Function projective synchronization for fractional-order chaotic systems

This letter investigates the function projective synchronization between fractional-order chaotic systems. Based on the stability theory of fractional-order systems and tracking control, a controller for the synchronization of two fractional-order chaotic systems is designed. This technique is applied to achieve synchronization between the fractional-order Lorenz systems with different orders, and achieve synchronization between the fractional-order Lorenz system and fractional-order Chen system. The numerical simulations demonstrate the validity and feasibility of the proposed method.     

Chaos synchronization between two different fractional-order hyperchaotic systems

This work investigates chaos synchronization between two different fractional-order hyperchaotic system (FOHS)s. A novel FOHS is also proposed in this paper. The Chen FOHS is controlled to be a new FOHS and the Lü FOHS, respectively. The analytical conditions for the synchronization of these pairs of different FOHSs are derived by utilizing Laplace transform. Furthermore, synchronization between two different FOHSs is achieved by utilizing feedback control method in a quite short period and both remain in chaotic states. Numerical simulations are used to verify the theoretical analysis using different values of the fractional-order parameter.     

Feedback control and hybrid projective synchronization of a fractional-order Newton-Leipnik system

In this work, stability analysis of the fractional-order Newton-Leipnik system is studied by using the fractional Routh-Hurwitz criteria. The fractional Routh-Hurwitz conditions are used to control chaos in the proposed fractional-order system to its equilibria. Based on the fractional Routh-Hurwitz conditions and using specific choice of linear feedback controllers, it is shown that the Newton-Leipnik system is controlled to its equilibrium points. Moreover, the theoretical basis of hybird projective synchronization of commensurate and incommensurate fractional-order Newton-Leipnik systems is investigated. Based on the stability theorems of fractional-order systems, the controllers for hybrid projective synchroniztion are derived. Numerical results show the effectiveness of the theoretical analysis.     

Chaos synchronization using single variable feedback based on backstepping method

In recent years, backstepping method has been developed in the field of nonlinear control, such as controller, observer and output regulation. In this paper, an effective backstepping design is applied to chaos synchronization. There are some advantages in this method for synchronizing chaotic systems, such as (a) the synchronization error is exponential convergent; (b) only one variable information of the master system is needed; (c) it presents a systematic procedure for selecting a proper controller. Numerical simulations for the Chua's circuit and the Rössler system demonstrate that this method is very effective.     

Chaos control of chaotic dynamical systems using backstepping design

This work presents chaos control of chaotic dynamical systems by using backstepping design method. This technique is applied to achieve chaos control for each of the dynamical systems Lorenz, Chen and Lü systems. Based on Lyapunov stability theory, control laws are derived. We used the same technique to enable stabilization of chaotic motion to a steady state as well as tracking of any desired trajectory to be achieved in a systematic way. Numerical simulations are shown to verify the results.     

Global fractional-order projective dynamical systems

This paper studies a class of global fractional-order projective dynamical systems. First, we show the existence and uniqueness of the solution of this type of system. Then, the existence of the equilibrium point of this class of dynamical systems is obtained. Further more, we obtain the α-exponential stability of the equilibrium point under suitable conditions. In addition, we use a predictor–corrector algorithm to find a solution to this kind of system. Finally a numerical example is provided to illustrate the results obtained in this paper.     

Chaos in fractional-order Genesio-Tesi system and its synchronization

In this paper we study the chaotic dynamics of fractional-order Genesio-Tesi system. Theoretically, a necessary condition for occurrence of chaos is obtained. Numerical investigations on the dynamics of this system have been carried out and properties of the system have been analyzed by means of Lyapunov exponents. It is shown that in case of commensurate system the lowest order of fractional-order Genesio-Tesi system to yield chaos is 2.79. Further, chaos synchronization of fractional-order Genesio-Tesi system is investigated via two different control strategies. Active control and sliding mode control are proposed and the stability of the controllers are studied. Numerical simulations have been carried out to verify the effectiveness of controllers.     

Chaos control and generalized projective synchronization of heavy symmetric chaotic gyroscope systems via Gaussian radial basis adaptive variable structure control

This paper proposes the chaos control and the generalized projective synchronization methods for heavy symmetric gyroscope systems via Gaussian radial basis adaptive variable structure control. Because of the nonlinear terms of the gyroscope system, the system exhibits chaotic motions. Occasionally, the extreme sensitivity to initial states in a system operating in chaotic mode can be very destructive to the system because of unpredictable behavior. In order to improve the performance of a dynamic system or avoid the chaotic phenomena, it is necessary to control a chaotic system with a periodic motion beneficial for working with a particular condition. As chaotic signals are usually broadband and noise like, synchronized chaotic systems can be used as cipher generators for secure communication. This paper presents chaos synchronization of two identical chaotic motions of symmetric gyroscopes. In this paper, the switching surfaces are adopted to ensure the stability of the error dynamics in variable structure control. Using the neural variable structure control technique, control laws are established which guarantees the chaos control and the generalized projective synchronization of unknown gyroscope systems. In the neural variable structure control, Gaussian radial basis functions are utilized to on-line estimate the system dynamic functions. Also, the adaptation laws of the on-line estimator are derived in the sense of Lyapunov function. Thus, the unknown gyro systems can be guaranteed to be asymptotically stable. Also, the proposed method can achieve the control objectives. Numerical simulations are presented to verify the proposed control and synchronization methods. Finally, the effectiveness of the proposed methods is discussed.     

