Different Types of Projective Synchronization in a Class of Time-Delayed Chaotic Systems Using Active Control Approach

We study different types of projective synchronization （projective-anticipating, projective, and projectivelag synchronization） in a class of time-delayed chaotic systems related to optical bistable or hybrid optical bistable devices. We relax some limitations of previous work, where the scaling factor a can not be any desired value. In this paper, we achieve projective-anticipating, projective, and projective-lag synchronization without the limitation of a. A suitable controller is chosen using active control approach. Based on the Lyapunov stability theory, we derive the sufficient stability condition through theoretical analysis. The analytical results are validated by the numerical simulations using Ikeda model and Mackey-Glass model.

Adaptive projective synchronization of unified chaotic systems and its application to secure communication

In this paper, a simple adaptive linear feedback control method is proposed for controlling the scaling factor between two coupled unified chaotic systems to a desired value, based on the invarianee principle of differential equations. Under this control strategy, one can arbitrarily select the scaling factor. Numerical simulations are given to support the effectiveness of the proposed method and show the robustness against noise. Furthermore, a secure communication scheme based on the adaptive projective synchronization of unified chaotic systems is presented and numerical simulation shows its feasibility.

A Comparative Study of Synchronization Methods of Rucklidge Chaotic Systems with Design of Active Control and Backstepping Methods

The performance of two widely used chaos synchronization approaches, active control and backstepping control, is investigated in this study. These two methods are projected to synchronize two chaotic systems (Master/Drive of Rucklidge Systems) that are identical but have different initial conditions. The paper’s significant feature is that based on error dynamics, controllers are designed using the appropriate variable and the time synchronization between master Rucklidge and drive Rucklidge systems using both methods. The control function of the active control method is designed on the proper selection of matrices. The chaotic behavior is controlled using a recursive backstepping design based on the Lyapunov stability theory with a validated Lyapunov function. The effectiveness of the controller in eradicating the chaotic behavior from the state trajectories is also revealed using numerical simulations with Matlab. The backstepping method is superior to the active control method for synchronization of the measured pair of systems, as it takes less time to synchronize while exhausting the first one than the second one with great performance, according to numerical simulation and graphical outcomes.

Cascade adaptive control of uncertain unified chaotic systems

The chaos control of uncertain unified chaotic systems is considered. Cascade adaptive control approach with only one control input is presented to stabilize states of the uncertain unified chaotic system at the zero equilibrium point. Since an adaptive controller based on dynamic compensation mechanism is employed, the exact model of the unified chaotic system is not necessarily required. By choosing appropriate controller parameters, chaotic phenomenon can be suppressed and the response speed is tunable. Sufficient condition for the asymptotic stability of the approach is derived. Numerical simulation results confirm that the cascade adaptive control approach with only one control signal is valid in chaos control of uncertain unified chaotic systems.

Synchronization of Chaotic Systems via Active Disturbance Rejection Control

This paper presents the use of active disturbance rejection control method (ADRC) to synchronize two different chaotic systems. The master system and slave systems have uncertainties and external disturbances. The numerical results are presented for the synchronization between the Duffing-Holmes system and the van der pol system. The numerical results presented show the effectiveness of the proposed method.

Synchronization of different chaotic systems via active radial basis functions sliding mode controller

This paper presents a new method to synchronize different chaotic systems with disturbances via an active radial basis function （RBF） sliding controller. This method incorporates the advantages of active control, neural network and sliding mode control. The main part of the controller is given based on the output of the RBF neural networks and the weights of these single layer networks are tuned on-line based on the sliding mode reaching law. Only several radial basis functions are required for this controller which takes the sliding mode variable as the only input. The proposed controller can make the synchronization error converge to zero quickly and can overcome external disturbances. Analysis of the stability for the controller is carried out based on the Lyapunov stability theorem. Finally, five examples are given to illustrate the robustness and effectiveness of the proposed synchronization control strategy.

