Exponential Synchronization of Chaotic Xian System Using Linear Feedback Control

Chaotic neural networks consisting of a great number of chaotic neurons are able to reproduce the rich dynamics observed in biological nervous systems. In recent years, the memristor has attracted much interest in the efficient implementation of artificial synapses and neurons. This work addresses adaptive synchronization of a class of memristor-based neural chaotic systems using a novel adaptive backstepping approach. A systematic design procedure is presented. Simulation results have demonstrated the effectiveness of the proposed adaptive synchronization method and its potential in practical application of memristive chaotic oscillators in secure communication.

Recently, a new six-term 3-D chaotic system with an exponential nonlinearity was discovered (Sundarapandian, 2013). This novel chaotic system has two saddle focus nodes and hence they are unstable. The maximal Lyapunov exponent (MLE) for the six-term Sundarapandian chaotic system was found as 3.2827 and the Lyapunov dimension of the Sundarapandian chaotic system was found as D <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</sub> = 2.1668. Complete synchronization of chaotic systems deals with the problem of asymptotically synchronizing the states of a pair of chaotic systems called as master and slave systems with the help of controllers attached to the slave system. Complete synchronization of chaotic systems has applications in many engineering areas such as secure communications, secure data encryption, cryptosystems, etc. This paper addresses the design of active and adaptive feedback controllers for completely synchronizing the states of identical six-term Sundarapandian chaotic systems with exponential nonlinearity. Active controllers are used when the system parameters a, b and c are available for measurement and the synchronization result is established using Lyapunov stability theory. Numerical simulations using MATLAB have been shown to demonstrate the proposed active and adaptive synchronization results for the novel chaotic system discussed in this paper.

Chaos synchronization is a phenomenon that occurs when a chaotic system drives another chaotic system. If a certain chaotic system is called the master system and another chaotic system is called the slave system, then anti-synchronization is said to occur when the state vectors of the master and slave systems have the same amplitude but opposite signs. This paper derives an important result for the anti-synchronization of identical chaotic systems using sliding mode control. This main result has been established using Lyapunov stability theory. In robust control systems, the sliding mode control (SMC) is often preferred in view of its inherent advantages such as easy realization, fast response, good transient performance and also insensitivity to parameter uncertainties and external disturbances. As an application of this general result, this paper solves the anti-synchronization problem for identical Wang-Chen chaotic systems (2012) via sliding mode control. Numerical simulations using MATLAB have been presented in detail to validate the sliding controller design for achieving anti-synchronization of identical Wang-Chen chaotic systems.

This study is concerned with the synchronization problem for a class of chaotic systems with multiple time-delays and transmitted noise. A dynamic compensator is proposed to achieve the synchronization between master and slave chaotic systems with multiple timedelays. First, the neural-network (NN) model is employed to approximate the chaotic systems with multiple time delays. Then, a linear differential inclusion (LDI) state-space representation is established for the dynamics of the NN model. Based on the LDI state-space representation, an exponential stability criterion of error dynamics derived in terms of Lyapunov's direct method is proposed to ensure that the trajectories of the slave system can approach those of master system. Subsequently, the stability condition of this criterion is reformulated into a linear matrix inequality (LMI). Based on the LMI, a fuzzy controller is synthesized to realize the exponential synchronization of the chaotic master-slave systems. Finally, this study provides a numerical example with simulations is given to illustrate the concepts discussed throughout this work.

In this research work, we announce a six-term novel 3-D dissipative chaotic system with two quadratic nonlinearities. First, this work describes the dynamic equations and qualitative properties of the novel chaotic system. We show that the novel chaotic system has three unstable equilibrium points. We also show that the novel chaotic system has a rotation symmetry about the \(x_3\) axis. The Lyapunov exponents of the novel chaotic system are obtained as \(L_1 = 1.2334, L_2 = 0\) and \(L_3 = -4.7329\). Since the sum of the Lyapunov exponents is negative, the novel chaotic system is dissipative. Also, the Kaplan-Yorke dimension of the novel chaotic system is derived as \(D_{KY} = 2.2606\). Next, this work describes the active synchronization of identical novel chaotic systems with known parameters. Furthermore, this work describes the adaptive synchronization of identical novel chaotic systems with unknown parameters. Both the active and adaptive synchronization results are established using Lyapunov stability theory. MATLAB simulations are depicted to illustrate all the main results derived in this work for the six-term novel 3-D novel chaotic system.

This paper proposes a novel nonlinear adaptive controller for the synchronization of two identical unknown chaotic systems. The proposed controller accomplishes fast convergence of the steady-state error to the origin and reduces the amplitude of the oscillations in the error signals during the transient. The rapid convergence increases the disturbance rejection attributes of the controller. The reduction in the oscillations’ magnitude decreases energy consumption and lowers the actuators’ degradation, reducing the probability of failure. This controller consists of linear adaptive and nonlinear control components, and they jointly synthesize control signals to penalize the synchronization error. The direct linear adaptive term keeps the closed-loop system stable; it ensures that the synchronization error converges to zero. The nonlinear terms heavily penalize the large state error vector in the transient, and its effect becomes minimal in the steady-state. The linear adaptive control is active in the steady-state, which synthesizes a smooth control signal in the vicinity of zero; this behavior causes oscillation free convergence of the error. Analysis based on the Lyapunov direct theorem proves the global asymptotic stabilization of the closed-loop at the origin. The paper also introduces parameters update laws that estimate the controller and unknown parameters of the master system and assure the convergence. This work also presents a comparative study of a numerical example of two identical Lorenz-Like unknown chaotic systems.

