Numerical Simulation of the Fractional-Order Lorenz Chaotic Systems with Caputo Fractional Derivative

Although some numerical methods of the fractional-order chaotic systems have been announced,high-precision numerical methods have always been the direction that researchers strive to pursue.Based on this problem,this paper introduces a high-precision numerical approach.Some complex dynamic behavior of fractional-order Lorenz chaotic systems are shown by using the present method.We observe some novel dynamic behavior in numerical experiments which are unlike any that have been previously discovered in numerical experiments or theoretical studies.We investigate the influence of α_(1),α_(2),α_(3) on the numerical solution of fractional-order Lorenz chaotic systems.The simulation results of integer order are in good agreement with those of othermethods.The simulation results of numerical experiments demonstrate the effectiveness of the present method.

A new kind of nonlinear fractional-order chaotic phenomenon in coupled systems： coexistence of anti-phase and complete synchronization

In this paper, we have found a kind of interesting nonlinear phenomenon hybrid synchronization in linearly coupled fractional-order chaotic systems. This new synchronization mechanism, i.e., part of state variables are anti- phase synchronized and part completely synchronized, can be achieved using a single linear controller with only one drive variable. Based on the stability theory of the fractional-order system, we investigated the possible existence of this new synchronization mechanism. Moreover, a helpful theorem, serving as a determinant for the gain of the controller, is also presented. Solutions of coupled systems are obtained numerically by an improved Adams Bashforth-Moulton algorithm. To support our theoretical analysis, simulation results are given.

Circuit implementation of a new hyperchaos in fractional-order system

This paper introduces a new four-dimensional （4D） hyperchaotic system, which has only two quadratic nonlinearity parameters but with a complex topological structure. Some complicated dynamical properties are then investigated in detail by using bifurcations, Poincare mapping, LE spectra. Furthermore, a simple fourth-order electronic circuit is designed for hardware implementation of the 4D hyperchaotic attractors. In particular, a remarkable fractional-order circuit diagram is designed for physically verifying the hyperchaotic attractors existing not only in the integer-order system but also in the fractional-order system with an order as low as 3.6.

Chaotic synchronization for a class of fractional-order chaotic systems

In this paper, a very simple synchronization method is presented for a class of fractional-order chaotic systems only via feedback control. The synchronization technique, based on the stability theory of fractional-order systems, is simple and theoretically rigorous.

Fractional-order permanent magnet synchronous motor and its adaptive chaotic control

In this paper we investigate the chaotic behaviors of the fractional-order permanent magnet synchronous motor（PMSM）.The necessary condition for the existence of chaos in the fractional-order PMSM is deduced.And an adaptivefeedback controller is developed based on the stability theory for fractional systems.The presented control scheme,which contains only one single state variable,is simple and flexible,and it is suitable both for design and for implementation in practice.Simulation is carried out to verify that the obtained scheme is efficient and robust against external interference for controlling the fractional-order PMSM system.

Nonlinear feedback synchronisation control between fractional-order and integer-order chaotic systems

This paper focuses on the synchronisation between fractional-order and integer-order chaotic systems. Based on Lyapunov stability theory and numerical differentiation, a nonlinear feedback controller is obtained to achieve the synchronisation between fractional-order and integer-order chaotic systems. Numerical simulation results are presented to illustrate the effectiveness of this method.

Artificial Bee Colony Algorithm-based Parameter Estimation of Fractional-order Chaotic System with Time Delay

It is an important issue to estimate parameters of fractional-order chaotic systems in nonlinear science, which has received increasing interest in recent years. In this paper, time delay and fractional order as well as system’s parameters are concerned by treating the time delay and fractional order as additional parameters. The parameter estimation is converted into a multi-dimensional optimization problem. A new scheme based on artificial bee colony(ABC) algorithm is proposed to solve the optimization problem. Numerical experiments are performed on two typical time-delay fractional-order chaotic systems to verify the effectiveness of the proposed method.

Robust modified projective synchronization of fractional-order chaotic systems with parameters perturbation and external disturbance

Based on fractional-order Lyapunov stability theory, this paper provides a novel method to achieve robust modified projective synchronization of two uncertain fractional-order chaotic systems with external disturbance. Simulation of the fractional-order Lorenz chaotic system and fractional-order Chen＇s chaotic system with both parameters uncertainty and external disturbance show the applicability and the efficiency of the proposed scheme.

Function projective synchronization between fractional-order chaotic systems and integer-order chaotic systems

This paper investigates the function projective synchronization between fractional-order chaotic systems and integer-order chaotic systems using the stability theory of fractional-order systems. The function projective synchronization between three-dimensional （3D） integer-order Lorenz chaotic system and 3D fractional-order Chen chaotic system are presented to demonstrate the effectiveness of the proposed scheme.

A general method for synchronizing an integer-order chaotic system and a fractional-order chaotic system

This paper investigates the synchronization between integer-order and fractional-order chaotic systems. By intro- ducing fractional-order operators into the controllers, the addressed problem is transformed into a synchronization one among integer-order systems. A novel general method is presented in the paper with rigorous proof. Based on this method, effective controllers are designed for the synchronization between Lorenz systems with an integer order and a fractional order, and for the synchronization between an integer-order Chen system and a fractional-order Liu system. Numerical results, which agree well with the theoretical analyses, are also given to show the effectiveness of this method.

