Synchronization Of Fractional-order Chaotic Systems Research Articles

Back-stepping projective synchronization of fractional-order unified systems based on the lower triangular structure

The synchronization of fractional-order chaotic systems (FOCSs) plays an important role in modern control theory, the projective synchronization (PS) as a class of synchronization problems, also has huge applications and has attracted much attention. It is, however, shown in the obtained literature that the results on the PS of FOCSs either loss the rigorous theoretical demonstration or verify from the viewpoint of numerical simulations. How to derive a necessary and sufficient condition to guarantee the PS of complex FOCSs by a simple controller is still open. To this end, this article is concerned with the PS of fractional-order unified systems (FO-USs) that are important in FOCSs covering fractional-order Lorenz, Chen and Lü systems, where the controller is presented based on the lower triangular structure by use of the back-stepping technique. The necessary and sufficient criterion for the PS of FO-USs is proposed by solving an algebraic equation, and the controller for the PS of FO-USs is derived based on the lower triangular structure combined with back-stepping approach. Finally, the simulation results are reported to verify the correctness and efficiency of the obtained results.

Finite-time synchronization of fractional-order chaotic system based on hidden attractors

A new 3D fractional-order chaotic system is obtained by improving the Sprott-A system and introducing the definition of fractional calculus to it. Then the new system is certified to be chaotic by studying and analyzing the phase diagram, Lyapunov exponents, and smaller alignment index tests. Then the analysis of equilibrium points finds that the new system has virtually no equilibrium points and hidden attractors. The new system is dynamically analyzed by bifurcation diagram, time-domain waveform and complexity, it is indicated that the system is susceptible to initial conditions, and with the changes of different parameters the system produced different scroll types of attractors. In addition, to verify the feasibility of the system, a simulation circuit design based on Multisim is therefore carried out. Finally, the finite-time synchronization of the fractional-order system is successfully achieved by taking advantage of the high security of the hidden attractors.

Synchronization of time-varying fractional-order chaotic systems in the presence of unknown disturbance using a fast disturbance identification

This paper studies the synchronization of fractional-order chaotic systems in the presence of unknown external disturbances. A new disturbance identification method using fractional-order algebraic identification is proposed to handle unknown disturbances. In order to guarantee system stability, the nonlinear state-feedback controller is designed based on the Lyapunov theory. All system parameters and orders, as well as the external disturbances, are considered time-varying in this paper. The simulation results show that the estimation of unknown disturbance is executed more quickly and accurately using the proposed method, which also leads to fast and precise synchronization of the chaotic systems. A circuit implementation is also performed to identify the unknown disturbances in a fractional-order electronic chaotic oscillator system to evaluate the practicability of the proposed approach.

Fixed-Time Multi-Switch Combined–Combined Synchronization of Fractional-Order Chaotic Systems with Uncertainties and External Disturbances

In this paper, the fixed-time multi-switch combination–combination synchronization (FTMSCCS) of fractional-order chaotic systems with uncertainties and external disturbances is studied. The appropriate sliding mode surface and controller are proposed based on a Lyapunov theorem, and fixed-time multi-switching combination–combination synchronizations between four fractional-order chaotic systems are realized. The Lyapunov function is designed to prove the feasibility of the controller theoretically, and the effectiveness and robustness of the synchronization mechanism are further verified by numerical simulations. The advantage of this article is that it extends fixed-time synchronization to multi-switch combination–combination synchronization, enabling synchronization for a limited time, while increasing the complexity of the synchronization mechanism and improving its confidentiality in communication applications.

