A nonlinear control scheme to anticipated and complete synchronization in discrete-time chaotic (hyperchaotic) systems

In this Letter, a new synchronization scheme is presented to study anticipated synchronization and complete synchronization in discrete-time chaotic and hyperchaotic systems based on the active control idea. The scheme is applied to investigate anticipated synchronization and complete synchronization between two identical 3D generalized Hénon maps, as well as 3D discrete-time Yeh–Kokotovic map and 3D generalized Hénon maps. Numerical simulations are used to verify the effectiveness of the proposed scheme.     

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.     

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.     

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.     

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.     

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.     

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.     

Adaptive Q-S (lag, anticipated, and complete) time-varying synchronization and parameters identification of uncertain delayed neural networks

In this paper, a new type of generalized Q-S (lag, anticipated, and complete) time-varying synchronization is defined. Adaptive Q-S (lag, anticipated, and complete) time-varying synchronization and parameters identification of uncertain delayed neural networks have been considered, where the delays are multiple time-varying delays. A novel control method is given by using the Lyapunov functional method. With this new and effective method, parameters identification and Q-S (lag, anticipated, and complete) time-varying synchronization can be achieved simultaneously. Simulation results are given to justify the theoretical analysis in this paper.     

Adaptive lag synchronization and parameters adaptive lag identification of chaotic systems

This Letter investigates the problem of adaptive lag synchronization and parameters adaptive lag identification of chaotic systems. In comparison with those of existing parameters identification schemes, the unknown parameters are identified by adaptive lag laws, and the delay time is also identified in this Letter. Numerical simulations are also given to show the effectiveness of the proposed method.     

A new four-dimensional hyperchaotic Chen system and its generalized synchronization

Based on the Chen chaotic system, a new four-dimensional hyperchaotic Chen system is constructed, and the basic dynamic behaviours of the system were studied, and the generalized synchronization has been observed in the coupled four-dimensional hyperchaotic Chen system with unknown parameters. The Routh--Hurwitz theorem is used to derive the conditions of stability of this system. Furthermore based on Lyapunov stability theory, the control laws and adaptive laws of parameters are obtained to make generalized synchronization of the coupled new four-dimensional hyperchaotic Chen systems. Numerical simulation results are presented to illustrate the effectiveness of this method.     

A practical approach to robust impulsive lag synchronization between different chaotic systems

In this paper, a practical impulsive lag synchronization scheme for different chaotic systems with parametric uncertainties is proposed. By virtue of the new definition of synchronization and the theory of impulsive differential equations, some new and less conservative sufficient conditions are established to guarantee that the error dynamics can converge to a predetermined level. The idea and approach developed in this paper can provide a more practical framework for the synchronization between identical and different chaotic systems in parameter perturbation circumstances. Simulation results finally demonstrate the effectiveness of the method.     

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.     

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.     

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.     

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.     

A new chaotic communication scheme based on adaptive synchronization

A new chaotic communication scheme using adaptive synchronization technique of two unified chaotic systems is proposed. Different from the existing secure communication methods, the transmitted signal is modulated into the parameter of chaotic systems. The adaptive synchronization technique is used to synchronize two identical chaotic systems embedded in the transmitter and the receiver. It is assumed that the parameter of the receiver system is unknown. Based on the Lyapunov stability theory, an adaptive control law is derived to make the states of two identical unified chaotic systems with unknown system parameters asymptotically synchronized; thus the parameter of the receiver system is identified. Then the recovery of the original information signal in the receiver is successfully achieved on the basis of the estimated parameter. It is noticed that the time required for recovering the information signal and the accuracy of the recovered signal very sensitively depends on the frequency of the information signal. Numerical results have verified the effectiveness of the proposed scheme.     

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.     

Adaptive lag synchronization and parameter identification of fractional order chaotic systems

This paper proposes a simple scheme for the lag synchronization and the parameter identification of fractional order chaotic systems based on the new stability theory. The lag synchronization is achieved and the unknown parameters are identified by using the adaptive lag laws. Moreover, the scheme is analytical and is simple to implement in practice. The well-known fractional order chaotic Lü system is used to illustrate the validity of this theoretic method.     

Generating and enhancing lag synchronization of chaotic systems by white noise

In this paper, we study the crucial impact of white noise on lag synchronous regime in a pair of time-delay unidirectionally coupled systems. Our result demonstrates that merely via white-noise-based coupling lag synchronization could be achieved between the coupled systems (chaotic or not). And it is also demonstrated that a conventional lag synchronous regime can be enhanced by white noise. Sufficient conditions are further proved mathematically for noise-inducing and noise-enhancing lag synchronization, respectively. Additionally, the influence of parameter mismatch on the proposed lag synchronous regime is studied, by which we announce the robustness and validity of the new strategy. Two numerical examples are provided to illustrate the validity and some possible applications of the theoretical result.