Abstract
Chaos synchronization of different fractional order time-delay chaotic systems is considered. Based on the Laplace transform theory, the conditions for achieving synchronization of different fractional order time-delay chaotic systems are analyzed by use of active control technique. Then numerical simulations are provided to verify the effectiveness and feasibility of the developed method. At last, effects of the fraction order and the time delay on synchronization are further researched.
1. Introduction
Since the pioneering work of Pecora and Carroll [1], there has been a significant interest in developing powerful techniques for chaos synchronization in the past 20 years. Some different regimes of chaos synchronization have been studied. They contain partial synchronization [2], complete synchronization [3, 4], phase synchronization [5, 6], generalized synchronization [7, 8], projective synchronization [9], and lag synchronization [10]. Of course, Robust synchronization is an important aspect of chaos synchronization. Based on the Lyapunov stability theory and LMI technique, a new sufficient criterion is established for chaos robust synchronization [11]. By use of the sliding mode control technique, a robust control scheme is established even when the parameter uncertainty and external perturbation are present [12]. A robust antisynchronization scheme is proposed according to multiple-kernel least squares support vector machine modeling for two uncertain chaotic systems [13]. Many potential applications of chaos synchronization, especially in secure communications of analog and digital signals and for developing safe and reliable cryptographic systems, chemical oscillators, and electronic systems, have been researched.
It is that delayed differential equations have been largely investigated in [14, 15] and references cited therein. Time delays and time-varying delays are recently introduced to chaotic systems; for example, see [16–19]. However, most of these publications are for integer-order or typical differential equations. Although fractional calculus is a 300-year-old mathematical topic, for many years it was not used in physics and engineering. During the last 10 years or so, fractional calculus starts to become focus of attention of physicists and engineers [20, 21]. There are many known systems to display fractional-order dynamics, such as dielectric polarization, electromagnetic waves, and quantum evolution of complex systems. Thus it has been a new trend, that is, the fractional power of the control system dynamics [22–35].
According to the Poincare-Bendixon theorem [22], an integer order chaotic nonlinear system must have a minimum order of 3 for chaos to appear. However, it is not the case in the fractional order nonlinear systems. Some examples in the respect are Wu et al. [23] (unified system), Lu and Chen [24] (Chen system), and Lu [25] (Ikeda system). By using Lambert function, the analytical stability bound is obtained for delayed second-order systems with repeatable poles and the bound is obtained delayed linear time-invariant fractional-order dynamic systems [26, 27]. Based on the Laplace transform theory, synchronization scheme to chaotic fractional-order Chen systems is derived in [28]; Deng et al. [29] introduced a characteristic equation for the n-dimensional linear fractional differential system with multiple time delays. In line with the stability theorem of linear fractional systems, a necessary condition is given to check the chaos synchronization of fractional systems with incommensurate order [30]. In [31], Shao et al. proposed a method to achieve projective synchronization of the fractional order chaotic Rossler system. In [32], an analytical justification was proposed for phase synchronization of fractional differential equations. A stability test procedure was proposed for linear nonhomogeneous fractional order systems with a pure time delay [33]. Taghvafard and Erjaee [34] studied the phase and antiphase synchronization between two identical and nonidentical fractional order chaotic systems using techniques from active control theory. The effect of delay on the chaotic behaviour has been investigated for the first time in the literature [35].
However, there are few results about chaos synchronization of the fractional order time-delay chaotic systems. In our work, we give an improved version of Adams-Bashforth-Moulton method. By use of active control technique, the conditions for achieving synchronization of different fractional order time-delay chaotic systems are analyzed based on the Laplace transform theory. Then effects of the fraction order and the time delay on synchronization are further analyzed.
The organization of this paper is as follows. In Section 2, some definitions and systems are given and then an improved version of Adams-Bashforth-Moulton method is introduced. In the following three sections, we synchronize the following fractional systems using active control method: (i) Liu with Liu, (ii) Lü with Chen, and (iii) Lorenz with Chen. The effectiveness of our work is verified through numerical simulations. In Section 6, effects of the fraction order and the time delay on synchronization are further researched. Finally concluding remark is given.
2. Definitions and Systems
2.1. Fractional Calculus
There are several definitions of a fractional-order differential system. In the following, we introduce the most common one of them: where ; that is, is the first integer which is not less than , is the -order derivative in the usual sense, and is the -order Reimann-Liouville integral operator with expression: Here stands for Gamma function, and the operator is generally called “-order Caputo differential operator” [33].
2.2. System Description
The fractional order Lorenz system has recently been studied in many references [36, 37]. Here we give a fractional order time-delay Lorenz system. The system is described by where is the Prandtl number, is the Rayleigh number over the critical Rayleigh number, gives the size of the region approximated by the system, and is time delay. System (3) displays a chaotic attractor, as shown in Figure 1.
In 1999, Chen and Ueta [38] introduced a new chaotic system, which is similar but not topologically equivalent to the Lorenz system. Without changing the model structure of the system, we consider the fractional order time-delay Chen system in the following form: where , , and are real parameters. When = 35, = 3, and = 28, system (4) displays a chaotic attractor through the suitable selection of time delay, as shown in Figure 2. The fractional order time-delay Lü system is given by where , , and . As shown in Figure 3, system (5) displays a chaotic attractor through the suitable selection of time delay.
In 2009, Liu et al. proposed a novel three-dimensional autonomous chaos system (called Liu system) [39]. Daftardar-Gejji and Bhalekar [40] studied a fractional version of the chaotic system. Here we investigate a fractional order time-delay Liu system as follows: where ,,,, = 4, and = 4. The system can be in the chaotic state as shown in Figure 4.
