Chaos control and chaos synchronization for multi-scroll chaotic attractors generated using hyperbolic functions

In this paper, we design a series of chaotic systems that can generate one-directional, two-directional and three-directional multi-scroll chaotic attractors. Then, based upon the properties of these chaotic systems, we construct appropriate Lyapunov functions and design simple linear feedback controls to globally exponentially stabilize and synchronize these chaotic systems. Numerical simulation results are also presented to show the applicability of the proposed control laws.     

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.     

Passive and impulsive synchronization of a new four-dimensional chaotic system

This paper studies the synchronization problem for a new chaotic four-dimensional system presented by Qi et al. Two different methods, the passive control method and the impulsive control method, are used to control the synchronization of the four-dimensional chaotic system. Numerical simulations show the effectiveness of the two different methods.     

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 control and synchronization of identical new chaotic flows with unknown parameters via single input

The single input linear feedback control for synchronizing two identical new 3D chaotic flows reported by Li et al. [X.F. Li, K.E. Chlouverakis, D.L. Xu, Nonlinear dynamics and circuit realization of a new chaotic flow: a variant of Lorenz, Chen and Lü, Nonlinear Analysis RWA 10 (4) (2009) 2357-2368] is proposed in this paper. Sufficient conditions of synchronization are obtained for both linear feedback and adaptive control approaches. The problem of adaptive synchronization between two nearly identical chaotic systems with unknown parameters is also studied. Based on the Lyapunov stability theory, two kinds of single input adaptive synchronization controllers are designed and the adaptive parameter update laws are developed.     

Global chaos synchronization of new chaotic systems via nonlinear control

Nonlinear control is an effective method for making two identical chaotic systems or two different chaotic systems be synchronized. However, this method assumes that the Lyapunov function of error dynamic (e) of synchronization is always formed as V (e) = 1/2e^Te. In this paper, modification based on Lyapunov stability theory to design a controller is proposed in order to overcome this limitation. The method has been applied successfully to make two identical new systems and two different chaotic systems (new system and Lorenz system) globally asymptotically synchronized. Since the Lyapunov exponents are not required for the calculation, this method is effective and convenient to synchronize two identical systems and two different chaotic systems. Numerical simulations are also given to validate the proposed synchronization approach.     

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.     

Chaos synchronization of a new hyperchaotic system

In this paper, two kinds of synchronization schemes for a new hyperchaotic system are presented. Firstly, on the basis of stability criterion of linear system, synchronization is achieved with the help of the active control theory. Secondly, a nonlinear controller is designed according to Lyapunov stability theory to assure that synchronization can be achieved. Furthermore, an adaptive control approach for synchronization of uncertain hyperchaotic systems is proposed. Finally numerical simulations are provided to show the effectiveness and feasibility of the developed methods.     

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.     

Master–slave synchronization criteria for chaotic hindmarsh–rose neurons using linear feedback control

This article is concerned with master–slave synchronization for two chaotic Hindmarsh–Rose neurons. The main contribution of this article is that three synchronization criteria are derived by using linear feedback control without the estimation of bounds of state variables of controlled slave neurons. Three simulation examples are used to illustrate the effectiveness of our results. © 2015 Wiley Periodicals, Inc. Complexity 21: 319–327, 2016     

Exponential synchronization for fractional‐order chaotic systems with mixed uncertainties

This article focuses on the problem of exponential synchronization for fractional‐order chaotic systems via a nonfragile controller. A criterion for α‐exponential stability of an error system is obtained using the drive‐response synchronization concept together with the Lyapunov stability theory and linear matrix inequalities approach. The uncertainty in system is considered with polytopic form together with structured form. The sufficient conditions are derived for two kinds of structured uncertainty, namely, (1) norm bounded one and (2) linear fractional transformation one. Finally, numerical examples are presented by taking the fractional‐order chaotic Lorenz system and fractional‐order chaotic Newton–Leipnik system to illustrate the applicability of the obtained theory. © 2014 Wiley Periodicals, Inc. Complexity 21: 114–125, 2015     

Multiswitching dual combination synchronization of time‐delay chaotic systems

This paper presents a novel synchronization scheme of multiswitching dual combination synchronization which is first of its kind. Multiswitching dual combination synchronization is achieved for 6 time‐delay chaotic systems. Asymptotically stable synchronization states are established by nonlinear control method and Lyapunov Krasovskii functional. To elaborate the proposed scheme, an example of time‐delay Rossler, Chen, and Shimizu Morioka systems is considered, where time‐delay Rossler system and Chen system are considered as drive systems and time‐delay Shimizu Morioka system is considered as response system. Theoretical analysis and computational results are in excellent agreement.     

