Chaos,feedback control and synchronization of a fractional-order modified Autonomous Van der Pol–Duffing circuit

In this work, stability analysis of the fractional-order modified Autonomous Van der Pol–Duffing (MAVPD) circuit is studied using the fractional Routh–Hurwitz criteria. A necessary condition for this system to remain chaotic is obtained. It is found that chaos exists in this system with order less than 3. Furthermore, 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 controllers, it is shown that the fractional-order MAVPD system is controlled to its equilibrium points; however, its integer-order counterpart is not controlled. Moreover, chaos synchronization of MAVPD system is found only in the fractional-order case when using a specific choice of nonlinear control functions. This shows the effect of fractional order on chaos control and synchronization. Synchronization is also achieved using the unidirectional linear error feedback coupling approach. Numerical results show the effectiveness of the theoretical analysis.     

On chaos control and synchronization of the commensurate fractional order Liu system

In this work, we study chaos control and synchronization of the commensurate fractional order Liu system. Based on the stability theory of fractional order systems, the conditions of local stability of nonlinear three-dimensional commensurate fractional order systems are discussed. The existence and uniqueness of solutions for a class of commensurate fractional order Liu systems are investigated. We also obtain the necessary condition for the existence of chaotic attractors in the commensurate fractional order Liu system. The effect of fractional order on chaos control of this system is revealed by showing that the commensurate fractional order Liu system is controllable just in the fractional order case when using a specific choice of controllers. Moreover, we achieve chaos synchronization between the commensurate fractional order Liu system and its integer order counterpart via function projective synchronization. Numerical simulations are used to verify the analytical results.     

Modified generalized projective synchronization of a new fractional-order hyperchaotic system and its application to secure communication

This paper presents a new fractional-order hyperchaotic system. The chaotic behaviors of this system in phase portraits are analyzed by the fractional calculus theory and computer simulations. Numerical results have revealed that hyperchaos does exist in the new fractional-order four-dimensional system with order less than 4 and the lowest order to have hyperchaos in this system is 3.664. The existence of two positive Lyapunov exponents further verifies our results. Furthermore, a novel modified generalized projective synchronization (MGPS) for the fractional-order chaotic systems is proposed based on the stability theory of the fractional-order system, where the states of the drive and response systems are asymptotically synchronized up to a desired scaling matrix. The unpredictability of the scaling factors in projective synchronization can additionally enhance the security of communication. Thus MGPS of the new fractional-order hyperchaotic system is applied to secure communication. Computer simulations are done to verify the proposed methods and the numerical results show that the obtained theoretic results are feasible and efficient.     

Dynamic behaviors of a fractional order two-species cooperative systems with harvesting

This paper introduces a fractional order two-species cooperative systems with harvesting. By using the Routh-Hurwitz Conditions and the Lyapunov method, we provide several sufficient conditions to ensure the stability of the equilibriums for the system. Finally, a numerical example is presented to demonstrate the validity and feasibility of the theoretical result.     

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.     

Robust stabilization and synchronization of a class of fractional-order chaotic systems via a novel fractional sliding mode controller

This paper proposes a novel fractional-order sliding mode approach for stabilization and synchronization of a class of fractional-order chaotic systems. Based on the fractional calculus a stable integral type fractional-order sliding surface is introduced. Using the fractional Lyapunov stability theorem, a single sliding mode control law is proposed to ensure the existence of the sliding motion in finite time. The proposed control scheme is applied to stabilize/synchronize a class of fractional-order chaotic systems in the presence of model uncertainties and external disturbances. Some numerical simulations are performed to confirm the theoretical results of the paper. It is worth noticing that the proposed fractional-order sliding mode controller can be applied to control a broad range of fractional-order dynamical systems.     

Control and adaptive modified function projective synchronization of Liu chaotic dynamical system

In this work, the feedback control method is proposed to control the behaviour of Liu chaotic dynamical system. The controlled system is stable under some conditions on the parameters of the system determined by Routh-Hurwitz criterion. This paper also presents the adaptive modified function projective synchronization (AMFPS) between two identical Liu chaotic dynamical systems. Based on the Lyapunov stability theorem, adaptive control laws are designed to achieving the AMFPS. Finally, some numerical simulations are obtained to validate the proposed methods.     