Adaptive generalized function projective synchronization of uncertain chaotic systems

This paper mainly investigates adaptive generalized function projective synchronization of two different uncertain chaotic systems, which is a further extension of many existing projection synchronization schemes, such as modified projection synchronization, function projective synchronization and so on. On the basis of Lyapunov stability theory, an adaptive controller for the synchronization of two different chaotic systems is designed, and some parameter update laws for estimating the unknown parameters of the systems are also gained. This technique is applied to achieve synchronization between Lorenz and Rössler chaotic systems. The numerical simulations demonstrate the validity and feasibility of the proposed method.     

Chaos synchronization of the fractional-order Chen’s system

In this paper, based on the stability theorem of linear fractional systems, a necessary condition is given to check the chaos synchronization of fractional systems with incommensurate order. Chaos synchronization is studied by utilizing the Pecora–Carroll (PC) method and the coupling method. The necessary condition can also be used as a tool to confirm results of a numerical simulation. Numerical simulation results show the effectiveness of the necessary condition.     

Adaptive type-2 fuzzy backstepping control of uncertain fractional-order nonlinear systems with unknown dead-zone

A new problem of adaptive type-2 fuzzy fractional control with pseudo-state observer for commensurate fractional order dynamic systems with dead-zone input nonlinearity is considered in presence of unmatched disturbances and model uncertainties; the control scheme is constructed by using the backstepping and adaptive technique. To avoid the complexity of backstepping design process, the dynamic surface control is used. Also, Interval type-2 Fuzzy logic systems (IT2FLS) are used to approximate the unknown nonlinear functions. By using the fractional adaptive backstepping, fractional control laws are constructed; this method is applied to a class of uncertain fractional-order nonlinear systems. In order to better control performance in reducing tracking error, the PSO algorithm is utilized for tuning the controller parameters. Stability of the system is proven by the Mittag–Leffler method. It is shown that the proposed controller guarantees the boundedness property for the system and also the tracking error can converge to a small neighborhood of the origin. The efficiency of the proposed method is illustrated with simulation examples.     

Chaos control of 4D chaotic systems using recursive backstepping nonlinear controller

This paper examines chaos control of two four-dimensional chaotic systems, namely: the Lorenz–Stenflo (LS) system that models low-frequency short-wavelength gravity waves and a new four-dimensional chaotic system (Qi systems), containing three cross products. The control analysis is based on recursive backstepping design technique and it is shown to be effective for the 4D systems considered. Numerical simulations are also presented.     

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.     

Modified function projective lag synchronization of chaotic systems with disturbance estimations

This paper addresses the modified function projective lag synchronization (MFPLS) for a class of chaotic systems with unknown external disturbances. The disturbances are supposed to be generated by the exogenous systems. By using the disturbance-observer-based control and the linear matrix inequality approach, the disturbance observers are developed to ensure the boundedness of the disturbance error dynamics. Then by employing the sliding mode control (SMC) technique, an active SMC law is established to guarantee the disturbance rejection and realize MFPLS between the master and slave systems. And the corresponding numerical simulation is provided to illustrate the effectiveness of the proposed method.     

Adaptive modified function projective synchronization of hyperchaotic systems with unknown parameters

This paper is involved with the adaptive modified function projective synchronization (MFPS) problem of hyperchaotic systems with unknown parameters. Based on the Lyapunov stability theorem and adaptive control method, adaptive controllers and parameters update laws can be presented for the MFPS not only between two identical hyperchaotic systems but particularly also between two different hyperchaotic systems with fully unknown or partially unknown parameters. Moreover, the coupling strength can be automatically adapted to a updated law. Numerical simulations are presented to show the effectiveness of the proposed synchronization schemes.     

Chaos synchronization between two different chaotic systems using active control

This work presents chaos synchronization between two different chaotic systems by using active control. This technique is applied to achieve chaos synchronization for each pair of the dynamical systems Lorenz, Lü and Chen. Numerical simulations are shown to verify the results.     

Bifurcations and synchronization of the fractional-order simplified Lorenz hyperchaotic systems

In this paper, dynamics of the fractional-order simplied Lorenz hyperchaotic system is investigated. Modied Adams-Bashforth-Moulton method is applied for numerical simulation. Chaotic regions and periodic windows are identied. Dierent types of motions are shown along the routes to chaos by means of phase portraits, bifurcation diagrams, and the largest Lyapunov exponent. The lowest fractional order to generate chaos is 3.8584. Synchronization between two fractional-order simplied Lorenz hyperchaotic systems is achieved by using active control method. The synchronization performances are studied by changing the fractional order, eigenvalues and eigenvalue standard deviation of the error system.     

Adaptive modified function projective synchronization of unknown chaotic systems with different order

This paper investigates the modified function projective synchronization (MFPS) between two different dimensional chaotic systems with fully unknown or partially unknown parameters via increased order. Based on the Lyapunov stability theorem and adaptive control method, a unified adaptive controller and parameters update law can be designed for achieving the MFPS of the two different chaotic systems with different orders. Numerical simulations are presented to show the effectiveness of the proposed synchronization scheme.     

Generalized function projective synchronization of two different hyperchaotic systems with unknown parameters

Combining adaptive control theory with an antisymmetric structure, an extended adaptive controller which is more generalized and simpler than some existing controllers is designed. Under the controller, generalized function projective synchronization of two different uncertain hyperchaotic systems is achieved, and the unknown parameters are also estimated. In numerical simulations, the scaling function factors discussed in this paper are more complicated, and they have not been discussed in other papers. Corresponding simulation results are presented to show that the controller works well.