Elsevier

ISA Transactions

Volume 64, September 2016, Pages 34-46
ISA Transactions

Function projective synchronization between integer-order and stochastic fractional-order nonlinear systems☆

https://doi.org/10.1016/j.isatra.2016.04.018Get rights and content

Highlights

  • •

    The stochastic processes are combined with the multidimensional fractional-order nonlinear systems.

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    The function projective synchronization between integer-order and stochastic fractional-order nonlinear systems is considered.

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    With the stability theory of fractional-order systems and tracking control, a controller is designed.

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    The method of transforming stochastic error system into an equivalent deterministic system is given.

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    The proposed method is investigated by integer-order Lorenz system and stochastic fractional-order Chen system.

Abstract

In this paper, the function projective synchronization between integer-order and stochastic fractional-order nonlinear systems is investigated. Firstly, according to the stability theory of fractional-order systems and tracking control, a controller is designed. At the same time, based on the orthogonal polynomial approximation, the method of transforming stochastic error system into an equivalent deterministic system is given. Thus, the stability of the stochastic error system can be analyzed through its equivalent deterministic one. Finally, to demonstrate the effectiveness of the proposed scheme, the function projective synchronization between integer-order Lorenz system and stochastic fractional-order Chen system is studied.

Introduction

Since the synchronization of chaotic dynamical systems was observed by Pecora and Carroll [1] in 1990, theoretical as well as experimental research on chaos synchronization has been carried out in a variety of nonlinear dynamic systems. It is because chaos synchronization can be applied in vast areas of physics, engineering science, and in particular in secure communication [2], [3]. A variety of approaches [4], [5], [6], [7] have been proposed for the synchronization of chaotic systems, such as complete synchronization [1], anti-synchronization [4], generalized synchronization [5] and projective synchronization (PS) [6]. Among all these kinds of synchronization schemes, PS is a generalized method of synchronization and more remarkable, because it can obtain faster communication with its proportional feature. In PS, the drive and response vectors evolve in a proportional scale. In application to secure communications, the proportional feature can be used to extend binary digital to M-nary digital communication [7] for achieving faster communication.

More recently, a novel PS, called function projective synchronization (FPS), has been raised by Chen et al. [8], [9], [10]. As compared with PS, FPS means that the master and slave systems could be synchronized up to a desired scaling function, instead of a constant. This feature could be used to get more secure communications in application to secure communications [11], [12], because the unpredictability of the scaling function in FPS can additionally enhance the security of communication. It is obvious that FPS is the more general definition of PS.

As we all know, stochastic processes are prevalent in nature. They affect all physical phenomena both from external and internal sources. The models in natural science and social life are almost nonlinear stochastic dynamical systems under the influence of several kinds of stochastic factors. Effectively reflecting the essence of these stochastic models can help us recognize the real world more reasonably. These stochastic models can be more accurate than the deterministic models. Some more prominence phenomena maybe be found, which may be neglected in deterministic one. Therefore, it is of importance to study the chaos synchronization of the stochastic system. However, at present, there exist many FPS methods, which mainly focus on the deterministic integer-order chaotic systems [13], [14], [15] or the deterministic fractional-order chaotic systems [16], [17], [18]. The FPS for different integer-order chaotic systems with uncertain parameters was discussed by Du et al. in the Ref. [19]. The generalized function projective synchronization of different integer-order chaotic systems with unknown parameters was addressed in the Ref. [20]. The function projective synchronization between different fractional-order chaotic systems with uncertain parameters was studied by modified adaptive control method in the Ref. [21].

