A new scheme to generalized (lag, anticipated, and complete) synchronization in chaotic and hyperchaotic systems

In this paper, a generalized (lag, anticipated, and complete) synchronization of a class of continuous-time systems is defined. A systematic, powerful and concrete scheme is developed to investigate the generalized (lag, anticipated, and complete) synchronization between the drive system and response system based on the active control idea. The hyperchaotic R?ssler system, transformed R?ssler and Chen system as well as two coupled nonidentical R?ssler oscillators are chosen to illustrate the proposed scheme. Numerical simulations are used to verify the effectiveness of the proposed scheme. The scheme can be also extended to research generalized (lag, anticipated, and complete) synchronization between other dynamical systems.     

Q-S (complete or anticipated) synchronization backstepping scheme in a class of discrete-time chaotic (hyperchaotic) systems: a symbolic-numeric computation approach

First, a type of Q-S (complete or anticipated) synchronization is defined in discrete-time dynamical systems. Second, based on backstepping design with a scalar controller, a systematic, concrete and automatic scheme is presented to investigate Q-S (complete or anticipated) synchronization between the discrete-time drive system and response system with strict-feedback form. Finally, the proposed scheme is used to illustrate Q-S (complete or anticipated) synchronization between the two-dimensional discrete-time Lorenz system and Fold system, as well as the three-dimensional hyperchaotic discrete-time Rossler system and generalized discrete-time Rossler system. Moreover numerical simulations are used to verify the effectiveness of the proposed scheme. Our scheme can be also extended to investigate Q-S (complete or anticipated) synchronization between other discrete-time dynamical systems with strict-feedback forms. With the aid of symbolic-numeric computation, the scheme can be performed to yield automatically the scalar controller and to verify its effectiveness in computer.     

Lag projective synchronization in fractional-order chaotic (hyperchaotic) systems

In this Letter, a new lag projective synchronization for fractional-order chaotic (hyperchaotic) systems is proposed, which includes complete synchronization, anti-synchronization, lag synchronization, generalized projective synchronization. It is shown that the slave system synchronizes the past state of the driver up to a scaling factor. A suitable controller for achieving the lag projective synchronization is designed based on the stability theory of linear fractional-order systems and the pole placement technique. Two examples are given to illustrate effectiveness of the scheme, in which the lag projective synchronizations between fractional-order chaotic Rössler system and fractional-order chaotic Lü system, between fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system, respectively, are successfully achieved. Corresponding numerical simulations are also given to verify the analytical results.     

High precision fast projective synchronization in chaotic (hyperchaotic) systems

A high precision fast projective synchronization method was proposed through introducing an impact factor. The numerical simulations indicate that the synchronization precision was increased three orders of magnitude higher than available methods in literature, under the same conditions of numerical method and hardware. The synchronization can achieved after first iteration. The synchronization speed has been increased by 13 484 times and 23 000 times when and , respectively, compared to general methods.     

Full state hybrid lag projective synchronization in chaotic (hyperchaotic) systems

This Letter introduces another novel type of chaos synchronization—full state hybrid lag projective synchronization (FSHLPS), which includes complete synchronization, anti-synchronization, lag synchronization, general projective synchronization and FSHPS in [M. Hu, Z. Xu, R. Zhang, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 456; M. Hu, Z. Xu, R. Zhang, A. Hu, Phys. Lett. A 361 (2007) 231]. Furthermore, systematic FSHLPS schemes are respectively proposed for the continuous and discrete systems. Finally, some numerical simulations are given to verify the effectiveness of the developed schemes.     

Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters

This paper studies the adaptive complete synchronization of chaotic and hyperchaotic systems with fully unknown parameters. In practical situations, some systems' parameters cannot be exactly known a priori, and the uncertainties often affect the stability of the process of synchronization of the chaotic oscillators. An adaptive scheme is proposed to compensate for the effects of parameters' uncertainty based on the structure of chaotic systems in this paper. Based on the Lyapunov stability theorem, an adaptive controller and a parameters update law can be designed for the synchronization of chaotic and hyperchaotic systems. The drive and response systems can be nonidentical, even with different order. Three illustrative examples are given to demonstrate the validity of this technique, and numerical simulations are also given to show the effectiveness of the proposed chaos synchronization method. In addition, this synchronization scheme is quite robust against the effect of noise.     

Control and synchronization of discrete-time chaotic systems via variable structure control technique

We present an input/output linearization control method incorporated in a discrete-time variable structure control technique to resolve the output tracking problem of a class of discrete-time nonlinear systems. The proposed control scheme is then applied to address the control and synchronization problems associated with the Hénon chaotic systems. Numerical simulations demonstrate the feasibility and robustness of the proposed control strategy.     

Extending synchronization scheme to chaotic fractional-order Chen systems

Nonlinear control theory is successfully extended to fractional-order Chen systems to achieve synchronization. The corresponding fraction-order numerical algorithms are established. The analytical results are derived based on the Laplace transformation theory. Moreover, numerical simulations are shown to verify the effectiveness of the proposed synchronization schemes.     

Q-S (lag or anticipated) synchronization backstepping scheme in a class of continuous-time hyperchaotic systems--a symbolic-numeric computation approach

First, a Q-S (lag or anticipated) synchronization of continuous-time dynamical systems is defined. Second, based on a backstepping design with one controller, a systematic, concrete, and automatic scheme is developed to investigate the Q-S (lag or anticipated) synchronization between the drive system and response system with a strict-feedback form. Two identical hyperchaotic Tamasevicius-Namajunas-Cenys(TNC) systems as well as the hyperchaotic TNC system and hyperchaotic Rossler system are chosen to illustrate the proposed scheme. Numerical simulations are used to verify the effectiveness of the proposed scheme. The scheme can also be extended to study Q-S (lag or anticipated) synchronization between other dynamical systems with strict-feedback forms. With the aid of symbolic-numeric computation, the scheme can be performed to yield automatically the scalar controller in computer.     

Robust synchronization for a class of fractional-order chaotic and hyperchaotic systems

In this paper, the issue of robust synchronization for a class of fractional-order chaotic and hyperchaotic systems with model uncertainties and disturbances is studied. A stability criterion for fractional-order nonlinear dynamic systems is introduced, and an adaptive scheme is contrived to accomplish synchronization of fractional-order chaotic and hyperchaotic systems. The controller contains only a single state variable, which is simple and flexible in implementation. Two corresponding numerical examples are given to confirm the theoretical results of the paper.     

Generalized projective synchronization of a class of chaotic (hyperchaotic) systems with uncertain parameters

In this paper is investigated the generalized projective synchronization of a class of chaotic (or hyperchaotic) systems, in which certain parameters can be separated from uncertain parameters. Based on the adaptive technique, the globally generalized projective synchronization of two identical chaotic (hyperchaotic) systems is achieved by designing a novel nonlinear controller. Furthermore, the parameter identification is realized simultaneously. A sufficient condition for the globally projective synchronization is obtained. Finally, by taking the hyperchaotic Lü system as example, some numerical simulations are provided to demonstrate the effectiveness and feasibility of the proposed technique.     

Generation and synchronization of N-scroll chaotic and hyperchaotic attractors in fourth-order systems

Based on our previous works and Lyapunov stability theory, this paper studies the generation and synchronization of N-scroll chaotic and hyperchaotic attractors in fourth-order systems. A fourth-order circuit, by introducing additional breakpoints in the modified Chua oscillator, is implemented for the study of generation and synchronization of N-scroll chaotic attractors.This confirms the consistency of theoretical calculation, numerical simulation and circuit experiment.Furthermore,we give a refined and extended study of generating and synchronizing N-scroll hyperchaotic attractors in the fourth-order MCK system and report the new theoretical result, which is verified by computer simulations.     

