of the proposed synchronization scheme.     

Control and adaptive modified function projective synchronization of Liu chaotic dynamical system

In this work, the feedback control method is proposed to control the behaviour of Liu chaotic dynamical system. The controlled system is stable under some conditions on the parameters of the system determined by Routh-Hurwitz criterion. This paper also presents the adaptive modified function projective synchronization (AMFPS) between two identical Liu chaotic dynamical systems. Based on the Lyapunov stability theorem, adaptive control laws are designed to achieving the AMFPS. Finally, some numerical simulations are obtained to validate the proposed methods.     

Generalized function projective synchronization of two different hyperchaotic systems with unknown parameters

Combining adaptive control theory with an antisymmetric structure, an extended adaptive controller which is more generalized and simpler than some existing controllers is designed. Under the controller, generalized function projective synchronization of two different uncertain hyperchaotic systems is achieved, and the unknown parameters are also estimated. In numerical simulations, the scaling function factors discussed in this paper are more complicated, and they have not been discussed in other papers. Corresponding simulation results are presented to show that the controller works well.     

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.     

Modified projective synchronization of chaotic systems with disturbances via active sliding mode control

We apply the active sliding mode control technique to realize the modified projective synchronization of the chaotic systems. The disturbances are considered both in the drive system and the response system. The sufficient conditions for the modified projective synchronization both the non-identical and identical chaotic systems are presented. The corresponding numerical simulations are provided to illuminate the effectiveness of the proposed active sliding mode controllers.     

Modified projective synchronization of chaotic system

A modified projective synchronization is proposed to acquire a general kind of proportional relationships between the drive and response systems. From rigorously control theory, a sufficient condition is attained for the stability of the error dynamics, and is applied to guiding the design of the controllers. Finally, we take Lorenz system as an example for illustration and verification.     

Adaptive feedback controller for projective synchronization

Due to the unpredictability of the scaling factor of projective synchronization in coupled partially linear systems, it is hard to know for sure the terminal state of the synchronized dynamics. In this paper, a simple adaptive linear feedback control method is proposed for controlling the scaling factor onto a desired value, based on the invariance principle of differential equations. Firstly, we prove the synchronizability of the proposed simple adaptive projective synchronization control method from the viewpoint of mathematics. Then, two numerical examples are presented to illustrate the applications of the derived results. Finally, we propose a communication scheme based on the adaptive projective synchronization of the Lorenz chaotic system. Numerical simulation shows its feasibility.     

Binary generalized synchronization

In this paper we report for the first time on the binary generalized synchronization, when for the certain values of the coupling strength two unidirectionally coupled dynamical systems generating the aperiodic binary sequences are in the generalized synchronization regime. The presence of the binary generalized synchronization has been revealed with the help of both the auxiliary system approach and the largest conditional Lyapunov exponent calculation. The mechanism resulting in the binary generalized synchronization has been explained. The finding discussed in this paper gives a strong potential for new applications under many relevant circumstances.     

Chaos control and function projective synchronization of fractional-order systems through the backstepping method

We study the chaos control and the function projective synchronization of a fractional-order T-system and Lorenz chaotic system using the backstepping method. Based on stability theory, we consider the condition for the local stability of nonlinear three-dimensional commensurate fractional-order system. Using the feedback control method, we control the chaos in the considered fractional-order T-system. We simulate the function projective synchronization between the fractional-order T-system and Lorenz system numerically using MATLAB and depict the results with plots.