Simplified method and synchronization for a class of complex chaotic systems

This paper is devoted to investigate a class of complex chaotic systems and a linear correlation between the real and imaginary component of complex variables in these systems is found. Based on this linear relationship, a simplified law is proposed. First, complex Lorenz system is given to show the linear correlation, then it is simplified. Second, a simplified law is proposed to determine whether the complex system can be simplified, and the complex Lü system and hyperchaotic complex Lü system are used to verify the simplified law. Finally, a new synchronization control is proposed to synchronize complex Lorenz system and real Lorenz system. The theoretical analysis and numerical simulation prove the feasibility and better performance of this method.     

A special full-state hybrid projective synchronization in symmetrical chaotic systems

A special full-state hybrid projective synchronization type is proposed in this paper. The anti-synchronization and complete synchronization can be achieved simultaneously in this new synchronization phenomenon. We point out how to realize this synchronization in chaotic systems: anti-synchronization in symmetrical coordinate subspace and complete synchronization in its normal coordinate subspace. Two illustrative examples, multi-scroll chaotic system by the partial Lyapunov stability theory, and a four-dimensional chaotic system by the invariance principle of differential equation are presented to exhibit this new synchronization.     

Parameter identification and synchronization of fractional-order chaotic systems

The knowledge about parameters and order is very important for synchronization of fractional-order chaotic systems. In this article, identification of parameters and order of fractional-order chaotic systems is converted to an optimization problem. Particle swarm optimization algorithm is used to solve this optimization problem. Based on the above parameter identification, synchronization of the fractional-order Lorenz, Chen and a novel system (commensurate or incommensurate order) is derived using active control method. The new fractional-order chaotic system has four-scroll chaotic attractors. The existence and uniqueness of solutions for the new fractional-order system are also investigated theoretically. Simulation results signify the performance of the work.     

Projective lag synchronization of spatiotemporal chaos via active sliding mode control

This paper investigates projective lag synchronization of spatiotemporal chaos with disturbances. A control scheme is designed via active sliding mode control. The synchronization of spatiotemporal chaos between a drive system and a response system with disturbances and time-delay is implemented by adding the active sliding mode controllers. The control law is applied to two identical spatiotemporal Gray-Scott systems. Numerical results demonstrate the feasibility and the effectiveness of the proposed approach.     

Dynamical analysis and numerical simulation of bloch chaotic system

In this article, some dynamics of Bloch chaotic system have been studied. Based on Lagrange multiplier method, optimization theory, and the generalized positively definite and radially unbound Lyapunov functions with respect to the parameters of the system, we derive the ultimate bound and a family of mathematical expressions of globally exponentially attractive sets for this system with respect to the parameters of system. The results obtained in this article provides theory basis for chaotic synchronization, chaotic control, Hausdorff dimension, and Lyapunov dimension of chaotic attractors of Bloch chaotic system. © 2016 Wiley Periodicals, Inc. Complexity 21: 201–206, 2016     

Hybrid chaos control of continuous unified chaotic systems using discrete rippling sliding mode control

In this paper, a new systematic design procedure to stabilize continuous unified chaotic systems based on discrete sliding mode control (DSMC) is presented. In contrast to the previous works, the concept of rippling control is newly introduced such that the design of DSMC can be simplified and only a single controller is needed to realize chaos suppression. As expected, under the proposed DSMC law, the unified system can be stabilized in a manner of ripple effect, even when the external uncertainty is present. Last, two examples are included to illustrate the effectiveness of the proposed rippling DSMC developed in this paper.     

On a new chaotic system

In this paper, a novel 3D nonlinear system with chaotic behavior is considered. Our proposed system is a generalized Lorenz‐like system, with no reflections about coordinates axes. Although the considered system depends on seven real parameters, its stability is completely analyzed. Also, the presence of Hopf bifurcation is pointed out. Copyright © 2014 John Wiley & Sons, Ltd.