Chaos control and function projective synchronization of fractional-order systems through the backstepping method

We study the chaos control and the function projective synchronization of a fractional-order T-system and Lorenz chaotic system using the backstepping method. Based on stability theory, we consider the condition for the local stability of nonlinear three-dimensional commensurate fractional-order system. Using the feedback control method, we control the chaos in the considered fractional-order T-system. We simulate the function projective synchronization between the fractional-order T-system and Lorenz system numerically using MATLAB and depict the results with plots.     

Projective synchronization of different fractional-order chaotic systems with non-identical orders

This paper investigates the projective synchronization (PS) of different fractional order chaotic systems while the derivative orders of the states in drive and response systems are unequal. Based on some essential properties on fractional calculus and the stability theorems of fractional-order systems, we propose a general method to achieve the PS in such cases. The fractional operators are introduced into the controller to transform the problem into synchronization problem between chaotic systems with identical orders, and the nonlinear feedback controller is proposed based on the concept of active control technique. The method is both theoretically rigorous and practically feasible. We present two examples that illustrate the effectiveness and applications of the method, which include the PS between two 3-D commensurate fractional-order chaotic systems and the PS between two 4-D fractional-order hyperchaotic systems with incommensurate and commensurate orders, respectively. Abundant numerical simulations are given which agree well with the analytical results. Our investigations show that PS can also be achieved between different chaotic systems with non-identical orders. We have further reviewed and compared some relevant methods on this topic reported in several recent papers. A discussion on the physical implementation of the proposed method is also presented in this paper.     

具有领导者的非线性分数阶多智能体系统的一致性分析

研究了利用非线性分数阶模型描述的具有领导者的多智能体系统的一致性问题.基于智能体之间的通讯拓扑图,设计了系统的控制协议和相应的控制增益矩阵.利用广义Gronwall不等式和分数阶微分方程的稳定性理论,得到了多智能体系统达到一致的充分条件.最后,数值仿真结果显示了理论结果的有效性.     

Parameter estimation and topology identification of uncertain fractional order complex networks

This paper focuses on a significant issue in the research of fractional order complex network, i.e., the identification problem of unknown system parameters and network topologies in uncertain complex networks with fractional-order node dynamics. Based on the stability analysis of fractional order systems and the adaptive control method, we propose a novel and general approach to address this challenge. The theoretical results in this paper have generalized the synchronization-based identification method that has been reported in several literatures on identifying integer order complex networks. We further derive the sufficient condition that ensures successful network identification. An uncertain complex network with four fractional-order Lorenz systems is employed to verify the effectiveness of the proposed approach. The numerical results show that this approach is applicable for online monitoring of the static or changing network topology. In addition, we present a discussion to explore which factor would influence the identification process. Certain interesting conclusions from the discussion are obtained, which reveal that large coupling strengths and small fractional orders are both harmful for a successful identification.     

Numerical simulation of the fractional-order control system

Multi-term fractional differential equations have been used to simulate fractional-order control system. It has been demonstrated the necessity of the such controllers for the more efficient control of fractionalorder dynamical system. In this paper, the multi-term fractional ordinary differential equations are transferred into equivalent a system of equations. The existence and uniqueness of the new system are proved. A fractional order difference approximation is constructed by a decoupled technique and fractional-order numerical techniques. The consistence, convergence and stability of the numerical approximation are proved. Finally, some numerical results are presented to demonstrate that the numerical approximation is a computationally efficient method. The new method can be applied to solve the fractional-order control system.     

