Scaling-Base Drive Function Projective Synchronization between Different Fractional-Order Chaotic Systems

and Applied Analysis 3 Thus λ + λ = ρ H { [M1 (x1, x2, y) + M2 (x1, x2, y)] +[M1 (x1, x2, y) + M2 (x1, x2, y)] H } ρ ρHρ . (14) UsingMij = −Mji (i ̸ = j, ∀bij ∈ R), so λ + λ = ρ H ( 2M11 0 ⋅ ⋅ ⋅ 0 0 2M22 ⋅ ⋅ ⋅ 0 0 0 ⋅ ⋅ ⋅ 0 0 0 0 2Mnn ) ρ ρHρ . (15) BecauseMii ≤ 0 (allMii are not equal to zero), we have λ + λ ≤ 0. (16) That is, 󵄨󵄨󵄨󵄨arg λ [M1 (x1, x2, y) + M2 (x1, x2, y)] 󵄨󵄨󵄨󵄨 ≥ 0.5π. (17) Therefore, 󵄨󵄨󵄨󵄨arg λ [M1 (x1, x2, y) + M2 (x1, x2, y)] 󵄨󵄨󵄨󵄨 ≥ qr π 2 . (18) According to the stability theorem for nonlinear fractional-order systems [9–11], (18) indicates that the equilibrium point e = (0, 0, . . . , 0) in system (10) is asymp-totically stable; that is, lim t→+∞ ‖e‖ = lim t→+∞ 󵄩󵄩󵄩󵄩[C1S1 (x1) + C2S2 (x2)] x1 − y 󵄩󵄩󵄩󵄩 = 0. (19) Equation (19) demonstrates that the SBDFPS between the scaling drive system (2), base drive systems (3), and response system (4) can be received. The proof is completed. 3. Illustrative Examples To illustrate the effectiveness of the proposed synchronization scheme, some examples are given and the numerical simulations are yielded. First, the improved version ofAdams-Bashforth-Moulton numerical algorithm [12] for fractional-order nonlinear systems is introduced. Now, consider the nonlinear fractionalorder system d qz1 dt1 = h1 (z1, z2) , d qz2 dt2 = h2 (z1, z2) , (20) with initial condition (z1(0), z2(0)). Let τ = T/N and tn = nτ (n = 0, 1, 2 . . . , N). Then, nonlinear fractional-order system (20) is discretized as follows: z1 (n + 1) = z1 (0) + τ q1 Γ (q1 + 2) × [ [ h1 (z p 1 (n + 1) , z p 2 (n + 1)) + n ∑ j=0 α1,j,n+1h1 (z1 (j) , z2 (j)) ]


Introduction
In the past twenty years, many synchronization schemes for chaotic systems have been presented [1][2][3][4][5][6][7][8][9].However, the function projective synchronization (FPS) scheme for chaotic systems is extensively considered due to its potential applications in secure communication.Because the drive and response systems could be synchronized with a scaling function matrix in FPS, the unpredictability of the scaling function matrix in FPS scheme can enhance the security in secure communication.In FPS, only two chaotic systems (one drive system and one response system) are considered, and the function matrix comes from one drive system.Therefore, more than one drive system (two or three drive systems or four drive systems, etc.) and one response system in FPS, and the scaling function matrix coming from multidrive systems, are general case.Moreover, multidrive systems 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 can be loaded in different drive systems, or the transmitted signals can be divided time into different intervals, and the signals in different intervals can be loaded in different drive systems [8].
Motivated by the previous part, we demonstrated a new function projective synchronization scheme between different fractional-order chaotic systems in this paper, which is called scaling-base drive function projective synchronization (briefly denoted by SBDFPS).In SBDFPS scheme, there are two drive systems, which are called the scaling drive system and the base drive system, respectively.The proposed SBDFPS technique is based on the stability theory of nonlinear fractional-order systems and is theoretically rigorous.The SBDFPS between two-driver chaotic systems (fractionalorder Lorenz chaotic system as scaling drive system and fractional-order Lu chaotic system as base drive system) and one response chaotic system (fractional-order Chen chaotic system) is achieved.Numerical experiments show the effectiveness of the SBDFPS scheme.
This paper is organized as follows.In Section 2, the SBDFPS scheme between different fractional-order chaotic systems is demonstrated.In Section 3, some examples are considered and show the effectiveness of the SBDFPS scheme.Finally, the conclusion ends the paper in Section 4.

The Scaling-Base Drive Function Projective Synchronization (SBDFPS) between Different Fractional-Order Chaotic Systems
The Caputo definition of the fractional derivative is used, which is where   is called the Caputo operator,  is the first integer which is not less than , and  () () is the -order derivative for (); that is,  () () =   ()/  .Now, the SBDFPS scheme between different fractionalorder chaotic systems will be established.Consider the fractional-order scaling drive chaotic system and base drive chaotic system and one response chaotic system described by systems (2), (3), and (4), respectively as follows: where 0 <   < 1 ( = 1, 2) and 0<  < 1 are fractional-order.
Remark 2. If  1 ̸ = 0,  2 = 0, then the SBDFPS scheme will be turned into FPS.If   = 0 ( = 1, 2), then the SBDFPS scheme will be turned into a chaos control problem.Remark 3. System (2) and systems (3) in SBDFPS scheme may be integer order systems.So, the SBDFPS between integer order chaotic system and fractional-order can be achieved.

Illustrative Examples
To illustrate the effectiveness of the proposed synchronization scheme, some examples are given and the numerical simulations are yielded.
The fractional-order Lu chaotic system [13] is described as Its chaotic attractor for  2 = 0.95 is illustrated in Figure 2.
The fractional-order Chen chaotic system [7] is The fractional-order Chen system (27) exhibits chaotic behavior for fractional-order   ≥ 0.83.The chaotic attractor of fractional-order Chen system (27) for   = 0.85 is displayed in Figure 3. Now, the fractional-order Lorenz chaotic system (25) is selected as the scaling drive system, the fractional-order Lu chaotic system (26) is selected as the base drive system, and the fractional-order Chen chaotic system (27) is selected as response system.Our goal is to realize the SBDFPS between the scaling drive system (25), the base drive system (26), and response system (27).
According to the results in Section 2, we derive that where  = Then, ) . (30) According to Theorem 4, we can obtain the following: This result means that lim Equation (32) implies that the SBDFPS between the scaling drive system (25), base drive systems (26), and response system (27) can be received.

Conclusions
In this paper, the scaling-base drive function projective synchronization (SBDFPS) is presented.The SBDFPS scheme is different from the FPS scheme because the scaling function matrix comes from more than one chaotic system (the scaling drive system and the base drive system).The SBDFPS between the fractional-order Lorenz chaotic system (scaling drive system), the fractional-order Lu chaotic system (base drive system), and the fractional-order Chen chaotic system (response system) is taken for example.Numerical experiments show the effectiveness of the SBDFPS scheme.