Suppression and synchronization of chaos in uncertain time-delay physical system

The mechanical horizontal platform(MHP)system exhibits a rich chaotic behavior.The chaotic MHP system has applications in the earthquake and offshore industries.This article proposes a robust adaptive continuous control(RACC)algorithm.It investigates the control and synchronization of chaos in the uncertain MHP system with time-delay in the presence of unknown state-dependent and time-dependent disturbances.The closed-loop system contains most of the nonlinear terms that enhance the complexity of the dynamical system;it improves the efficiency of the closed-loop.The proposed RACC approach(a)accomplishes faster convergence of the perturbed state variables(synchronization errors)to the desired steady-state,(b)eradicates the effect of unknown state-dependent and time-dependent disturbances,and(c)suppresses undesirable chattering in the feedback control inputs.This paper describes a detailed closed-loop stability analysis based on the Lyapunov-Krasovskii functional theory and Lyapunov stability technique.It provides parameter adaptation laws that confirm the convergence of the uncertain parameters to some constant values.The computer simulation results endorse the theoretical findings and provide a comparative performance.

Design of an adaptive finite-time controller for synchronization of two identical/different non-autonomous chaotic flywheel governor systems

The centrifugal flywheel governor （CFG） is a mechanical device that automatically controls the speed of an engine and avoids the damage caused by sudden change of load torque. It has been shown that this system exhibits very rich and complex dynamics such as chaos. This paper investigates the problem of robust finite-time synchronization of non-autonomous chaotic CFGs. The effects of unknown parameters, model uncertainties and external disturbances are fully taken into account. First, it is assumed that the parameters of both master and slave CFGs have the same value and a suitable adaptive finite-time controller is designed. Second, two CFGs are synchronized with the parameters of different values via a robust adaptive finite-time control approach. Finally, some numerical simulations are used to demonstrate the effectiveness and robustness of the proposed finite-time controllers.

A new kind of nonlinear fractional-order chaotic phenomenon in coupled systems： coexistence of anti-phase and complete synchronization

In this paper, we have found a kind of interesting nonlinear phenomenon hybrid synchronization in linearly coupled fractional-order chaotic systems. This new synchronization mechanism, i.e., part of state variables are anti- phase synchronized and part completely synchronized, can be achieved using a single linear controller with only one drive variable. Based on the stability theory of the fractional-order system, we investigated the possible existence of this new synchronization mechanism. Moreover, a helpful theorem, serving as a determinant for the gain of the controller, is also presented. Solutions of coupled systems are obtained numerically by an improved Adams Bashforth-Moulton algorithm. To support our theoretical analysis, simulation results are given.

A new color image encryption scheme based on chaos synchronization of time-delay Lorenz system

In this paper, a new image encryption scheme is presented based on time-delay chaos synchronization. Compared with existing methods, a new method is pro- posed and a lot of coupled items can be taken as zero items to simplify the whole system. A simple linear controller is introduced to realize time-delay chaos synchronization and image encryption. The positions of the image pixels are firstly shuffled and then be hidden in the cartier image. The address codes of the chaotic sequences are adopted to avoid the disturbances induced by the initial value and computer accuracy error. Simulation results for color image are provided to illustrate the effectiveness of the proposed method. It can be seen clearly that the system can converge quickly and the image can be encrypted rapidly.

Generalized synchronization between different chaotic maps via dead-beat control

This paper presents a new scheme to achieve generalized synchronization（GS） between different discrete-time chaotic（hyperchaotic） systems.The approach is based on a theorem,which assures that GS is achieved when a structural condition on the considered class of response systems is satisfied.The method presents some useful features：it enables exact GS to be achieved in finite time（i.e.,dead-beat synchronization）;it is rigorous,systematic,and straightforward in checking GS;it can be applied to a wide class of chaotic maps.Some examples of GS,including the Grassi-Miller map and a recently introduced minimal 2-D quadratic map,are illustrated.

Generalized synchronization of two different chaotic systems

In this paper, generalized synchronization of two different chaotic dynamical systems is investigated. An active control is adopted to construct a response system which synchronizes with a given drive system for a function relation. Based on rigorous analysis, the error system is asymptotically stable at the equilibrium. Numerical simulations illustrate the effectiveness of the proposed theory.

Prescribed performance synchronization for fractional-order chaotic systems

In this paper the synchronization for two different fractional-order chaotic systems, capable of guaranteeing synchronization error with prescribed performance, is investigated by means of the fractional-order control method. By prescribed performance synchronization we mean that the synchronization error converges to zero asymptotically, with convergence rate being no less than a certain prescribed function. A fractional-order synchronization controller and an adaptive fractional-order synchronization controller, which can guarantee the prescribed performance of the synchronization error,are proposed for fractional-order chaotic systems with and without disturbances, respectively. Finally, our simulation studies verify and clarify the proposed method.

