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.     

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.

     

FPGA implementation of novel fractional-order chaotic systems with two equilibriums and no equilibrium and its adaptive sliding mode synchronization

This paper introduces two novel fractional-order chaotic systems with cubic nonlinear resistor and investigates its adaptive sliding mode synchronization. Firstly the novel two equilibrium chaotic system with cubic nonlinear resistor (NCCNR) is derived and its dynamic properties are investigated. The fractional-order cubic nonlinear resistor system (FONCCNR) is then derived from the integer-order model and its stability and fractional-order bifurcation are discussed. Next a novel no-equilibrium chaotic cubic nonlinear resistor system (NECNR) is derived from NCCNR system. Dynamic properties of NECNR system are investigated. The fractional-order no equilibrium cubic nonlinear resistor system (FONECNR) is derived from NECNR. Stability and fractional-order bifurcation are investigated for the FONECNR system. The non-identical adaptive sliding mode synchronization of FONCCNR and FONECNR systems are achieved. Finally the proposed systems, adaptive control laws, sliding surfaces and adaptive controllers are implemented in FPGA.     

Lyapunov-based fractional-order controller design to synchronize a class of fractional-order chaotic systems

In this paper, a novel adaptive fractional-order feedback controller is first developed by extending an adaptive integer-order feedback controller. Then a simple but practical method to synchronize almost all familiar fractional-order chaotic systems has been put forward. Through rigorous theoretical proof by means of the Lyapunov stability theorem and Barbalat lemma, sufficient conditions are derived to guarantee chaos synchronization. A wide range of fractional-order chaotic systems, including the commensurate system and incommensurate case, autonomous system, and nonautonomous case, is just the novelty of this technique. The feasibility and validity of presented scheme have been illustrated by numerical simulations of the fractional-order Chen system, fractional-order hyperchaotic Lü system, and fractional-order Duffing system.     

Complete (anti-)synchronization of chaotic systems with fully uncertain parameters by adaptive control

This paper addresses a unified mathematical expression describing a class of chaotic systems, for which the problem of synchronization and anti-synchronization between different chaotic systems with fully uncertain parameters and different structure are studied. Based on the Lyapunov stability theory, a novel, simple, and systemic adaptive synchronization controller is designated, the analytic expression of the controller and the adaptive laws of parameters are developed. Moreover, the proposed scheme can be extended to anti-synchronize a class of chaotic systems. Two chaotic systems with different structure and fully uncertain parameters are employed as the examples to show the effectiveness of the proposed adaptive synchronization and anti-synchronization schemes. Additionally, the robustness and noise immunity of the adaptive synchronization scheme is investigated by measuring the mean squared error of the systems.     

Synchronization between integer-order chaotic systems and a class of fractional-order chaotic system based on fuzzy sliding mode control

In this paper, we focus on the synchronization between integer-order chaotic systems and a class of fractional-order chaotic system using the stability theory of fractional-order systems. A new fuzzy sliding mode method is proposed to accomplish this end for different initial conditions and number of dimensions. Furthermore, three examples are presented to illustrate the effectiveness of the proposed scheme, which are the synchronization between a fractional-order Lü chaotic system and an integer-order Liu chaotic system, the synchronization between a fractional-order hyperchaotic system based on Chen??s system and an integer-order hyperchaotic system based upon the Lorenz system, and the synchronization between a fractional-order hyperchaotic system based on Chen??s system, and an integer-order Liu chaotic system. Finally, numerical results are presented and are in agreement with theoretical analysis.     

Synchronization of nonidentical chaotic fractional-order systems with different orders of fractional derivatives

This paper addresses the problem of synchronization of chaotic fractional-order systems with different orders of fractional derivatives. Based on the stability theory of fractional-order linear systems and the idea of tracking control, suitable controllers are correspondingly proposed for two cases: the first is synchronization between two identical chaotic fractional-order systems with different fractional orders, and the other is synchronization between two nonidentical fractional-order chaotic systems with different fractional orders. Three numerical examples illustrate that fast synchronization can be achieved even between a chaotic fractional-order system and a hyperchaotic fractional-order system.     