This paper proposes a new finite-time controller that realizes multi-switching synchronization of chaotic systems with bounded disturbances using the drive and response system synchronization arrangement. The finite-time controller derives the synchronization error to zero within a specified time. The proposed controller consists of three basic terms; each of them accomplishes a distinct objective: (1) stability of the control loop, (2) smooth and fast convergence behavior of the synchronization error, and (3) disturbance rejection. This study also devises a methodology for designing the finite-time controller and describes a general approach that furnishes a systematic procedure for the analysis of the closed loop. The smooth behavior terminology introduced in (2) refers to the over-damped convergence of the synchronization error signals to zero and the synthesis of chattering-free control effort. The analysis, which assures the global stability of the closed loop, uses the second stability theorem of the Lyapunov, while the finite-time stability technique determines the finite-time convergence. This paper includes simulations of two numerical examples for the validation of the theoretical findings and discusses the comparative analysis. The proposed methodology is suitable to design controllers for a wide range of hyper(chaotic) systems. The contributions of the paper are: (1) describe an innovative generalize analytical methodology for the multi-switching combination synchronization of chaotic systems and (2) propose a novel controller design that insures the finite-time synchronization.

Output synchronization of chaotic systems under nonvanishing perturbations

The current research provides the potential role of active nonlinear control functions in the asymptotic convergence of the synchronized errors in a comprehensive manner. Firstly, we design a systematic procedure to investigate hybrid synchronization in identical chaotic chemical reactor systems via active control method (ACM). In view of master-slave configuration and Lyapunov’s theory of stability, the considered methodology determines the global asymptotic stability of the error dynamics of the given system. Finally, numerical simulations using MATLAB toolbox are implemented for illustrating the efficacy and feasibility of the discussed strategy.

It is well known that the dynamics of the chaotic system is very sensitive to the initial conditions of the state, and the synchronization of two identical chaotic systems is only obtained, in general, with the high gain control law once their initial conditions are in a certain large deviation. Furthermore, the initial conditions are commonly unknown in practice, which causes difficulty in synchronizing two chaotic systems. This paper deals with the synchronization of two unified chaotic systems with input constraint. First, the scalar sign function is utilized to approximate the constrained non-smooth input function so that a continuous smooth nonlinear input function and an approximated nonlinear synchronized error system are obtained. Then, an optimal linear quadratic regulator (LQR) continuous-time control law is designed based on the optimal linear model, which is constructed at the sampled operating point of the afore-mentioned approximated nonlinear synchronized error system. To reduce the high magnitude of the obtained control law, the continuous-time control law is digitally redesigned for the implementation and an iterative procedure is proposed to adjust the weighting matrices in the LQR performance index so as to avoid input saturation occurs. Finally, three illustrative examples of the Lorenz, the Chen and the L chaotic systems decomposed from the unified chaotic system are given to demonstrate the effectiveness of the proposed method.

In this paper, an adaptive integral sliding mode control scheme is developed to study the control and synchronization of the hyper chaotic Zhou system with unknown parameters. First, an integral sliding mode control is designed which can stabilize the hyper chaotic Zhou system with known parameters to its unstable equilibrium at the origin. In the next step, the aim is synchronization of two identical hyper chaotic master and slave Zhou systems with unknown parameters. The master and slave system can be considered as a simple dynamical network with two nodes and one link. We propose an adaptive scheme which is led to a new type adaptive controller for this dynamical network with hyper chaotic Zhou system as oscillators which can identify unknown parameters of the master system and synchronize master and slave systems. Simulation results have shown that the proposed method has an excellent convergence from both speed and accuracy points of view and it outperforms Sundarapandian Vaidyanathan's scheme, a well-recognized scheme in the area.

Synchronization of memristor-based chaotic systems by a simplified control and its application to image en-/decryption using DNA encoding

This paper introduces a new type of synchronization method of adaptive modified projective synchronization with complex scaling matrix (ACMPS) for two \(n\)-dimensional complex chaotic (hyperchaotic) systems with uncertain complex parameters. By choosing appropriate Lyapunov functions dependent on complex variables, and employing adaptive control technique, sufficient criteria on ACMPS are derived. Moreover, in the complex space, the slave system can be asymptotically synchronized up to nonidentical or identical master system by a desired complex scaling matrix, and all of unknown parameters in both master and slave systems are achieved to be identified by virtue of the complex update laws. Finally, two examples are worked out to verify the effectiveness and feasibility of the theoretical results.

Time delays are frequently appearing in many real-life phenomena and the presence of time delays in chaotic systems enriches its complexities. The analysis of fractional-order chaotic real nonlinear systems with time delays has a plenty of interesting results but the research on fractional-order chaotic complex nonlinear systems with time delays is in the primary stage. This paper studies the problem of hybrid projective synchronization (HPS) of fractional-order chaotic complex nonlinear systems with time delays. HPS is one of the extensions of projective synchronization, in which different state vectors can be synchronized up to different scaling factors. Based on Laplace transformation and the stability theory of linear fractional-order systems, a suitable nonlinear controller is designed to achieve synchronization between the master and slave fractional-order chaotic complex nonlinear systems with time delays in the sense of HPS with different scaling factors. Finally, the HPS between fractional-order delayed complex Lorenz system and fractional-order delayed complex Chen system and that of fractional-order delayed complex Lorenz system and fractional-order delayed complex Lu system are taken into account to demonstrate the effectiveness and feasibility of the proposed HPS techniques in the numerical example section.

Anticipation in the synchronization of chaotic time-delay systems