2.3. Numerical Method
According to, [41–43] the predictor-corrector scheme for system (4) is derived. Here we give an improved version of Adams-Bashforth-Moulton method [43]. Based on the fractional order time-delay Chen system, let us illustrate this scheme.
The following differential equation: is equivalent to the Volterra integral equation [42] Let ,, and. Then (8) can be discretized as follows: where The error estimate is where .
Applying the above method, (4) can be discretized as follows: where
3. Chaos Synchronization between Fractional Order Time-Delay Liu and Chen System
In this section we study the synchronization between Liu and Chen systems. Assuming that the Liu system drives the Chen system, we define the drive (master) and response (slave) systems as follows:
The unknown terms , , and in (14) are active control functions to be determined. Define the error functions as Equation (15) together with (13) and (14) yields the error system
By using Matignon’s theorem [44], so we here define active control function as Substituting (17) into (16), we achieve
Then we start to prove that the error system (18) is asymptotically stable; that is, the system (13) and the system (14) with the active control law (17) are synchronized.
Taking the Laplace transformation in two sides of (18), letting , and utilizing (see [21]), we obtain Equation (19) can be rewritten as follows: By use of the final-value theorem of the Laplace transformation, we have At the same time, from (20), we can get The above analysis implies that the fractional-order drive system (13) and the fractional-order response system (14) with the active control law (17) are synchronized.
3.1. Simulation Research
Here, let , , , , , in the Liu system and parameters of the Chen system are taken as , , . The fractional order is taken to be 0.985 and assume . The initial conditions for drive and response system are , , and , , , respectively. Thus initial conditions for the error system are , , . Figures 5(a)–5(c) shows the synchronization between Chen and Liu system. The errors for the drive and response system are shown in Figure 5(d).
(a)
(b)
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4. Chaos Synchronization between Fractional Order Time-Delay Lü and Chen System
In this section we consider Lü system as the drive systems and the Chen system as the response system
The unknown terms , , in (24) are active control functions to be determined. Define the error functions as Equation (25) together with (23) and (24) yields the error system
By using of the Matignon’s theorem [44], so we here define active control function as Substituting (27) into (26), we achieve
Then we start to prove that the error system (28) is asymptotically stable, that is, the system (23) and the system (24) with the active control law (27) are synchronized.
Taking the Laplace transformation in two sides of (28), letting , and utilizing (see [18]), we obtain Equation (29) can be rewritten as follows: By use of the final-value theorem of the Laplace transformation, we have At the same time, from (30), we can get The above analysis implies that the fractional-order drive system (23) and the fractional-order response system (24) with the active control law (27) are synchronized.
4.1. Simulation Research
Here, let , , and in the Lü system and parameters of the Chen system are the same as in Section 3.1. The fractional order is taken to be 0.78 and assume. The initial conditions for drive and response system are , , , and , , , respectively. Thus initial conditions for the error system (28) are , , and . Figures 6(a)–6(c) show the synchronization between Chen and Lü system. The errors for the drive and response system are shown in Figure 6(d).
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(b)
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5. Chaos Synchronization between Fractional Order Time-Delay Lorenz and Chen System
Assuming that Chen system is synchronized with Lorenz system, define the drive system as and the Chen system as the response system
Let , , and be error functions. For synchronization it is essential that the errors as . Then we can get
By using Matignon’s theorem [44], so we here define active control function as Substituting (36) into (35), we achieve Then we start to prove that the error system (37) is asymptotically stable; that is, the system (33) and the system (34) with the active control law (36) are synchronized.
Taking the Laplace transformation in two sides of (37), letting , and utilizing (see [18]), we obtain Equation (38) can be rewritten as follows: By use of the final-value theorem of the Laplace transformation, we have At the same time, from (39), we can get So the error system (37) is asymptotically stable and chaos synchronization between the two systems is achieved.
5.1. Simulation Research
Here, we take parameters for fractional order time-delay Lorenz system as , , and and parameters of the Chen system are the same as in Section 3.1. The fractional order is taken to be 0.95 and assume . The initial conditions for the two systems are , , , and , , and , respectively. Hence the initial conditions for the error system (37) are , , and . Figures 7(a)–7(c) show the synchronization between Chen and Lorenz system. The errors for the drive and response system are shown in Figure 7(d).
(a)
(b)
(c)
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6. Effects of Order and Time Delay on Synchronization
Recently, fractional order chaotic systems have been focus of attention. Generally speaking, there is a lowest order in a fractional order chaotic system. From the literature [23–25], we know that the order affects the behavior of fractional chaotic dynamical systems. As shown in [35, 45], the time delay also affects the behavior of chaotic dynamical systems. In the section, we pay attention to the effects of the order and the delay on synchronization. In Table 1, some results are given about the error functions for different values of order in system (13) and system (14). Here experiments are done for fixed value of the delay . It is obvious that the error in synchronization firstly increases and then decreases as the order is decreased, which is not the case in [46]. At the same time, we have summarized some results about the error functions for different values of the time delay in system (23) and system (24). In these observations, we take the order . From Table 2, it is clear that the error functions and become bigger as the delay is increased. But the error function firstly decreases and then increases.
7. Conclusions
In the paper, chaos synchronization of different fractional order time-delay chaotic systems is considered. Based on the Laplace transform theory, the conditions for achieving synchronization of different fractional order time-delay chaotic systems are analyzed by use of active control technique. Then numerical simulations are provided to verify the effectiveness and feasibility of the developed method. Finally, effects of the fraction order and the time delay on synchronization are further discussed.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the Natural Science Foundation of China (Grant nos. 60872073, 51075068, and 60975017), by the Foundation of Huaqiao University (Grant nos. 12BS228 and 13BS103), and by the Natural Science Foundation of Fujian Province (Grant no. 2013H2002). The authors would like to thank the reviewers for their valuable suggestions and comments.