A practical synchronization approach for fractional-order chaotic systems

A practical synchronization approach is proposed for a class of fractional-order chaotic systems to realize perfect \(\delta \)-synchronization, and the nonlinear functions in the fractional-order chaotic systems are all polynomials. The \(\delta \)-synchronization scheme in this paper means that the origin in synchronization error system is stable. The reliability of \(\delta \)-synchronization has been confirmed on a class of fractional-order chaotic systems with detailed theoretical proof and discussion. Furthermore, the \(\delta \)-synchronization scheme for the fractional-order Lorenz chaotic system and the fractional-order Chua circuit is presented to demonstrate the effectiveness of the proposed method.     

Adaptive impulsive synchronization of nonlinear chaotic systems

In this paper, an adaptive impulse control with only one restriction criterion is derived to achieve synchronization of nonlinear chaotic systems in the exponential rate of convergence. It is assumed that the system satisfies the local Lipschitz condition and the Lipschitz constant is estimated by an augmented adaptation equation. The significance of the related control parameters in the criterion is also discussed in detail. The Duffing and the Lorenz systems have been simulated to illustrate the theoretical analysis.     

Controlled synchronization of discrete-time chaotic systems under communication constraints

This paper investigates the controlled synchronization problem for a class of nonlinear discrete-time chaotic systems subject to limited communication capacity. A general chaotic master system and its slave system with a controller are connected via a limited capacity channel. In this case, the effect of quantization errors is considered. A practical quantized scheme is proposed so that the synchronization error is input-to-state stable with respect to the transmission error. Meanwhile, the transmission error decays to zero exponentially. This implies that the synchronization error converges to zero under a limited communication channel. A?simulation example for the Fold chaotic system is presented to illustrate the effectiveness of the proposed method.     

Adaptive impulsive synchronization of uncertain drive-response complex-variable chaotic systems

In this paper, impulsive synchronization of drive-response complex-variable chaotic systems is investigated. The drive-response systems with known parameters is considered via impulsive control and adaptive scheme as well as systems with unknown parameters. Noticeably, adaptive strategy is adopted to relax the restriction on the impulsive interval, and the system parameters need not to be known beforehand. According to the Lyapunov stability theory, some synchronization criteria are derived and verified by several numerical simulations.     

Robust synchronization of uncertain fractional-order chaotic systems with time-varying delay

This paper presents a new technique using a recurrent non-singleton type-2 sequential fuzzy neural network (RNT2SFNN) for synchronization of the fractional-order chaotic systems with time-varying delay and uncertain dynamics. The consequent parameters of the proposed RNT2SFNN are learned based on the Lyapunov–Krasovskii stability analysis. The proposed control method is used to synchronize two non-identical and identical fractional-order chaotic systems, with time-varying delay. Also, to demonstrate the performance of the proposed control method, in the other practical applications, the proposed controller is applied to synchronize the master–slave bilateral teleoperation problem with time-varying delay. Simulation results show that the proposed control scenario results in good performance in the presence of external disturbance, unknown functions in the dynamics of the system and also time-varying delay in the control signal and the dynamics of system. Finally, the effectiveness of proposed RNT2SFNN is verified by a nonlinear identification problem and its performance is compared with other well-known neural networks.     

A novel secure image transmission scheme based on synchronization of fractional-order discrete-time hyperchaotic systems

In this paper, a secure image transmission scheme based on synchronization of fractional-order discrete-time hyperchaotic systems is proposed. In this scheme, a fractional-order modified-Hénon map is considered as a transmitter, the system parameters and fractional orders are considered as secret keys. As a receiver, a step-by-step delayed observer is used, and based on this one, an exact synchronization is established. To make the transmission scheme secure, an encryption function is used to cipher the original information using a key stream obtained from the chaotic map sequences. Moreover, to further enhance the scheme security, the ciphered information is inserted by inclusion method in the chaotic map dynamics. The first contribution of this paper is to propose new results on the observability and the observability matching condition of nonlinear discrete-time fractional-order systems. To the best of our knowledge, these features have not been addressed in the literature. In the second contribution, the design of delayed discrete observer, based on fractional-order discrete-time hyperchaotic system, is proposed. The feasibility of this realization is demonstrated. Finally, different analysis are introduced to test the proposed scheme security. Simulation results are presented to highlight the performances of our method. These results show that, our scheme can resist different kinds of attacks and it exhibits good performance.     

Adaptive synchronization of fractional-order chaotic systems via a single driving variable

This letter investigates the synchronization of a class of three-dimensional fractional-order chaotic systems. Based on sliding mode variable structure control theory and adaptive control technique, a single-state adaptive-feedback controller containing a novel fractional integral sliding surface is developed to synchronize a class of fractional-order chaotic systems. The present controller, which only contains a single driving variable, is simple both in design and implementation. Simulation results for three fractional-order chaotic systems are provided to illustrate the effectiveness of the proposed scheme.     

Analysis and synchronization for a new fractional-order chaotic system with absolute value term

A new fractional-order chaotic system with absolute value term is introduced. Some dynamical behaviors are investigated and analyzed. Furthermore, synchronization of this system is achieved by utilizing the drive-response method and the feedback method. The suitable parameters for achieving synchronization are studied. Both the theoretical analysis and numerical simulations show the effectiveness of the two methods.     

