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

Abstract

A new function projective synchronization scheme between different fractional-order chaotic systems, called scaling-base drive function projective synchronization (SBDFPS), is discussed. In this SBDFPS scheme, one fractional-order chaotic system is chosen as scaling drive system, one fractional-order chaotic system is chosen as base drive systems, and another fractional-order chaotic system is chosen as response system. The SBDFPS technique scheme is based on the stability theory of nonlinear fractional-order systems, and the synchronization technique is theoretically rigorous. Numerical experiments are presented and show the effectiveness of the SBDFPS scheme.

1. Introduction

In the past twenty years, many synchronization schemes for chaotic systems have been presented [1–9]. 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 (fractional-order 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.

2. 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 𝐷^𝑞𝑓(𝑡)=1Γ𝑚−𝑞^𝑡₀𝑓^(𝑚)(𝜏)(𝑡−𝜏)^{𝑞+1−𝑚}𝑑𝜏,𝑚−1<𝑞<𝑚, (1) where 𝐷^𝑞 is called the Caputo operator, 𝑚 is the first integer which is not less than 𝑞, and 𝑓^(𝑚)(𝑡) is the 𝑚-order derivative for 𝑓(𝑡); that is, 𝑓^(𝑚)(𝑡)=𝑑^𝑚𝑓(𝑡)/𝑑𝑡^𝑚.

Now, the SBDFPS scheme between different fractional-order 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: 𝐷^𝑞𝑑1𝑥₁=𝑓_𝑑1𝑥₁,𝐷^𝑞𝑑2𝑥₂=𝑓_𝑑2𝑥₂,𝐷^𝑞𝑟𝑦=𝑓_𝑟𝑦+𝑀𝑥₁,𝑥₂,𝑦, (2)(3)(4) where 0<𝑞_𝑑𝑖<1(𝑖=1,2) and 0<𝑞_𝑟<1 are fractional-order. 𝑥₁=(𝑥₁₁,𝑥₁₂,…,𝑥_1𝑛)^𝑇, 𝑥₂=(𝑥₂₁,𝑥₂₂,…,𝑥_2𝑛)^𝑇, and 𝑦=(𝑦₁,𝑦₂,…𝑦_𝑛)^𝑇 are state vectors of fractional-order chaotic systems (2)–(4). 𝑓_𝑑𝑖(𝑖=1,2)∶𝑅^𝑛→𝑅^𝑛 and 𝑓_𝑟∶𝑅^𝑛→𝑅^𝑛 are differential nonlinear functions. 𝑀(𝑥₁,𝑥₂,𝑦)∈𝑅^𝑛×1 is a vector controller and will be designed.

Definition 1. Give the scaling drive system (2), the base drive systems (3), and the response system (4). It is said to scaling-base drive function projective synchronization (SBDFPS) if there exist real nonzero constant matrix 𝐶_𝑖∈𝑅^𝑛×𝑛(𝑖=1,2) and nonzero scaling function matrix 𝑆_𝑖(𝑥_𝑖)∈𝑅^𝑛×𝑛(𝑖=1,2) such that lim_𝑡→+∞‖𝑒‖=lim_𝑡→+∞𝐶₁𝑆₁𝑥₁+𝐶₂𝑆₂𝑥₂𝑥₁−𝑦=0, (5) where ‖⋅‖ represents the Euclidean norm.

Remark 2. If 𝐶₁≠0, 𝐶₂=0, then the SBDFPS scheme will be turned into FPS. If 𝐶_𝑖=0(𝑖=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.

Let the SBDFPS error between the scaling drive system (2), base drive systems (3), and response system (4) be defined as 𝑒=𝑦−𝐶₁𝑆₁𝑥₁+𝐶₂𝑆₂𝑥₂𝑥₁, (6) where 𝑒=(𝑒₁,𝑒₂,…,𝑒_𝑛)^𝑇.

Now, choose vector controller 𝑀(𝑥₁,𝑥₂,𝑦)∈𝑅^𝑛×1 as 𝑀𝑥₁,𝑥₂,𝑦=𝐷^𝑞𝑟𝐶₁𝑆₁𝑥₁+𝐶₂𝑆₂𝑥₂𝑥₁−𝑓_𝑟𝐶₁𝑆₁𝑥₁+𝐶₂𝑆₂𝑥₂𝑥₁+𝑀₁𝑥₁,𝑥₂,𝑦𝑒, (7) where feedback controller 𝑀₁(𝑥₁,𝑥₂,𝑦)∈𝑅^𝑛×𝑛 will be designed later.