Adaptive Fuzzy Backstepping Control of Fractional-Order Chaotic System Synchronization Using Event-Triggered Mechanism and Disturbance Observer

The synchronization of fractional-order chaotic systems is investigated using command-filtered adaptive fuzzy control with a disturbance observer, where an event-triggered mechanism and backstepping control technique are employed. In order to relieve the pressure of the continuous update of the controller and improve the resource utilization, an event-triggered control strategy is constructed to reduce the amount of communication for the actuator. Under the framework of adaptive fuzzy backstepping recursive design, fuzzy logical systems and disturbance observers are proposed to estimate the unknown parametric uncertainties and external disturbances, respectively. Moreover, a tracking differentiator is introduced to eliminate the drawback of the explosion of complexity in traditional backstepping. By applying the fractional-order stability theory, all closed-loop signals are bounded and chaos synchronization is achieved. Finally, a simulation example is provided to confirm the effectiveness of the designed method.

Parameter Identification and the Finite-Time Combination–Combination Synchronization of Fractional-Order Chaotic Systems with Different Structures under Multiple Stochastic Disturbances

This paper researches the issue of the finite-time combination-combination (C-C) synchronization (FTCCS) of fractional order (FO) chaotic systems under multiple stochastic disturbances (SD) utilizing the nonsingular terminal sliding mode control (NTSMC) technique. The systems we considered have different characteristics of the structures and the parameters are unknown. The stochastic disturbances are considered parameter uncertainties, nonlinear uncertainties and external disturbances. The bounds of the uncertainties and disturbances are unknown. Firstly, we are going to put forward a new FO sliding surface in terms of fractional calculus. Secondly, some suitable adaptive control laws (ACL) are found to assess the unknown parameters and examine the upper bound of stochastic disturbances. Finally, combining the finite-time Lyapunov stability theory and the sliding mode control (SMC) technique, we propose a fractional-order adaptive combination controller that can achieve the finite-time synchronization of drive-response (D-R) systems. In this paper, some of the synchronization methods, such as chaos control, complete synchronization, projection synchronization, anti-synchronization, and so forth, have become special cases of combination-combination synchronization. Examples are presented to verify the usefulness and validity of the proposed scheme via MATLAB.

A simple fractional-order chaotic system based on memristor and memcapacitor and its synchronization application

Memelements play an important part in the design of high density memory systems and low power memory. In this paper, we created a novel fractional-order chaotic circuit with a memristor and a memcapacitor with a linear inductor. The various dynamical properties of the fractional-order system are investigated by using some dynamic analysis methods like Lyapunov exponents and bifurcation after the numerical solution for the fractional-order system. In addition, to show the applicative advantages of the proposed chaotic system, we have realized about the synchronization of fractional-order chaotic systems and used it in secure communication systems first time in the literature according to our knowledge. The results of the theoretical analysis and simulation show that the simple fractional-order chaotic system has very rich dynamic properties and can be used in different engineering applications.

Fuzzy synchronization of fractional-order chaotic systems using finite-time command filter

For conventional backstepping control, the analytical calculation of certain virtual controller is required in each step, so when system order increases, the computational burden also increases. An adaptive command filtered fuzzy synchronization approach is implemented for strict feedback fractional-order chaotic systems (FCSs) in this paper. The computational explosion in the backstepping is solved through a fractional-order command filter (FCF). The filtering approximation errors arising from the FCF are overcame by a error compensation mechanism (ECM). In every step, a fuzzy system is introduced to process system uncertainties. The stability of synchronization errors of a strict feedback FCS is ensured based on fractional Lyapunov stability criteria. Finally, numerical simulation examples illustrate the effectiveness of the proposed scheme.

Synchronization of Fractional-Order Chaotic Systems with Model Uncertainty and External Disturbance

This paper is concerned with complete synchronization of fractional-order chaotic systems with both model uncertainty and external disturbance. Firstly, we propose a new dynamic feedback control method for complete synchronization of fractional-order nominal systems (without both uncertainty and disturbance). Then, a new uncertainty and disturbance estimator (UDE)-based dynamic feedback control method for the fractional-order systems with both uncertainty and disturbance is presented, by which the synchronization problem of such fractional-order chaotic systems is realized. Finally, the fractional-order Lorenz system is used to demonstrate the practicability of the proposed results.