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.     

Complete synchronization of chaotic complex nonlinear systems with uncertain parameters

Our main objective in this work is to investigate complete synchronization (CS) of n-dimensional chaotic complex systems with uncertain parameters. An adaptive control scheme is designed to study the synchronization of chaotic attractors of these systems. We applied this scheme, as an example, to study complete synchronization of chaotic attractors of two identical complex Lorenz systems. The adaptive control functions and the parameters estimation laws are calculated analytically based on the complex Lyapunov function. We show that the error dynamical systems are globally stable. Numerical simulations are computed to check the analytical expressions of adaptive controllers.     

Stabilization of fractional-order chaotic system via a single state adaptive-feedback controller

This letter investigates the stabilization of three-dimensional fractional-order chaotic systems, and proposes a single state adaptive-feedback controller for fractional-order chaos control based on Lyapunov stability theory, fractional order differential inequality, and adaptive control theory. The present controller which only contains a single state variable is simple both in design and implementation. Simulation results for several fractional-order chaotic systems are provided to illustrate the effectiveness of the proposed scheme.     

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.     

Chaos detection and parameter identification in fractional-order chaotic systems with delay

The paper first applies the 0–1 test for chaos to detecting chaos exhibited by fractional-order delayed systems. The results of the test reveal that there exists chaos in some fractional-order delayed systems with specific parameter values, which coincides with previous reports based on the phase portrait. In addition, it is very important to identify exactly the unknown specific parameters of fractional-order chaotic delayed systems in chaos control and synchronization. Thus, a method for parameter identification of fractional-order chaotic delayed systems based on particle swarm optimization (PSO) is presented. By treating the orders as parameters, the parameters and orders are identified through minimizing an objective function. PSO can efficiently find the optimal feasible solution of the objective function. Finally, numerical simulations on fractional-order chaotic logistic delayed system and fractional-order chaotic Chen delayed system show that the proposed method has effective performance of parameter identification.     

Generalized finite-time synchronization between coupled chaotic systems of different orders with unknown parameters

This letter investigates the adaptive finite-time synchronization of different coupled chaotic (or hyperchaotic) systems with unknown parameters. The sufficient conditions for achieving the generalized finite-time synchronization of two chaotic systems are derived based on the theory of finite-time stability of dynamical systems. By the adaptive control technique, the control laws and the corresponding parameters update laws are proposed such that the generalized finite-time synchronization of nonidentical chaotic (or hyperchaotic) systems is to be obtained. These results obtained are in good agreement with the existing one in open literature and it is shown that the technique introduced here can be further applied to various finite-time synchronizations between dynamical systems. Finally, numerical simulations are given to demonstrate the effectiveness of the proposed scheme.     

Adaptive generalized function projective synchronization of uncertain chaotic complex systems

The complex nonlinear systems appear in many important fields of physics and engineering, which are very useful for cryptography and secure communication. This paper investigates adaptive generalized function projective synchronization (AGFPS) between two different dimensional chaotic complex systems with fully or partially unknown parameters via both reduced order and increased order. Based on the Lyapunov stability theorem and adaptive control technique, a general adaptive controller with corresponding parameter update rule is constructed to achieve AGFPS between two nonidentical chaotic complex systems with distinct orders, and identify the unknown parameters simultaneously. This scheme is then applied to obtain AGFPS between the hyperchaotic complex Lü system and the chaotic complex Lorenz system with fully unknown parameters, and between the uncertain chaotic complex Chen system and the uncertain hyperchaotic complex Lorenz system, respectively. Corresponding simulations results are performed to show the feasibility and effectiveness of the proposed synchronization method.     

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.     