By (6) and (7), system (4) can be changed as follows: 𝐷^𝑞𝑟𝑒=𝑓_𝑟𝑦−𝑓_𝑟𝐶₁𝑆₁𝑥₁+𝐶₂𝑆₂𝑥₂𝑥₁+𝑀₁𝑥₁,𝑥₂,𝑦𝑒. (8)

In this paper, we assume that 𝑓_𝑟𝑦−𝑓_𝑟𝐶₁𝑆₁𝑥₁+𝐶₂𝑆₂𝑥₂𝑥₁=𝑀₂𝑥₁,𝑥₂,𝑦𝑒, (9) where 𝑀₂(𝑥₁,𝑥₂,𝑦)∈𝑅^𝑛×𝑛. In fact, many fractional-order chaotic (hyperchaotic) systems satisfy this assumption.

By (9), system (8) can be rewritten as 𝐷^𝑞𝑟𝑒=𝑀₁𝑥₁,𝑥₂,𝑦+𝑀₂𝑥₁,𝑥₂,𝑦𝑒. (10)

By (10), the SBDFPS between the scaling drive system (2), base drive systems (3), and response system (4) is turned into the following problem: select suitable 𝑀₁(𝑥₁,𝑥₂,𝑦)∈𝑅^𝑛×𝑛 such that the system (10) asymptotically converges to zero.

Theorem 4. Select suitable matrix 𝑀₁(𝑥₁,𝑥₂,𝑦)∈𝑅^𝑛×𝑛 such that 𝑀₁(𝑥₁,𝑥₂,𝑦)+𝑀₂(𝑥₁,𝑥₂,𝑦) satisfy the following conditions: (1)   𝑀_𝑖𝑗=−𝑀_𝑗𝑖(𝑖≠𝑗) (2)  𝑀_𝑖𝑖≤0 (all 𝑀_𝑖𝑖 are not equal to zero),
where 𝑀_𝑖𝑗(𝑖,𝑗=1,2,…𝑛,∀𝑀_𝑖𝑗∈𝑅) are the entries of 𝑀₁(𝑥₁,𝑥₂,𝑦)+𝑀₂(𝑥₁,𝑥₂,𝑦). Then the SBDFPS between the scaling drive system (2), base drive systems (3), and response system (4) can be reached.

