Adaptive synchronization of hyperchaotic Lü system with uncertainty

This paper presents a novel adaptive control scheme for synchronization of the latest hyperchaotic Lü system. Based on the Lyapunov stability theory, a feedback controller and a parameter update law are designed for the synchronization of hyperchaotic Lfi systems with uncertainty. Numerical simulations are given to demonstrate the validity of the synchronization technique.     

A unified approach to fuzzy modelling and robust synchronization of different hyperchaotic systems

In this paper, a Takagi Sugeno （T-S） fuzzy model-based method is proposed to deal with the problem of synchronization of two identical or different hyperchaotic systems. The T S fuzzy models with a small number of fuzzy IF-THEN rules are employed to represent many typical hyperchaotic systems exactly. The benefit of employing the T-S fuzzy models lies in mathematical simplicity of analysis. Based on the T-S fuzzy hyperchaotic models, two fuzzy controllers arc designed via parallel distributed compensation （PDC） and exact linearization （EL） techniques to synchronize two identical hyperchaotic systems with uncertain parameters and two different hyperchaotic systems, respectively. The sufficient conditions for the robust synchronization of two identical hyperchaotic systems with uncertain parameters and the asymptotic synchronization of two different hyperchaotic systems are derived by applying the Lyapunov stability theory. This method is a universal one of synchronizing two identical or different hyperchaotic systems. Numerical examples are given to demonstrate the validity of the proposed fuzzy model and hyperchaotic synchronization scheme.     

Synchronization of hyperchaotic Chen systems: a class of the adaptive control approach

The synchronization of hyperchaotic Chen systems is considered. An adaptive synchronization approach and a cascade adaptive synchronization approach are presented to synchronize a drive system and a response system. By utilizing an adaptive controller based on the dynamic compensation mechanism, exact knowledge of the systems is not necessarily required, and the synchronous speed is controllable by tuning the controller parameters. Sufficient conditions for the asymptotic stability of the two synchronization schemes are derived. Numerical simulation results demonstrate that the adaptive synchronization scheme with four control inputs and the cascade adaptive synchronization scheme with only one control signal are effective and feasible in chaos synchronization of hyperchaotic systems.     

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.     

The synchronization of a fractional order hyperchaotic system based on passive control

This paper investigates the synchronization of a fractional order hyperchaotic system using passive control. A passive controller is designed, based on the properties of a passive system. Then the synchronization between two fractional order hyperchaotic systems under different initial conditions is realized, on the basis of the stability theorem for fractional order systems. Numerical simulations and circuitry simulations are presented to verify the analytical results.     

Reliability of linear coupling synchronization of hyperchaotic systems with unknown parameters

Complete synchronization could be reached between some chaotic and/or hyperchaotic systems under linear coupling. More generally, the conditional Lyapunov exponents are often calculated to confirm the stability of synchronization and reliability of linear controllers. In this paper, detailed proof and measurement of the reliability of linear controllers are given by constructing a Lyapunov function in the exponential form. It is confirmed that two hyperchaotic systems can reach complete synchronization when two linear controllers are imposed on the driven system unidirectionally and the unknown parameters in the driving systems are estimated completely. Finally, it gives the general guidance to reach complete synchronization under linear coupling for other chaotic and hyperchaotic systems with unknown parameters.     

Modified adaptive controller for synchronization of incommensurate fractional-order chaotic systems

We investigate the synchronization of a class of incommensurate fractional-order chaotic systems,and propose a modified adaptive controller for fractional-order chaos synchronization based on the Lyapunov stability theory,the fractional order differential inequality,and the adaptive strategy.This synchronization approach is simple,universal,and theoretically rigorous.It enables the synchronization of0 fractional-order chaotic systems to be achieved in a systematic way.The simulation results for the fractional-order Qi chaotic system and the four-wing hyperchaotic system are provided to illustrate the effectiveness of the proposed scheme.     

Anti-synchronization Between Lorenz and Liu Hyperchaotic Systems

Anti-synchronization between different hyperchaotic systems is presented using Lorenz and Liu systems. When the parameters of two systems are known, one can use active synchronization. When the parameters are unknown or uncertain, the adaptive synchronization is applied. The simulation results verify the effectiveness of the proposed two schemes for anti-synchronization between different hyperchaotic systems.     

Generalized-Type Synchronization of Hyperchaotic Oscillators Using a Vector Signal

In this paper, a systematic and powerful scheme is proposed to address a generalized-type synchronization of a class of continuous-time systems, which includes generalized lag synchronization, generalized anticipated synchronization, and generalized synchronization. The presented scheme is used to investigate the generalized-type synchronization of the 4D hyperchaotic oscillator and the hyperchaotic oscillator with gyrators. Numerical simulations are used to verify the effectiveness of the proposed scheme. The scheme is more powerful than the scalar signal scheme due to Grassi and Mascolo.     

Control and synchronization of a hyperchaotic system based on passive control

In this paper, a new hyperchaotic system is proposed, and the basic properties of this system are analyzed by means of the equilibrium point, a Poincar map, the bifurcation diagram, and the Lyapunov exponents. Based on the passivity theory, the controllers are designed to achieve the new hyperchaotic system globally, asymptotically stabilized at the equilibrium point, and also realize the synchronization between the two hyperchaotic systems under different initial values respectively. Finally, the numerical simulation results show that the proposed control and synchronization schemes are effective.     