Finite-Time Synchronization of Fractional-Order Chaotic Systems with Different Structures under Stochastic Disturbances

This paper studies the finite-time synchronization of fractional-order chaotic systems with different structures under parameter disturbance and external disturbance. We put forward a fractional-order controller that can achieve the finite-time synchronization of any-order fractional-order chaotic systems under stochastic disturbances. This controller has good robustness and anti-interference performance. With the concept of the finite-time stability theory given, some judgment criterions for the synchronization of fractional-order chaotic systems are proved. This method can not only make the error systems have a faster convergence rate but also can be implemented in engineering easily. The numerical simulations of two specific examples demonstrate the effectiveness of the method. At the same time, the synchronised time of finite-time synchronization is shorter and faster than the complete synchronization and the time can be adjusted according to the parameters in the controller.

Fixed-time terminal synergetic observer for synchronization of fractional-order chaotic systems.

In this paper, a fixed-time terminal synergetic observer for synchronization of fractional-order nonlinear chaotic systems is proposed. First, fixed-time terminal attractors for fractional-order nonlinear systems are introduced on the basis of fixed-time stability of integer-order nonlinear differential equations and on defining particular fractional-order macro-variables. Second, a new synergetic observer dedicated to the synchronization of fractional-order chaotic systems is developed. The proposed observer converges in a predefined fixed-time uniformly bounded with respect to initial conditions. Thanks to the step-by-step procedure, only one communication channel is used to achieve the synchronization. Third, a fixed-time synergetic extended observer with unknown input is constructed to simultaneously estimate the state variables and to recover the unknown input. Finally, computer simulations are performed to illustrate the efficiency of the proposed synchronization method and its application in a secure communication scheme.

Modified Projective Synchronization of Fractional-order Chaotic Systems with Different Dimensions

In this paper, the modified projective synchronization between two fractional-order chaotic systems with different dimensions is investigated. The added-order scheme and the reduced-order scheme are proposed, respectively. Based on the Laplace transformation and feedback control theory, controllers are designed such that two chaotic systems with different dimensions could be synchronized asymptotically under the presented schemes. Corresponding numerical simulations are given to show the effectiveness of the proposed schemes.

The fractional-order unified chaotic system: A general cascade synchronization method and application

In this paper, we extend the cascade synchronization of integer-order chaotic systems to function cascade synchronization of fractional-order chaotic systems. The nice feature of our method is that a fractional-order chaotic system will synchronize to another system via a scaling function. The rich choices of the scaling function turn our method more general than some existing methods. As applications, we take a fractional-order unified chaotic system as an illustrative example to test the effectiveness.

Finite-time projective synchronization of fractional-order chaotic systems via soft variable structure control

The finite-time projective synchronization of two fractional-order chaotic systems with control constraints is investigated in this study. A soft variable structure control scheme is first introduced for finite-time synchronization. A controller is then proposed for finite-time generalized projective synchronization. The finite-time stability of the error systems is rigorously proven. The control parameters of the controller are limited by the soft variable structure method considering the constraints of the controller. Finally, numerical simulation results are presented to demonstrate the effectiveness and feasibility of the proposed strategy and verify the theoretical results.

Global Mittag-Leffler boundedness and synchronization for fractional-order chaotic systems

This paper deals with the issues of the global Mittag-Leffler boundedness and synchronization for three-dimensional fractional order chaotic systems (FOCSs). First, by constructing proper generalized Lyapunov function and using the extremum principle of function, some new 3D ellipsoid estimations and a cylindrical domain estimation of the bounds are derived for the addressed fractional system without existence assumptions, which improve the earlier publications and can deduce some new estimations. Besides, linear feedback control strategies with a single state and one input or two inputs are proposed to achieve synchronization. Some new sufficient synchronization criteria are derived by inequality techniques. Comparison of our results with that of the existing ones shows that the computation of the bounds and the control gain constants is easier and simpler and the controllers here have more simple structures. Simulation results are given to show the validity of the proposed scheme.