Proof. Let 𝜆 be one of the eigenvalues of matrix 𝑀₁(𝑥₁,𝑥₂,𝑦)+𝑀₂(𝑥₁,𝑥₂,𝑦) and 𝜌 the corresponding nonzero eigenvector. So, we have 𝑀₁𝑥₁,𝑥₂,𝑦+𝑀₂𝑥₁,𝑥₂,𝑦𝜌=𝜆𝜌. (11)
By (11), taking conjugate transpose on both sides of (11), one can obtain 𝑀₁𝑥₁,𝑥₂,𝑦+𝑀₂𝑥₁,𝑥₂,𝑦𝜌^𝐓=𝜆𝜌^𝐇, (12) where 𝐇 denotes conjugate transpose.
Now, (12) multiplied right by 𝜌 plus (11) multiplied left by 𝜌^𝐇. Thus 𝜌^𝐇𝑀₁𝑥₁,𝑥₂,𝑦+𝑀₂𝑥₁,𝑥₂+𝑀,𝑦₁𝑥₁,𝑥₂,𝑦+𝑀₂𝑥₁,𝑥₂,𝑦^𝐇𝜌=𝜌^𝐇𝜌𝜆+𝜆. (13)
Thus 𝜆+𝜆=𝜌^𝐇𝑀₁𝑥₁,𝑥₂,𝑦+𝑀₂𝑥₁,𝑥₂+𝑀,𝑦₁𝑥₁,𝑥₂,𝑦+𝑀₂𝑥₁,𝑥₂,𝑦^𝐇𝜌𝜌^𝐇𝜌. (14)
Using 𝑀_𝑖𝑗=−𝑀_𝑗𝑖(𝑖≠𝑗,∀𝑏_𝑖𝑗∈𝑅), so 𝜆+𝜆=𝜌^𝐇2𝑀₁₁0⋯002𝑀₂₂⋯000⋯00002𝑀_𝑛𝑛𝜌𝜌^𝐇𝜌. (15)
Because 𝑀_𝑖𝑖≤0 (all 𝑀_𝑖𝑖 are not equal to zero), we have 𝜆+𝜆≤0. (16)
That is, arg𝜆𝑀₁𝑥₁,𝑥₂,𝑦+𝑀₂𝑥₁,𝑥₂,𝑦≥0.5𝜋. (17)
Therefore, arg𝜆𝑀₁𝑥₁,𝑥₂,𝑦+𝑀₂𝑥₁,𝑥₂,𝑦≥𝑞_𝑟𝜋2. (18)
According to the stability theorem for nonlinear fractional-order systems [9–11], (18) indicates that the equilibrium point 𝑒=(0,0,…,0)^𝑇 in system (10) is asymptotically stable; that is, lim_𝑡→+∞‖𝑒‖=lim_𝑡→+∞𝐶₁𝑆₁𝑥₁+𝐶₂𝑆₂𝑥₂𝑥₁−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 of Adams-Bashforth-Moulton numerical algorithm [12] for fractional-order nonlinear systems is introduced. Now, consider the nonlinear fractional-order system 𝑑^𝑞1𝑧₁𝑑𝑡^𝑞1=ℎ₁𝑧₁,𝑧₂,𝑑^𝑞2𝑧₂𝑑𝑡^𝑞2=ℎ₂𝑧₁,𝑧₂, (20) with initial condition (𝑧₁(0),𝑧₂(0)). Let 𝜏=𝑇/𝑁 and 𝑡_𝑛=𝑛𝜏 (𝑛=0,1,2…,𝑁). Then, nonlinear fractional-order system (20) is discretized as follows: 𝑧₁(𝑛+1)=𝑧₁(0)+𝜏^𝑞1Γ𝑞₁×ℎ+2₁𝑧^𝑝₁(𝑛+1),𝑧^𝑝₂()+𝑛+1_𝑛𝑗=0𝛼_{1,𝑗,𝑛+1}ℎ₁𝑧₁𝑗,𝑧₂𝑗,𝑧₂(𝑛+1)=𝑧₂(0)+𝜏^𝑞2Γ𝑞₂×ℎ+2₂𝑧^𝑝₁(𝑛+1),𝑧^𝑝₂(+𝑛+1)_𝑛𝑗=0𝛼_{2,𝑗,𝑛+1}ℎ₂𝑧₁𝑗,𝑧₂𝑗, (21) where 𝑧^𝑝₁(𝑛+1)=𝑧₁(0)+1Γ𝑞₁^𝑛_𝑗=0𝛽_{1,𝑗,𝑛+1}ℎ₁𝑧₁𝑗,𝑧₂𝑗,𝑧^𝑝₂(𝑛+1)=𝑧₂(0)+1Γ𝑞₂^𝑛_𝑗=0𝛽_{2,𝑗,𝑛+1}ℎ₂𝑧₁𝑗,𝑧₂𝑗,𝛼_{𝑖,𝑗,𝑛+1}=𝑛^𝑞𝑖+1−𝑛−𝑞_𝑖(𝑛+1)^𝑞𝑖,𝑗=0𝑛−𝑗+2^𝑞𝑖+1+𝑛−𝑗^𝑞𝑖+1−2𝑛−𝑗+1^𝑞𝑖+1(),𝛽,1≤𝑗≤𝑛1,𝑗=𝑛+1,𝑖=1,2_{𝑖,𝑗,𝑛+1}=𝜏^𝑞𝑖𝑞_𝑖𝑛−𝑗+1^𝑞𝑖−𝑛−𝑗^𝑞𝑖,0≤𝑗≤𝑛(𝑖=1,2). (22)(23) The error of this approximation is 𝑧_𝑖𝑡_𝑛−𝑧_𝑖(𝑛)=𝑜𝜏^𝑝𝑖,𝑝_𝑖=min2,1+𝑞_𝑖(𝑖=1,2). (24) The fractional-order Lorenz chaotic system [7] is depicted as 𝐷^𝑞𝑑1𝑥₁₁=10𝑥₁₂−𝑥₁₁,𝐷^𝑞𝑑1𝑥₁₂=28𝑥₁₁−𝑥₁₂−𝑥₁₁𝑥₁₃,𝐷^𝑞𝑑1𝑥₁₃=𝑥₁₁𝑥₁₂−8𝑥₁₃3. (25)

The fractional-order Lorenz system (25) exhibits chaotic behavior for fractional-order 𝑞≥0.993. The chaotic attractor for 𝑞_𝑑1=0.994 is shown in Figure 1.