Dynamics analysis and hybrid function projective synchronization of a new chaotic system

This paper introduces a novel three-dimensional autonomous chaotic system by adding a quadratic cross-product term to the first equation and modifying the state variable in the third equation of a chaotic system proposed by Cai et al. (Acta Phys. Sin. 56:6230, 2007). By means of theoretical analysis and computer simulations, some basic dynamical properties, such as Lyapunov exponent spectrum, bifurcations, equilibria, and chaotic dynamical behaviors of the new chaotic system are investigated. Furthermore, hybrid function projective synchronization (HFPS) of the new chaotic system is studied by employing three different synchronization methods, i.e., adaptive control, system coupling and active control. The proposed approaches are applied to achieve HFPS between two identical new chaotic systems with fully uncertain parameters, HFPS in coupled new chaotic systems, and HFPS between the integer-order new chaotic system and the fractional-order Lü chaotic system, respectively. Corresponding numerical simulations are provided to validate and illustrate the analytical results.     

Adaptive bidirectionally coupled synchronization of?chaotic?systems with unknown parameters

This paper investigates the chaos synchronization of two bidirectionally coupled chaotic systems. In comparison with previous methods (identical bidirectionally coupled synchronization), the present control scheme is different bidirectionally coupled synchronization, which includes different complete bidirectionally coupled synchronization and different partial bidirectionally coupled synchronization. Based on the Lasalle invariance principle, adaptive schemes are designed to make two different bidirectionally coupled chaotic systems asymptotically synchronized, and unknown parameters are identified simultaneously in the process of synchronization. Theoretical analysis and numerical simulations are shown to verify the results.     

Lag synchronization of a class of chaotic systems with unknown parameters

Research on chaos synchronization of dynamical systems has been largely reported in literature. However, synchronization of different structure—uncertain dynamical systems—has received less attention. This paper addresses synchronization of a class of time-delay chaotic systems containing uncertain parameters. A unified scheme is established for synchronization between two strictly different time-delay uncertain chaotic systems. The synchronization is successfully achieved by designing an adaptive controller with the estimates of the unknown parameters and the nonlinear feedback gain. The result is rigorously proved by the Lyapunov stability theorem. Moreover, we illustrate the application of the proposed scheme by numerical simulation, which demonstrates the effectiveness and feasibility of the proposed synchronization method.     

Application of Takagi–Sugeno fuzzy model to a class of chaotic synchronization and anti-synchronization

In this study, we investigate a class of chaotic synchronization and anti-synchronization with stochastic parameters. A controller is composed of a compensation controller and a fuzzy controller which is designed based on fractional stability theory. Three typical examples, including the synchronization between an integer-order Chen system and a fractional-order Lü system, the anti-synchronization of different 4D fractional-order hyperchaotic systems with non-identical orders, and the synchronization between a 3D integer-order chaotic system and a 4D fractional-order hyperchaos system, are presented to illustrate the effectiveness of the controller. The numerical simulation results and theoretical analysis both demonstrate the effectiveness of the proposed approach. Overall, this study presents new insights concerning the concepts of synchronization and anti-synchronization, synchronization and control, the relationship of fractional and integer order nonlinear systems.     

Robust adaptive full state hybrid synchronization of chaotic complex systems with unknown parameters and external disturbances

This paper studies the robust adaptive full state hybrid projective synchronization (FSHPS) scheme for a class of chaotic complex systems with uncertain parameters and external disturbances. By introducing a compensator and using nonlinear control and adaptive control, the robust adaptive FSHPS scheme is derived, which can eliminate the influence of uncertainties effectively and achieve adaptive FSHPS of the chaotic (hyperchaotic) complex systems asymptotically with a small error bound. The adaptive laws of the unknown parameters are given, and the sufficient conditions of realizing FSHPS are derived as well. Moreover, we also discuss the case that parameters of chaotic complex system are complex. Finally, the complex Chen system and Lü system, and the hyperchaotic complex Lorenz system are taken as two examples and the numerical simulations are provided to verify the effectiveness and robustness of the proposed control scheme.