Chaos communication based on synchronization of discrete-time chaotic systems

A novel chaos communication method is proposed based on synchronization of discrete-time chaotic systems. This method uses a full-order state observer to achieve synchronization and secure communication between the transmitter and the receiver. Further, we present a multiple-access chaotic digital communication method by combining the observer with the on-line least square method. Simulation results are also given for illustration.     

Adaptive impulsive synchronization for a class of fractional-order chaotic and hyperchaotic systems

In this paper, the adaptive impulsive synchronization for a class of fractional-order chaotic and hyperchaotic systems with unknown Lipschitz constant is investigated. Firstly, based on the adaptive control theory and the impulsive differential equations theory, the impulsive controller, the adaptive controller and the parametric update law are designed, respectively. Secondly, by constructing the suitable response system, the original fractional-order error system can be converted into the integral-order one. Finally, the new sufficient criterion is derived to guarantee the asymptotical stability of synchronization error system by the Lyapunov stability theory and the generalized Barbalat's lemma. In addition, numerical simulations demonstrate the effectiveness and feasibility of the proposed adaptive impulsive control method.     

Impulsive synchronization of nonlinear coupled chaotic systems

The issue of impulsive synchronization of the nonlinear coupled chaotic systems is investigated. A new framework for impulsive synchronization between such chaotic systems is presented, which makes the synchronization error system a linear impulsive control system. Therefore, it is easy to derive the impulsive synchronization law. To illustrate the effectiveness of the new scheme, a numerical example is given.     

Temporary lag and anticipated synchronization and anti-synchronization of uncoupled time-delayed chaotic systems

Without any control scheme and coupling terms, temporary lag and anticipated synchronization and temporary lag and anticipated anti-synchronization are newly discovered in two identical double Mackey–Glass systems with different initial conditions. When all initial conditions are positive, the lag synchronization is obtained. The negative initial values make the time history inverse and temporary lag anti-synchronization occur. The phenomena both appear intermittently.     

A new kind of nonlinear phenomenon in coupled fractional-order chaotic 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.     

Impulsive control for synchronization of nonlinear Rossler chaotic systems

This paper reports that an impulsive control theory for synchronization of nonlinear R?ssler chaotic systems is developed. A new framework for impulsive synchronization between such chaotic systems is presented, which makes the synchronization error system a linear impulsive control system. Therefore, it is easy to derive the impulsive synchronization law. The proposed impulsive control scheme is illustrated by nonlinear R?ssler chaotic systems and the simulation results demonstrate the effectiveness of the method.     

General hybrid projective complete dislocated synchronization with non-derivative and derivative coupling based on parameter identification in several chaotic and hyperchaotic systems

According to the Lyapunov stability theorem, a new general hybrid projective complete dislocated synchronization scheme with non-derivative and derivative coupling based on parameter identification is proposed under the framework of drive-response systems. Every state variable of the response system equals the summation of the hybrid drive systems in the previous hybrid synchronization. However, every state variable of the drive system equals the summation of the hybrid response systems while evolving with time in our method. Complete synchronization, hybrid dislocated synchronization, projective synchronization, non-derivative and derivative coupling, and parameter identification are included as its special item. The Lorenz chaotic system, Rssler chaotic system, memristor chaotic oscillator system, and hyperchaotic Lü system are discussed to show the effectiveness of the proposed methods.     

单模激光驱动的超混沌系统的完全同步

利用主动-被动同步法实现了由单模激光驱动的超混沌系统的完全同步.根据稳定性理论,对系统进行了一般分解.以单模激光Lorenz系统和四维超混沌Chen系统为例,验证了这种方法的有效性.仿真模拟结果表明,这种同步方法是快速有效的.利用这种同步方法可以实现对任意高维混沌系统的混沌同步,具有一定的普适性.