Stability analysis of fractional-order complex-valued neural networks with time delays

In this paper, we consider the problem of stability analysis of fractional-order complex-valued Hopfield neural networks with time delays, which have been extensively investigated. Moreover, the fractional-order complex-valued Hopfield neural networks with hub structure and time delays are studied, and two types of fractional-order complex-valued Hopfield neural networks with different ring structures and time delays are also discussed. Some sufficient conditions are derived by using stability theorem of linear fractional-order systems to ensure the stability of the considered systems with hub structure. In addition, some sufficient conditions for the stability of considered systems with different ring structures are also obtained. Finally, three numerical examples are given to illustrate the effectiveness of our theoretical results.     

Projective synchronization between different fractional‐order hyperchaotic systems with uncertain parameters using proposed modified adaptive projective synchronization technique

In the present article, the authors have proposed a modified projective adaptive synchronization technique for fractional‐order chaotic systems. The adaptive projective synchronization controller and identification parameters law are developed on the basis of Lyapunov direct stability theory. The proposed method is successfully applied for the projective synchronization between fractional‐order hyperchaotic Lü system as drive system and fractional‐order hyperchaotic Lorenz chaotic system as response system. A comparison between the effects on synchronization time due to the presence of fractional‐order time derivatives for modified projective synchronization method and proposed modified adaptive projective synchronization technique is the key feature of the present article. Numerical simulation results, which are carried out using Adams–Boshforth–Moulton method show that the proposed technique is effective, convenient and also faster for projective synchronization of fractional‐order nonlinear dynamical systems. Copyright © 2013 John Wiley & Sons, Ltd.     

一类分数阶神经网络模型的稳定性与Hopf分支分析

分析了一类分数阶神经网络的稳定性与Hopf分支问题.基于分数阶稳定性判据,得到了分数阶神经网络模型局部渐近稳定的条件.并以q为分支参数,得到了分数阶系统产生Hopf的条件.最后数值仿真证明了我们的结论.     

Synchronization of a chaotic finance system

Synchronization strategies of a three-dimensional chaotic finance system are investigated in this paper. Based on Lyapunov stability theory and Routh-Hurwitz criteria, some effective controllers are designed for the global asymptotic synchronization on different conditions. When the system parameters are known, the hybrid feedback control and a method based on special matrix structure are adopted respectively, to realize the synchronization of the chaotic finance system. When the parameters are unknown, the active control is extended and introduced to realize the synchronization. Numerical simulations show the validity and feasibility of the synchronization schemes.     

Synchronization of N-coupled incommensurate fractional-order chaotic systems with ring connection

In this paper, based on the stability theorem of linear fractional systems, a necessary condition is given to synchronize N-coupled incommensurate fractional-order chaotic systems with ring connection. Chaos synchronization is studied by utilizing the coupling method that includes unidirectional and bidirectional coupling. The main contribution of this work is to design the coupling parameters in which it is shown that our proposed controller does stabilize error dynamics around all of equilibriums of master system. Finally, the claims are justified through a set of simulations and results coincide with the theoretical analysis.     

Achieving synchronization between the fractional-order hyperchaotic Novel and Chen systems via a new nonlinear control technique

In this work, we discuss the stability conditions for a nonlinear fractional-order hyperchaotic system. The fractional-order hyperchaotic Novel and Chen systems are introduced. The existence and uniqueness of solutions for two classes of fractional-order hyperchaotic Novel and Chen systems are investigated. On the basis of the stability conditions for nonlinear fractional-order hyperchaotic systems, we study synchronization between the proposed systems by using a new nonlinear control technique. The states of the fractional-order hyperchaotic Novel system are used to control the states of the fractional-order hyperchaotic Chen system. Numerical simulations are used to show the effectiveness of the proposed synchronization scheme.     

Fractional terminal sliding mode control design for a class of dynamical systems with uncertainty

A novel type of control strategy combining the fractional calculus with terminal sliding mode control called fractional terminal sliding mode control is introduced for a class of dynamical systems subject to uncertainties. A fractional-order switching manifold is proposed and the corresponding control law is formulated based on the Lyapunov stability theory to guarantee the sliding condition. The proposed fractional-order terminal sliding mode controller ensures the finite time stability of the closed-loop system. Finally, numerical simulation results are presented and compared to illustrate the effectiveness of the proposed method.