Modified projective synchronization of fractional-order chaotic systems via active sliding mode control

This paper is devoted to study the problem of modified projective synchronization of fractional-order chaotic system. Base on the stability theorems of fractional-order linear system, active sliding mode controller is proposed to synchronize two different fractional-order systems. Moreover, the controller is robust to the bounded noise. Numerical simulations are provided to show the effectiveness of the analytical results.     

Parameter identification and adaptive impulsive synchronization of uncertain complex-variable chaotic systems

Synchronization of nonlinear dynamical systems with complex variables has attracted much more attention in various fields of science and engineering. In this paper, the problem of parameter identification and adaptive impulsive synchronization for a class of chaotic (hyperchaotic) complex nonlinear systems with uncertain parameters is investigated. Based on the theories of adaptive control and impulsive control, a synchronization scheme is designed to make a class of chaotic and hyperchaotic complex systems asymptotically synchronized, and uncertain parameters are identified simultaneously in the process of synchronization. Particularly, the proposed adaptive–impulsive control laws for synchronization are simple and can be readily applied in practical applications. The synchronization of two identical chaotic complex Chen systems and two identical hyperchaotic complex Lü systems are taken as two examples to verify the feasibility and effectiveness of the proposed controllers and identifiers.     

Robust chaos synchronization of fractional-order chaotic systems with unknown parameters and uncertain perturbations

Chaotic systems in practice are always influenced by some uncertainties and external disturbances. This paper investigates the problem of practical synchronization of fractional-order chaotic systems. Based on Lyapunov stability theory and a fractional-order differential inequality, a modified adaptive control scheme and adaptive laws of parameters are developed to robustly synchronize coupled fractional-order chaotic systems with unknown parameters and uncertain perturbations. This synchronization approach is simple, global and theoretically rigorous. Simulation results for two fractional-order chaotic systems are provided to illustrate the effectiveness of the proposed scheme.     

Robust synchronization of two different uncertain fractional-order chaotic systems via adaptive sliding mode control

This paper proposes a novel robust fractional-order sliding mode approach for the synchronization of two fractional-order chaotic systems in the presence of system parameter uncertain and external disturbance. An adaptive sliding mode controller is constructed resorted to the designed fractional integral type sliding surface. Based on the Lyapunov stability theorem, the stability of the closed error system is proved. Finally, a numerical simulation is performed to illustrate the effectiveness of the proposed method.     

Circuit simulation for synchronization of a fractional-order and integer-order chaotic system

We design a new three-dimensional double-wing fractional-order chaotic system with three quadratic terms, confirmed by numerical simulation and circuit implementation. We then study the synchronization between the new double-wing fractional-order chaotic system and different Lorenz systems with different structures. In the process of the synchronization, the definition of ‘the simplest response system’ and the practical method of designing the circuit have been originally proposed. The circuit of ‘the simplest response system’ (even the simplest incommensurate-order response system), holding different structures with the drive system, of any one integral or fractional drive system now can be designed effectively and sufficiently. Our results are supported by numerical simulation and circuit implementation.     

The synchronization between two discrete-time chaotic systems using active robust model predictive control

In this paper, we consider the synchronization of two discrete-time chaotic systems. A novel active robust model predictive strategy is proposed to guarantee the synchronization of two discrete-time systems in the presence of model uncertainty. The proposed approach reduces the synchronization to a convex optimization involving linear matrix inequalities. The numerical simulations illustrate the effectiveness of the proposed method.     

Different impulsive effects on synchronization of fractional-order memristive BAM neural networks

This paper considered exponential synchronization in fractional-order memristive BAM neural networks (FMBAMNNs) with time delay via switching jumps mismatch. Exponential function is introduced for studying fractional-order differential system. According to double-layer structure of FMBAMNNs, two controllers are designed for the response FMBAMNNs. Particularly, more wide ranges of impulsive effects, which are affected by fractional-order \(\alpha \), are discussed in detail. One case is that the impulsive effect contributes to system convergence, and the other is that the impulsive effect destroys the system convergence. Based on the fractional stability theory and the definition of average impulsive interval, several criteria for achieving synchronization of FMBAMNNs are established. For different impulsive effects, the rate of convergence is precisely expressed. Finally, numerical examples verify the validity of the theoretical results.     

Robust synchronization of two different fractional-order chaotic systems with unknown parameters using adaptive sliding mode approach

In this paper, an adaptive sliding mode control method is introduced to ensure robust synchronization of two different fractional-order chaotic systems with fully unknown parameters and external disturbances. For this purpose, a fractional integral sliding surface is defined and an adaptive sliding mode controller is designed. In this method, no knowledge of the bounds of parameters and perturbation is required in advance and the parameters are updated through an adaptive control process. The proposed scheme is global and theoretically rigorous. Two examples are given to illustrate effectiveness of the scheme, in which the synchronizations between fractional-order chaotic Chen system and fractional-order chaotic Rössler 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.