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Chaos in fractional-order generalized Lorenz system and its synchronization circuit simulation

Zhang Ruo-Xun,Yang Shi-Ping,

中国物理 B , 2009,
Abstract: The chaotic behaviours of a fractional-order generalized Lorenz system and its synchronization are studied in this paper. A new electronic circuit unit to realize fractional-order operator is proposed. According to the circuit unit, an electronic circuit is designed to realize a 3.8-order generalized Lorenz chaotic system. Furthermore, synchronization between two fractional-order systems is achieved by utilizing a single-variable feedback method. Circuit experiment simulation results verify the effectiveness of the proposed scheme.
Chaos Synchronization between Two Different Fractional Systems of Lorenz Family
A. E. Matouk
Mathematical Problems in Engineering , 2009, DOI: 10.1155/2009/572724
Abstract: This work investigates chaos synchronization between two different fractional order chaotic systems of Lorenz family. The fractional order Lü system is controlled to be the fractional order Chen system, and the fractional order Chen system is controlled to be the fractional order Lorenz-like system. The analytical conditions for the synchronization of these pairs of different fractional order chaotic systems are derived by utilizing Laplace transform. Numerical simulations are used to verify the theoretical analysis using different values of the fractional order parameter.
Chaos Control and Synchronization in Fractional-Order Lorenz-Like System  [PDF]
Sachin Bhalekar
International Journal of Differential Equations , 2012, DOI: 10.1155/2012/623234
Abstract: The present paper deals with fractional-order version of a dynamical system introduced by Chongxin et al. (2006). The chaotic behavior of the system is studied using analytic and numerical methods. The minimum effective dimension is identified for chaos to exist. The chaos in the proposed system is controlled using simple linear feedback controller. We design a controller to place the eigenvalues of the system Jacobian in a stable region. The effectiveness of the controller in eliminating the chaotic behavior from the state trajectories is also demonstrated using numerical simulations. Furthermore, we synchronize the system using nonlinear feedback. 1. Introduction A variety of problems in engineering and natural sciences are modeled using chaotic dynamical systems.A chaotic system is a nonlinear deterministic system possessing complex dynamical behaviors such as being extremely sensitive to tiny variations of initial conditions, unpredictability, and having bounded trajectories in the phase space [1]. Controlling the chaotic behavior in the dynamical systems using some form of control mechanism has recently been the focus of much attention. So many approaches are proposed for chaos control namely, OGY method [2], backstepping design method [3], differential geometric method [4], inverse optimal control [5], sampled-data feedback control [6], adaptive control [7], and so on. One simple approach is the linear feedback control [8]. Linear feedback controllers are easy to implement, they can perform the job automatically, and stabilize the overall control system efficiently [9]. The controllers can also be used to synchronize two identical or distinct chaotic systems [10–13]. Synchronization of chaos refers to a process wherein two chaotic systems adjust a given property of their motion to a common behavior due to a coupling. Synchronization has many applications in secure communications of analog and digital signals [14] and for developing safe and reliable cryptographic systems [15]. Fractional calculus deals with derivatives and integration of arbitrary order [16–18] and has deep and natural connections with many fields of applied mathematics, engineering, and physics. Fractional calculus has wide range of applications in control theory [19], viscoelasticity [20], diffusion [21–25], turbulence, electromagnetism, signal processing [26, 27], and bioengineering [28]. Study of chaos in fractional order dynamical systems and related phenomena is receiving growing attention [29, 30]. I. Grigorenko and E. Grigorenko investigated fractional ordered Lorenz
Function projective synchronization between fractional-order chaotic systems and integer-order chaotic systems