Chaos synchronization between two different 4D hyperchaotic Chen systems

This paper presents chaos synchronization between two different four-dimensional (4D) hyperchaotic Chen systems by nonlinear feedback control laws. A modified 4D hyperchaotic Chen system is obtained by changing the nonlinear function of the 4D hyperchaotic Chen system, furthermore, an electronic circuit to realize two different 4D hyperchaotic Chen systems is designed. With nonlinear feedback control method, chaos synchronization between two different 4D hyperchaotic Chen systems is achieved. Based on the stability theory, the functions of the nonlinear feedback control for synchronization of two different 4D hyperchaotic Chen systems is derived, the range of feedback gains is determined. Numerical simulations are shown to verify the theoretical results.     

Modified Projective Synchronization of a New Hyperchaotic System via Nonlinear Control

In this paper, a nonlinear control scheme of two identical hyperchaotic Chert systems is developed to realize their modified projective synchronization. We achieve modified projective synchronization between the two identical hyperchaotic systems by directing the scaling factor onto the desired value. With symbolic computation system Maple and Lyapunov stability theory, numerical simulations are given to perform the process of the synchronization.     

Global Exponential Projective Synchronization and Lag Synchronization of Hyperchaotic Lii System

This paper investigates the projective synchronization and lag synchronization of a new hyperchaotic system [Physica A 364 （2006） 103]. On the basis of Lyapunov stability theory, two novel nonlinear controllers are respectively designed to guarantee the global exponential projective synchronization （including complete synchronization and antisynchronization） and lag synchronization. Finally, numerical simulations are given to show the effectiveness of the main results.     

Synchronization between two different noise-perturbed chaotic systems with unknown parameters

In this paper, a general method of synchronizing noise-perturbed chaotic systems with unknown parameters is proposed. Based on the LaSalle-type invariance principle for stochastic differential equations and by employing a combination of feedback control and adaptive control, some sufficient conditions of chaos synchronization between these noise-perturbed systems with unknown parameters are established. The model used in the research is the chaotic system, but the method is also applicable to the hyperchaotic systems. Unified system and noise-perturbed RSssler system, hyperchaotic Chen system and nolse-perturbed hyperchaotic RSssler system are taken for illustrative examples to demonstrate this technique.[第一段]     

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.     

Fuzzy modeling and impulsive control of hyperchaotic Lü system

This paper presents a novel approach to hyperchaos control of hyperchaotic systems based on impulsive control and the Takagi--Sugeno (T--S) fuzzy model. In this study, the hyperchaotic Lü system is exactly represented by the T--S fuzzy model and an impulsive control framework is proposed for stabilizing the hyperchaotic Lü system, which is also suitable for classes of T--S fuzzy hyperchaotic systems, such as the hyperchaotic R?ssler, Chen, Chua systems and so on. Sufficient conditions for achieving stability in impulsive T--S fuzzy hyperchaotic systems are derived by using Lyapunov stability theory in the form of the linear matrix inequality, and are less conservative in comparison with existing results. Numerical simulations are given to demonstrate the effectiveness of the proposed method.     

Projective synchronization of a hyperchaotic system via periodically intermittent control

We further study the projective synchronization of a new hyperchaotic system. Different from the most existing methods, intermittent control is applied to chaotic synchronization in the present paper. We formulate the intermittent control system that governs the dynamics of the projective synchronization error, then derive the sufficient conditions for the exponential stability of intermittent control system by using the Lyapunov stability theory, and finally establish the periodically intermittent controller according to the stability criterion by which the projective synchronization is expected to be achieved. The analytical results are also demonstrated by several numerical simulations.     

Modified Cascade Synchronization Scheme for Discrete-Time Hyperchaotic Systems

In this paper, the modified cascade synchronization scheme is proposed to investigate the synchronization in discrete-time hyperchaotic systems. By choosing a general kind of proportional scaling error functions and based on rigorous control theory, we take the discrete-time hyperchaotic system due to Wang and 3D generalized Henon map as two examples to achieve the modified cascade synchronization, respectively. Numerical simulations are used to verify the effectiveness of the proposed technique.     

Circuitry implementation of a novel four-dimensional nonautonomous hyperchaotic Liu system and its experimental studies on synchronization control

Based on the three-dimensional Liu chaotic system, this paper appends a feedback variable to construct a novel hyperchaotic Liu system. Then, a control signal is further added to construct a novel nonautonomous hyperchaotic Liu system. Through adjusting the frequency of the control signal, the chaotic property of the system can be controlled to show some different dynamic behaviors such as periodic, quasi-periodic, chaotic and hyperchaotic dynamic behaviours. By numerical simulations, the Lyapunov exponent spectrums, bifurcation diagrams and phase diagrams of the two new systems are studied, respectively. Furthermore, the synchronizing circuits of the nonautonomous hyperchaotic Liu system are designed via the synchronization control method of single variable coupling feedback. Finally, the hardware circuits are implemented, and the corresponding waves of chaos are observed by an oscillograph.     

Adaptive synchronization of a hyperchaotic L system based on extended passive control

This paper investigates the adaptive synchronization of hyperchaotic Lü systems based on the method of extended passive control. By combining the feedback control, the extended passive control method with two output variables is developed, which can synchronize hyperchaotic Lü systems asymptotically and globally more easily without knowing the bound of state of the hyperchaotic system. Adaptive laws are introduced to estimate the unknown parameters as well. Simulation results show the effectiveness and flexibility of the proposed control scheme.