Synchronization of uncertain fractional-order chaotic systems with disturbance based on a fractional terminal sliding mode controller

This paper provides a novel method to synchronize uncertain fractional-order chaotic systems with external disturbance via fractional terminal sliding mode control. Based on Lyapunov stability theory, a new fractional-order switching manifold is proposed, and in order to ensure the occurrence of sliding motion in finite time, a corresponding sliding mode control law is designed. The proposed control scheme is applied to synchronize the fractional-order Lorenz chaotic system and fractional-order Chen chaotic system with uncertainty and external disturbance parameters. The simulation results show the applicability and efficiency of the proposed scheme.

Robust sliding mode control for fractional-order chaotic economical system with parameter uncertainty and external disturbance

This paper presents a modified sliding mode control for fractional-order chaotic economical systems with parameter uncertainty and external disturbance. By constructing the suitable sliding mode surface with fractional-order integral, the effective sliding mode controller is designed to realize the asymptotical stability of fractional-order chaotic economical systems. Comparing with the existing results, the main results in this paper are more practical and rigorous. Simulation results show the effectiveness and feasibility of the proposed sliding mode control method.

No-chattering sliding mode control in a class of fractional-order chaotic systems

A no-chattering sliding mode control strategy for a class of fractional-order chaotic systems is proposed in this paper. First, the sliding mode control law is derived to stabilize the states of the commensurate fractional-order chaotic system and the non-commensurate fractional-order chaotic system, respectively. The designed control scheme guarantees the asymptotical stability of an uncertain fractional-order chaotic system. Simulation results are given for several fractional-order chaotic examples to illustrate the effectiveness of the proposed scheme.

A novel study on the impulsive synchronization of fractional-order chaotic systems

The stability of impulsive fractional-order systems is discussed. A new synchronization criterion of fractional-order chaotic systems is proposed based on the stability theory of impulsive fractional-order systems. The synchronization criterion is suitable for the case of the order 0 〈 q ≤ 1. It is more general than those of the known results. Simulation results are given to show the effectiveness of the proposed synchronization criterion.

Fractional-Order Control for a Novel Chaotic System Without Equilibrium

The control problem is discussed for a chaotic system without equilibrium in this paper.On the basis of the linear mathematical model of the two-wheeled self-balancing robot,a novel chaotic system which has no equilibrium is proposed.The basic dynamical properties of this new system are studied via Lyapunov exponents and Poincar′e map.To further demonstrate the physical realizability of the presented novel chaotic system,a chaotic circuit is designed.By using fractional-order operators,a controller is designed based on the state-feedback method.According to the Gronwall inequality,Laplace transform and Mittag-Leffler function,a new control scheme is explored for the whole closed-loop system.Under the developed control scheme,the state variables of the closed-loop system are controlled to stabilize them to zero.Finally,the numerical simulation results of the chaotic system with equilibrium and without equilibrium illustrate the effectiveness of the proposed control scheme.

Comparison between two different sliding mode controllers for a fractional-order unified chaotic system

Two different sliding mode controllers for a fractional order unified chaotic system are presented. The controller for an integer-order unified chaotic system is substituted directly into the fractional-order counterpart system, and the fractional-order system can be made asymptotically stable by this controller. By proving the existence of a sliding manifold containing fractional integral, the controller for a fractional-order system is obtained, which can stabilize it. A comparison between these different methods shows that the performance of a sliding mode controller with a fractional integral is more robust than the other for controlling a fractional order unified chaotic system.

Circuit realization of the fractional-order unified chaotic system

This paper studies the chaotic behaviours of the fractional-order unified chaotic system. Based on the approximation method in frequency domain, it proposes an electronic circuit model of tree shape to realize the fractional-order operator. According to the tree shape model, an electronic circuit is designed to realize the 2.7-order unified chaotic system. Numerical simulations and circuit experiments have verified the existence of chaos in the fraction-order unified system.

Prescribed performance synchronization for fractional-order chaotic systems

In this paper the synchronization for two different fractional-order chaotic systems, capable of guaranteeing synchronization error with prescribed performance, is investigated by means of the fractional-order control method. By prescribed performance synchronization we mean that the synchronization error converges to zero asymptotically, with convergence rate being no less than a certain prescribed function. A fractional-order synchronization controller and an adaptive fractional-order synchronization controller, which can guarantee the prescribed performance of the synchronization error,are proposed for fractional-order chaotic systems with and without disturbances, respectively. Finally, our simulation studies verify and clarify the proposed method.

Image encryption based on a delayed fractional-order chaotic logistic system

A new image encryption scheme is proposed based on a delayed fractional-order chaotic logistic system.In the process of generating a key stream,the time-varying delay and fractional derivative are embedded in the proposed scheme to improve the security.Such a scheme is described in detail with security analyses including correlation analysis,information entropy analysis,run statistic analysis,mean-variance gray value analysis,and key sensitivity analysis.Experimental results show that the newly proposed image encryption scheme possesses high security.

One new fractional-order chaos system and its circuit simulation by electronic workbench

This paper proposes a new chaotic system and its fractional-order chaotic system. The necessary condition for the existence of chaotic attractors in this new fractional-order system is obtained. It finds that this new fractional-order system is chaotic for q 〉 0.783 if the system parameter m=6. The chaotic attractors for q=0.8, and q=0.9 are obtained. A circuit is designed to realize its fractional-order chaos system for q=0.9 by electronic workbench.