Zhou Ping,Cao Yu-Xia,

中国物理 B , 2010,
Abstract: 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.
Synchronization of Coupled Nonidentical Fractional-Order Hyperchaotic Systems  [PDF]
Zhouchao Wei
Discrete Dynamics in Nature and Society , 2011, DOI: 10.1155/2011/430724
Abstract: Synchronization of coupled nonidentical fractional-order hyperchaotic systems is addressed by the active sliding mode method. By designing an active sliding mode controller and choosing proper control parameters, the master and slave systems are synchronized. Furthermore, synchronizing fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system is performed to show the effectiveness of the proposed controller. 1. Introduction Fractional calculus has been known since the development of the regular calculus, with the first reference probably being associated with Leibniz and L’H?spital in 1695. Although it is a mathematical topic with more than 300 years old history, the applications of fractional calculus to physics and engineering are just a recent focus of interest [1, 2]. Nowadays, by utilizing fractional calculus technique, many investigations were devoted to the chaotic and hyperchaotic behaviors of fractional-order systems, such as fractional-order Chua circuit [3], fractional-order Lorenz system [4], fractional-order R?ssler system [5], fractional-order Chen system [6], and fractional-order conjugate Lorenz system [7]. Over the last two decades, synchronization of chaotic systems has become more and more interesting to researchers in different fields. Since the synchronization of fractional-order chaotic systems was firstly investigated in [8], it has recently attracted increasing attention due to its potential applications in secure communication and control processing [9–13]. Moreover, many theoretical analysis and numerical simulation results about the synchronization of fractional-order chaotic systems are obtained [14–19]. Such synchronization may be safer than those of the classical chaotic systems in secure communications. This can be seen from two aspects: (i) the order of fractional derivatives can be regarded as a parameter and (ii) the fractional derivatives are nonlocal thus more complicated than the regular derivatives. This paper focuses on synchronization of coupled fractional-order hyperchaotic nonidentical systems. The active sliding mode synchronization method is chosen to achieve this goal. The active sliding mode synchronization technique is a discontinuous control strategy, which relies on two stages of design. The first stage is to select an appropriate active controller to facilitate the design of the sequent sliding mode controller. The second stage is to design a sliding mode controller to achieve the synchronization. This process is verified when active sliding mode synchronization method is
Synchronization of fractional-order unified chaotic system via linear control

