Compound Generalized Function Projective Synchronization for Fractional-Order Chaotic Systems

A modified function projective synchronization for fractional-order chaotic system, called compound generalized function projective synchronization (CGFPS), is proposed theoretically in this paper. There are one scaling-drive system, more than one base-drive system, and one response system in the scheme of CGFPS, and the scaling function matrices come from multidrive systems. The proposed CGFPS technique is based on the stability theory of fractional-order system. Moreover, we achieve the CGFPS between three-driver chaotic systems, that is, the fractional-order Arneodo chaotic system, the fractional-order Chen chaotic system, and the fractional-order Lu chaotic system, and one response chaotic system, that is, the fractional-order Lorenz chaotic system. Numerical experiments are demonstrated to verify the effectiveness of the CGFPS scheme.


Introduction
Chaotic phenomenon in nonlinear system has attracted more and more attentions in recent years [1,2].In 1990, chaos synchronization was introduced by Pecora and Carroll for the first time [3], and after that, chaos synchronization has attracted a lot of attention over the last two decades.This is due to its potential applications in many areas of physics and engineering science [3], such as complex networks control [4], signal processing [5], information transmission [6], authenticated encryption [7], and secure communication [8,9].On the other hand, as physical interpretation of the fractional derivative becomes clear, many real-world physical systems can be more accurately described by fractionalorder differential equations [10,11].Chaotic phenomenon has been also observed in many physical fractional-order systems, for example, microelectromechanical systems [12], gyroscopes [13], and electronic circuits [14,15].Meanwhile, more and more attentions [4,7,16,17] have been paid on synchronization and control of fractional-order chaotic system in both theoretical and applied perspectives, since it is usually a prerequisite of many practical applications in chaos engineering, especially in authenticated encryption [7] and chaos communications [18].
Up to now, various types of chaos synchronization have been proposed, such as complete synchronization (CS) [3,8], generalized synchronization (GS) [9,19], projective synchronization (PS) [20], function projective synchronization (FPS) [21][22][23], combination synchronization [24], and compound synchronization [25].Among all these schemes, PS and FPS have been extensively studied in recent years because the two schemes can not only obtain faster communication but also enhance the security of communications with its proportional feature.The PS scheme indicates that the drive system and response system could be arrived to synchronize with a scaling constant factor.The FPS scheme, generalized from the PS scheme, means that the driver system and response system could be synchronized with a scaling function matrix instead of the scaling factor.Since the scaling function matrix can be also used as additional secret codes, the FPS scheme can enhance the security in secure communication.
However, the response system could be synchronized to the drive system with a function matrix in FPS scheme, and there are only one drive system and one response system.Therefore, the FPS between multidrive systems and one response system, where the scaling function matrix comes from multidrive systems, is interesting and general topic.It is obvious that multidrive systems and one response system synchronization in FPS scheme can additionally enhance the security of communication; this is due to the fact that the transmitted signals can be split into several parts, and each part may be loaded in different drive systems; or the transmitted signals can be divided in time into different intervals, and the signals in different intervals may be loaded in different drive systems [24,25].Motivated by the abovementioned, a new function projective synchronization scheme, called compound generalized function projective synchronization (briefly denoted by CGFPS), is proposed for fractional-order chaotic systems in this paper.The CGFPS scheme means that there are multidrive systems and one response system, and the scaling function matrix comes from multidrive systems.This CGFPS technique is based on the stability theory of fractional-order systems.Some examples are demonstrated to verify the effectiveness of the CGFPS scheme.
The layout of this paper is organized as follows.In Section 2, the CGFPS scheme for fractional-order chaotic systems is presented.In Section 3, numerical experiments are used to verify the effectiveness of the CGFPS scheme.Finally, the conclusion ends the paper in Section 4.
Before we give a detailed definition of the CGFPS scheme, let us first recall the definitions of the FPS scheme and the compound synchronization scheme.
Remark 7. The scaling-drive system (2) and the base-drive systems (3) in CGFPS scheme are maybe integer order systems.Therefore, the CGFPS between integer order chaotic systems and fractional-order systems can be achieved.Moreover, the CGFPS can be applied for integer order chaotic systems; that is, the scaling-drive system (2), the base-drive systems (3), and response system (4) are integer order chaotic systems.

Illustrative Examples
In this section, numerical experiments are used to verify the effectiveness of the CGFPS scheme in our paper.Some examples are given and the numerical simulations are performed.Now, choose the fractional-order Arneodo chaotic system as the scaling-drive system, which is given by [30] The chaotic attractor of system (19) with  0 = 0.995 is shown in Figure 1.
Then, choose the fractional-order Chen chaotic system and the fractional-order Lu chaotic system as the base-drive systems.The fractional-order Chen chaotic system [22] and the chaotic attractor of system (20) with  1 = 0.9 is shown as in Figure 2. The fractional-order Lü chaotic system [31] is defined as and the chaotic attractor of system (21) with  2 = 0.96 is shown in Figure 3. Next, we choose the fractional-order Loren chaotic system as response system, which is given by [22]     1 = 10 ( 2 −  1 ) , The chaotic attractor of system (22) with   = 0.997 is shown in Figure 4. Figure 4: The chaotic attractor of system (22) for   = 0.997.
Equations ( 27)- (28) indicate that the compound generalized function projective synchronization (CGFPS) between drive systems ( 19)-(21) and response system (22) can be achieved.Now, two cases with numerical simulations are given to verify the effectiveness of proposed scheme.

Conclusions
In this paper, we present a new synchronization scheme for fractional-order chaotic system and called this type of synchronization compound generalized function projective synchronization, or briefly denoted it by CGFPS.In this scheme, there are one scaling-drive system, more than one base-drive system, and one response system, and the scaling function matrix comes from multidrive systems.So, the CGFPS in this paper is different from all the previous synchronization reported before.To verify its effectiveness, we achieve the CGFPS between three drive systems and one response system, where the scaling-drive system is the fractionalorder Arneodo chaotic system, the base-drive systems are the fractional-order Chen chaotic system and the fractionalorder Lu chaotic system, and the response system is the fractional-order Loren chaotic system.Numerical simulations suggest that the presented CGFPS scheme works well.It is worth mentioning that the CGFPS scheme in our paper can also be used for other fractional-order chaotic systems or integer order chaotic systems.