Robust Finite-time Synchronization of Non-identical Fractional-order Hyperchaotic Systems and Its Application in Secure Communication

This paper proposes a novel adaptive sliding mode control(SMC) method for synchronization of non-identical fractional-order(FO) chaotic and hyper-chaotic systems. Under the existence of system uncertainties and external disturbances,finite-time synchronization between two FO chaotic and hyperchaotic systems is achieved by introducing a novel adaptive sliding mode controller(ASMC). Here in this paper, a fractional sliding surface is proposed. A stability criterion for FO nonlinear dynamic systems is introduced. Sufficient conditions to guarantee stable synchronization are given in the sense of the Lyapunov stability theorem. To tackle the uncertainties and external disturbances, appropriate adaptation laws are introduced. Particle swarm optimization(PSO) is used for estimating the controller parameters. Finally, finite-time synchronization of the FO chaotic and hyper-chaotic systems is applied to secure communication.

Arbitrary full-state hybrid projective synchronization for chaotic discrete-time systems via a scalar signal

In this paper we present a new projective synchronization scheme, where two chaotic （hyperchaotic） discrete-time systems synchronize for any arbitrary scaling matrix. Specifically, each drive system state synchronizes with a linear combination of response system states. The proposed observer-based approach presents some useful features： i） it enables exact synchronization to be achieved in finite time （i.e., dead-beat synchronization）; ii） it exploits a scalar synchronizing signal; iii） it can be applied to a wide class of discrete-time chaotic （hyperchaotic） systems; iv） it includes, as a particular case, most of the synchronization types defined so far. An example is reported, which shows in detail that exact synchronization is effectively achieved in finite time, using a scalar synchronizing signal only, for any arbitrary scaling matrix.

Impulsive synchronization and control of directed transport in chaotic ratchets

The impulsive synchronization problem of two identical chaotic ratchets is investigated in this paper. We demonstrate that the impulsive method to control directed transport is applicable when there are multiple co-existing attractors in phase space transporting particles in different directions. Numerical simulations are carried out to illustrate the effectiveness of the proposed method.

统一混沌系统的active control同步性研究

驱动响应同步、反馈同步和自适应同步是目前研究混沌同步的三种常用方法。但这三种方法对处理具有综合复杂性的系统(如统一混沌系统)有一定的局限性。本文采用active-control同步控制方法对统一混沌系统的同结构和异结构同步问题进行了研究,并用计算机进行了数值模拟。研究结果表明,active-control同步方法能使统一混沌系统达到同步,且其同构同步可以在计算机上模拟实现,即active-control同步控制方法对研究统一混沌这类具有综合复杂性的系统有效。

Synchronization of Modified Chua＇s Circuit with x｜x｜ Function

This paper considers the chaos synchronization of the modified Chua 's circuit with x|x| function. We firstly show that a couple of the modified Chua systems with different parameters and initial conditions can be synchronized using active control when the values of parameters both in drive system and response system are known aforehand.Furthermore, based on Lyapunov stability theory we propose an adaptive active control approach to make the states of two identical Chua systems with unknown constant parameters asymptotically synchronized. Moreover the designed controller is independent of those unknown parameters. Numerical simulations are given to validate the proposed synchronization approach.

Anti-synchronization of Four-wing Chaotic Systems via Sliding Mode Control

Sliding mode control is an important method used in nonlinear control systems. In robust control systems, the sliding mode control is often adopted due to its inherent advantages of easy realization, fast response and good transient performance as well as its insensitivity to parameter uncertainties and disturbances. In this paper, we derive new results based on the sliding mode control for the anti-synchronization of identical Qi three-dimensional (3D) four-wing chaotic systems (2008) and identical Liu 3D four-wing chaotic systems (2009). The stability results for the anti-synchronization schemes derived in this paper using sliding mode control (SMC) are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the SMC method is very effective and convenient to achieve global chaos anti-synchronization of the identical Qi four-wing chaotic systems and identical Liu four-wing chaotic systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper.

A general method for modified function projective synchronization

This paper investigates the modified function projective synchronization,which means that the drive system and the response system are synchronized up to a desired scale matrix of function. By the active control scheme,a general method for modified function projective synchronization is proposed. Numerical simulations on chaotic Rssler system and hyper-chaotic Chen system are presented to verify the effectiveness of the proposed scheme.