As we can find from the above Refs. [13], [14], [15], [16], [17], [18], [19], [20], [21], these papers mainly focus on synchronization between integer-order chaotic systems or fractional-order chaotic systems. However, to the best of our knowledge, there are few works about the FPS between integer-order nonlinear systems and fractional-order nonlinear systems with multistochastic disturbances, available from the literature. Because multistochastic processes and fractional calculus have complex theoretical definition in mathematics and need more analysis tools as well as mass of calculation, the FPS between integer-order nonlinear systems and fractional-order nonlinear systems with multistochastic disturbances is more difficult than the synchronization between determinate integer-order chaotic systems. From a practical perspective, it is very important to synchronize integer-order nonlinear systems and stochastic fractional-order systems. Because it can generate hybrid chaotic transient signals before the final states and is hard to decryption. Besides, as compared with other synchronization, the FPS between integer-order nonlinear systems and stochastic fractional-order nonlinear systems can be used in secure communications since the stochastic fractional-order nonlinear system has more adjustable variables and stochastic factors than the integer-order nonlinear systems. So it can additionally enhance the security of communication. In 2010, the FPS between fractional-order chaotic systems and integer-order chaotic systems was studied in the Ref. [22]. But the stochastic disturbances are not considered. Motivated by the above discussion, the function projective synchronization between integer-order nonlinear systems and fractional-order nonlinear systems with multistochastic disturbances is investigated in this paper. It means two or more parameters of the fractional-order nonlinear system are designed as random ones. Firstly, in order to achieve the FPS between integer-order nonlinear systems and fractional-order nonlinear systems with multistochastic disturbances, a controller is designed based on the stability theory of fractional-order systems and tracking control. The error system with multistochastic disturbances is obtained. Then the method of simplifying the error system with multistochastic disturbances into an equivalent deterministic one is presented. Thirdly, the stability analysis of the error system with multistochastic disturbances is received by that of its equivalent deterministic one. Finally, for illustrating purposes, the FPS between integer-order Lorenz system and fractional-order Chen system with multistochastic disturbances is studied. Numerical simulation shows that the proposed scheme is effective.

The rest of the paper is arranged as follows. The FPS between integer-order and stochastic fractional-order nonlinear systems is shown in Section 2, and the method of transforming the stochastic error system into an equivalent deterministic one is presented in this section. The FPS between integer-order Lorenz system and stochastic fractional-order Chen system is achieved in Section 3. Finally, conclusion closes the paper in Section 4.

Section snippets

FPS between integer-order and stochastic fractional-order nonlinear systems

There are several definitions of the fractional-order differential system [23]. In the following, the most common one is presented asDαx(t)=Jα−mx(m)(t)(α>0),where the operator Dα is generally called α-order Caputo differential operator, m is the least integer which is not less than α, x(m) is the m-order derivative in common sense, and J−β(β>0) is the β-order Reimann–Liouville integral operator which satisfiesJ−βy(t)=1Γ(β)∫at(t−τ)β−1y(τ)dτ.Γrepresents Gamma function. Caputo differential

FPS between integer-order Lorenz system and fractional-order Chen system with multi-stochastic disturbances

The integer-order Lorenz system [28], as the first chaotic attractor model in 3D nonlinear autonomous, is given by{dx1/dt=a(x2−x1)dx2/dt=cx1−x1x3−x2dx3/dt=x1x2−bx3.When the parameters (a,b,c)=(10,8/3,28), the system displays a chaotic attractor shown in Fig. 1.

The fractional-order Chen system is described by{dqy1/dtq=a⁎(y2−y1)dqy2/dtq=(c⁎−a⁎)y1−y1y3+c⁎y2dqy3/dtq=y1y2−b⁎y3.Mohammad and Mohammad [29] pointed out that the fractional-order Chen system exhibits chaotic behaviors for fraction-order q≥

Conclusion

In this paper, the FPS between integer-order nonlinear systems and stochastic fractional-order nonlinear systems has been investigated. With the stability criterion of fractional-order systems and tracking control, a synchronization approach is proposed and proved. This proposed scheme is theoretically rigorous. Then the stochastic FPS error system is got. According to the orthogonal polynomial approximation, the method of converting the stochastic FPS error system into an equivalent

Conflict of Interests

None declared.

Acknowledgments

This work is supported by the National Nature Science Foundation of China under Grant no. 11371049 and the Fundamental Research Funds for the Central Universities under Grant no. 2016JBM070.

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