Zhang Ruo-Xun,Yang Shi-Ping,Liu Yong-Li,

物理学报 , 2010,
Abstract: Chaos synchronization in fractional-order unified chaotic system is disscussed in this paper. Based on the stability theory of fractional-order system, the control law is presented to achieve chaos synchronization. The advantage of the proposed controllers is that they are linear and have lower dimensions than that of the states. With this technique it is very easy to find the suitable feedback constant. Simulation results for fractional-order Lorenz, Lü and Chen chaotic systems are provided to illustrate the effectiveness of the proposed scheme.
Modified Function Projective Synchronization of Fractional Order Chaotic Systems with Different Dimensions  [PDF]
Hong-Juan Liu,Zhi-Liang Zhu,Hai Yu,Qian Zhu
Discrete Dynamics in Nature and Society , 2013, DOI: 10.1155/2013/763564
Abstract: The modified function projective synchronization of different dimensional fractional-order chaotic systems with known or unknown parameters is investigated in this paper. Based on the stability theorem of linear fractional-order systems, the adaptive controllers with corresponding parameter update laws for achieving the synchronization are given. The fractional-order chaotic system and hyperchaotic system are applied to achieve synchronization in both reduced order and increased order. The corresponding numerical results coincide with theoretical analysis. 1. Introduction Synchronization has attracted a great deal of interest due to its important applications in ecological systems [1], physical systems [2], chemical systems [3], modeling brain activity, system identification, pattern recognition phenomena and secure communications [4], and so forth. Since the pioneering work of Pecora and Carroll [5], various synchronization scenarios have been studied for chaotic systems, including complete synchronization [6], phase synchronization [7], lag synchronization [8], Q-S synchronization [9], and projective synchronization [10]. As a much more universal synchronization manner, the modified function projective synchronization (MFPS) means that the drive and response systems could be synchronized up to a scaling function matrix, but not a constant matrix. Obviously, the unpredictability of the scaling functions in MFPS can additionally enhance the security of communication [11–13]. In recent years, the study on the nonlinear dynamics and synchronization control of fractional-order chaotic systems has become a hot topic in nonlinear research area. It is demonstrated that many fractional-order differential systems behave chaotically or hyperchaotically, such as the fractional-order Chua circuit [14], the fractional-order Arneodo system [15], the fractional-order Chen system [16], the fractional-order hyperchaotic Lorenz system [17], the fractional-order hyperchaotic Lü system [18], and so forth. Studies show that a fractional-order controller can provide better performances than an integer order one and lead to more robust control performance [19]. According to different definitions of the fractional-order differential equation from the integer order differential equation, most of the methods and results of chaos synchronization in the ordinary differential systems cannot be simply extended to the case of the fractional-order systems. Some approaches have been proposed to achieve chaos synchronization in fractional-order chaotic systems, such as active control
Active Sliding Mode for Synchronization of a Wide Class of Four-Dimensional Fractional-Order Chaotic Systems  [PDF]
Bin Wang,Yuangui Zhou,Jianyi Xue,Delan Zhu
ISRN Applied Mathematics , 2014, DOI: 10.1155/2014/472371
Abstract: We focus on the synchronization of a wide class of four-dimensional (4-D) chaotic systems. Firstly, based on the stability theory in fractional-order calculus and sliding mode control, a new method is derived to make the synchronization of a wide class of fractional-order chaotic systems. Furthermore, the method guarantees the synchronization between an integer-order system and a fraction-order system and the synchronization between two fractional-order chaotic systems with different orders. Finally, three examples are presented to illustrate the effectiveness of the proposed scheme and simulation results are given to demonstrate the effectiveness of the proposed method. 1. Introduction Chaos synchronization is the concept of closeness of the frequencies between different periodic oscillations generated by two chaotic systems, one of which is the master and the other is the slave. Since the pioneering work of Pecora and Carroll [1] who proposed a method to synchronize two identical chaotic systems, chaos synchronization has attracted a lot of attention in a variety of research fields over the last two decades. This is because chaos synchronization can be used in many areas such as physics, engineering, and particularly in secure communication [2–5]. Many methods have been proposed to synchronize chaotic systems including active control [6], back-stepping control [7], linear feedback control [8], adaptive control theory [9], sliding mode control [10, 11], and fuzzy control [12]. For example, Bhalekar and Daftardar-Gejji [13] used active control for the problem of synchronization of fractional-order Liu system with fractional-order Lorenz system. Based on the idea of tracking control and stability theory of fractional-order systems, Zhou and Ding [14] designed a controller to synchronize the fractional-order Lorenz chaotic system via fractional-order derivative. Zhang and Yang [15] dealt with the lag synchronization of fractional-order chaotic systems with uncertain parameters. Projective synchronization of a class of fractional-order hyperchaotic system with uncertain parameters was studied by Bai et al. [16] as well, but the derivative orders of the state in response system was the same with drive system. Chen et al. [17] designed a sliding mode controller for a class of fractional-order chaotic systems. However, most of the above-mentioned work on chaos synchronization has focused on fractional-order chaos and integer-order systems, respectively. To the best of our knowledge, there has been little information available about the synchronization between
Chaos in fractional-order Liu system and a fractional-order unified system and the synchronization between them

Zhang Cheng-Fen,Gao Jin-Feng,Xu Lei,

物理学报 , 2007,
Abstract: This paper studies the chaotic behaviors of the fractional-order Liu system and the fractional-order unified system which were presented several years ago. It is found that chaos exists in the both systems with order less than three and the lowest order to have chaos is 0.3. The calculation results of the maximum Lyapunov exponents when the system is chaotic illustrate the existence of chaos. Chaos synchronization between fractional-order Liu system, fractional-order Lorenz system and fractional-order Lü sytem is realized by employing active control technique. Theoretieal analysis and numerical simulations demonstrate the effectiveness of the proposed method.
A general method for synchronizing an integer-order chaotic system and a fractional-order chaotic system

Si Gang-Quan,Sun Zhi-Yong,Zhang Yan-Bin,

中国物理 B , 2011,
Abstract: This paper investigates the synchronization between integer-order and fractional-order chaotic systems. By introducing 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.
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