Impulsive observer with predetermined finite convergence time for synchronization of fractional-order chaotic systems based on Takagi–Sugeno fuzzy model

This paper is devoted to the problem of observer design for synchronization of nonlinear fractional-order chaotic systems described by the Takagi–Sugeno fuzzy model. We propose a new method of designing an observer that converges in finite time. The novelty of the proposed observer compared to those developed in the literature is that the estimated state exactly converges to the true state in a finite time which can be chosen beforehand. This is made possible by updating the estimated state at a defined time instant. The abrupt update shows up an impulsive behavior of the observer’s dynamics. Two cases are considered. In the first case, the system is not affected by an unknown input. The second case considers the system subjected to the action of an unknown input. Finite-time convergence of the two proposed observers for both cases is established using linear matrix inequality formulation. Simulation results on the synchronization of fractional-order chaotic systems clearly illustrate the impulsive behavior of the proposed observer and the predefined finite-time synchronization. The advantage of predefined finite-time synchronization is highlighted by the ability of recovering an encrypted message without loss of information in a fractional-order chaos-based secure communication application.

The Robust Control and Synchronization of a Class of Fractional-Order Chaotic Systems with External Disturbances via a Single Output

This paper investigates the stabilization and synchronization of a class of fractional-order chaotic systems which are affected by external disturbances. The chaotic systems are assumed that only a single output can be used to design the controller. In order to design the proper controller, some observer systems are proposed. By using the observer systems some sufficient conditions for achieving chaos control and synchronization of fractional-order chaotic systems are derived. Numerical examples are presented by taking the fractional-order generalized Lorenz chaotic system as an example to show the feasibility and validity of the proposed method.

Secure Multiple-Input Multiple-Output Communications Based on F–M Synchronization of Fractional-Order Chaotic Systems with Non-Identical Dimensions and Orders

This paper investigates the F – M synchronization between non-identical fractional-order systems characterized by different dimensions and different orders. F – M synchronization combines the inverse generalized synchronization with the matrix projective synchronization. In particular, the proposed approach enables the F – M synchronization to be achieved between an n-dimensional master system and an m-dimensional slave system. The developed approach is applied to chaotic and hyperchaotic fractional systems with the aim of illustrating its applicability and suitability. A multiple-input multiple-output (MIMO) secure communication system is also developed by using the F – M synchronization and verified through computer simulations.

Synchronization of Different Uncertain Fractional-Order Chaotic Systems with External Disturbances via T-S Fuzzy Model

This paper presents an adaptive fuzzy synchronization control strategy for a class of different uncertain fractional-order chaotic/hyperchaotic systems with unknown external disturbances via T-S fuzzy systems, where the parallel distributed compensation technology is provided to design adaptive controller with fractional adaptation laws. T-S fuzzy models are employed to approximate the unknown nonlinear systems and tracking error signals are used to update the parametric estimates. The asymptotic stability of the closed-loop system and the boundedness of the states and parameters are guaranteed by fractional Lyapunov theory. This approach is also valid for synchronization of fractional-order chaotic systems with the same system structure. One constructive example is given to verify the feasibility and superiority of the proposed method.

Chaos Synchronization of Fractional-Order Chaotic Systems With Input Saturation

This paper deals with the problem of master-slave synchronization of fractional-order chaotic systems with input saturation. Sufficient stability conditions for achieving the synchronization are derived from the basis of a fractional-order extension of the Lyapunov direct method, a new lemma of the Caputo fractional derivative, and a local sector condition. The stability conditions are formulated in linear matrix inequality (LMI) forms and therefore are readily solved. The fractional-order chaotic Lorenz and hyperchaotic Lü systems with input saturation are utilized as illustrative examples. The feasibility of the proposed synchronization scheme is demonstrated through numerical simulations.