Figure 1 

Chaotic attractors of fractional-order Lorenz system (25) for 𝑞𝑑1=0.994.

The fractional-order Lu chaotic system [13] is described as 𝐷^𝑞𝑑2𝑥₂₁=36𝑥₂₂−𝑥₂₁,𝐷^𝑞𝑑2𝑥₂₂=20𝑥₂₂−𝑥₂₁𝑥₂₃,𝐷^𝑞𝑑2𝑥₂₃=𝑥₂₁𝑥₂₂−3𝑥₂₃. (26) Its chaotic attractor for 𝑞_𝑑2=0.95 is illustrated in Figure 2.

Figure 2 

Chaotic attractors of fractional-order Lu system (26) for 𝑞𝑑2=0.95.

The fractional-order Chen chaotic system [7] is 𝐷^𝑞𝑟𝑦₁=35𝑦₂−𝑦₁,𝐷^𝑞𝑟𝑦₂=−7𝑦₁+28𝑦₂−𝑦₁𝑦₃,𝐷^𝑞𝑟𝑦₃=𝑦₁𝑦₂−3𝑦₃. (27)

The fractional-order Chen system (27) exhibits chaotic behavior for fractional-order 𝑞_𝑟≥0.83. The chaotic attractor of fractional-order Chen system (27) for 𝑞_𝑟=0.85 is displayed in Figure 3.

Figure 3 

Chaotic attractors of fractional-order Chen system (27) for 𝑞𝑟=0.85.

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 𝑀₂𝑥₁,𝑥₂,𝑦=−35350−7−𝑦₃𝑦28−𝛽₂𝛽−3, (28) where𝛽=𝑥₁₁[∑²_𝑖=1𝐶_𝑖𝑆_𝑖(𝑥_𝑖)]₁₁+𝑥₁₂[∑²_𝑖=1𝐶_𝑖𝑆_𝑖(𝑥_𝑖)]₁₂+𝑥₁₃[∑²_𝑖=1𝐶_𝑖𝑆_𝑖(𝑥_𝑖)]₁₃, [∑²_𝑖=1𝐶_𝑖𝑆_𝑖(𝑥_𝑖)]_1𝑗(𝑗=1,2) are the elements of matrix [∑²_𝑖=1𝐶_𝑖𝑆_𝑖(𝑥_𝑖)]. 𝐶_𝑖∈𝑅^3×3(𝑖=1,2) are real nonzero constant matrix.

Now, we select matrix 𝑀₁(𝑥₁,𝑥₂,𝑦) as follows: 𝑀₁𝑥₁,𝑥₂,𝑦=0−28+𝑦₃−𝑦₂0−300000. (29)

Then, 𝑀₁𝑥₁,𝑥₂,𝑦+𝑀₂𝑥₁,𝑥₂,𝑦=−357+𝑦₃−𝑦₂−7−𝑦₃𝑦−2−𝛽₂𝛽−3. (30)

According to Theorem 4, we can obtain the following: arg𝜆𝑀₁𝑥₁,𝑥₂,𝑦+𝑀₂𝑥₁,𝑥₂,𝑦≥𝑞_𝑟𝜋2. (31)

This result means that lim_𝑡→+∞‖𝑒‖=lim_𝑡→+∞𝐶₁𝑆₁𝑥₁+𝐶₂𝑆₂𝑥₂𝑥₁−𝑦=0. (32)

Equation (32) implies that the SBDFPS between the scaling drive system (25), base drive systems (26), and response system (27) can be received.

For example, let 𝐶_𝑖(),𝑆=diag1,1,1)(𝑖=1,2₁𝑥₁=diag35𝑥₁₂−𝑥₁₁,−7𝑥₁₁−𝑥₁₁𝑥₁₃,𝑥₁₁𝑥₁₂,𝑆₂𝑥₂=diag10𝑥₂₂−𝑥₂₁,−𝑥₂₂−𝑥₂₁𝑥₂₃,𝑥₂₁𝑥₂₂−𝑥₂₃. (33)

The initial conditions are (𝑥₁₁,𝑥₁₂,𝑥₁₃)=(2,2,2), (𝑥₂₁,𝑥₂₂,𝑥₂₃)=(1,1,1), and (𝑦₁,𝑦₂,𝑦₃)=(5,−24,8), respectively. The numerical experiments are illustrated in Figure 4.

Figure 4 

The SBDFPS errors between the scaling drive system (25), base drive systems (26), and response system (27).

4. 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.