Modified Function Projective Synchronization between Different Dimension Fractional-Order Chaotic Systems

Abstract

A modified function projective synchronization (MFPS) scheme for different dimension fractional-order chaotic systems is presented via fractional order derivative. The synchronization scheme, based on stability theory of nonlinear fractional-order systems, is theoretically rigorous. The numerical simulations demonstrate the validity and feasibility of the proposed method.

1. Introduction

Fractional-order calculus, which can be dated back to the 17th century [1, 2]. However, only in the last few decades, its application to physics and engineering has been addressed. So, the fractional-order calculus has attracted increasing attention only recently. On the other hand, complex bifurcation and chaotic phenomena have been found in many fractional-order dynamical systems. For example, the fractional-order Lorenz chaotic system [3], the fractional-order unified chaotic system [4], the fractional-order Chua chaotic circuit [5], the fractional-order modified Duffing chaotic system [6], and the fractional-order Rössler chaotic system [7, 8], and so on.

Nowadays, synchronization of chaotic systems and fractional-order chaotic systems has attracted much attention because of its applications in secure communication and control processing [9–21]. Many approaches have been reported for the synchronization of chaotic systems and fractional-order chaotic systems [12–19]. In 1999, Mainieri and Rehacek proposed projective synchronization (PS) [12] for chaotic systems, which has been extensively investigated in recent years because of its proportional feature in secure communications. Recently, a modified projective synchronization, which is called function projective synchronization (FPS) [13–15] has been reported. In FPS, the master and slave systems could be synchronized up to a scaling function, but not a constant. So, the unpredictability of the scaling function in FPS can additionally enhance the security of communication.

To the best of our knowledge, most of the existing FPS scheme for the fractional-order chaotic systems only discuss the same dimension. However, in many real physics systems, the synchronization is carried out through the oscillators with different dimension, especially the systems in biological science and social science [16–21]. Moreover, in some previous works [16, 17], all the nonlinear terms of response system or error system was absorbed. Referring to chaotic synchronization via fractional-order controller, there are a few results reported until now. Inspired by the above discussion, in this paper, we present a modified function projective synchronization (MFPS) scheme between different dimension fractional-order chaotic systems via fractional-order controller. The fractional-order controller is easily designed. The synchronization technique, based on tracking control and stability theory of nonlinear fractional-order systems, is theoretically rigorous. Our modified function projective synchronization (MFPS) scheme need not absorb all the nonlinear terms of response system. This is different from some previous works. Two examples are presented to demonstrate the effectiveness of the proposed MFPS scheme.

This paper is organized as follows. In Section 2, a modified function projective synchronization (MFPS) scheme is presented. In Section 3, two groups of examples are used to verify the effectiveness of the proposed scheme. The conclusion is finally drawn in Section 4.

2. The MFPS Scheme for Different Dimension Fractional-Order Chaotic Systems

The fractional-order chaotic drive and response systems with different dimension are defined as follows, respectively: 𝑑^𝑞𝑑𝑥𝑑𝑡^𝑞𝑑=𝐹_𝑑(𝑥),𝑑^𝑞𝑟𝑦𝑑𝑡^𝑞𝑟=𝐹_𝑟𝑦+𝐶𝑥,𝑦, (2.1)(2.2) where 𝑞_𝑑 (0<𝑞_𝑑<1) and 𝑞_𝑟 (0<𝑞_𝑟<1) are fractional order, and 𝑞_𝑑 may be different with 𝑞_𝑟. 𝑥∈𝑅^𝑛, 𝑦∈𝑅^𝑚 (𝑛≠𝑚) are state vectors of the drive system (2.1) and response system (2.2), respectively. 𝐹_𝑑∶𝑅^𝑛→𝑅^𝑛, 𝐹_𝑟∶𝑅^𝑚→𝑅^𝑚 are two continuous nonlinear vector functions, and 𝐶(𝑥,𝑦)∈𝑅^𝑚 is a controller which will be designed later.

Definition 2.1. For the drive system (2.1) and response system (2.2), it is said to be modified function projective synchronization (MFPS) if there exist a controller 𝐶(𝑥,𝑦) such that: lim_𝑡→+∞‖𝑒‖=lim_𝑡→+∞𝑦−𝑀(𝑥)𝑥=0, (2.3) where ‖⋅‖ is the Euclidean norm, 𝑀(𝑥) is a 𝑚×𝑛 real matrix, and matrix element 𝑀_𝑖𝑗(𝑥)(𝑖=1,2,…𝑚,𝑗=1,2,…,𝑛) are continuous bounded functions. 𝑒_𝑖=𝑦_𝑖−∑^𝑛_𝑗=1𝑀_𝑖𝑗𝑥_𝑗(𝑖=1,2,…𝑚) are called MFPS error.

Remark 2.2. According to the view of tracking control, 𝑀(𝑥)𝑥 can be chosen as a reference signal. The MFPS in our paper is transformed into the problem of tracking control, that is the output signal 𝑦 in system (2.2) follows the reference signal 𝑀(𝑥)𝑥.
In order to achieve the output signal 𝑦 follows the reference signal 𝑀(𝑥)𝑥. Now, we define a compensation controller 𝐶₁(𝑥)∈𝑅^𝑚 for response system (2.2) via fractional-order derivative 𝑑^𝑞𝑟(𝑀(𝑥)𝑥)/𝑑𝑡^𝑞𝑟. The compensation controller is shown as follows: 𝐶₁(𝑥)=𝑑^𝑞𝑟(𝑀(𝑥)𝑥)𝑑𝑡^𝑞𝑟−𝐹_𝑟(𝑀(𝑥)𝑥), (2.4) and let controller 𝐶(𝑥,𝑦) as follows: 𝐶𝑥,𝑦=𝐶₁(𝑥)+𝐶₂𝑥,𝑦, (2.5) where 𝐶₂(𝑥,𝑦)∈𝑅^𝑚 is a vector function which will be designed later.
By controller (2.5) and compensation controller (2.4), the response system (2.2) can be changed as follows: 𝑑^𝑞𝑟𝑒𝑑𝑡^𝑞𝑟=𝐷₁𝑥,𝑦𝑒+𝐶₂𝑥,𝑦, (2.6) where 𝐷₁(𝑥,𝑦)𝑒=𝐹_𝑟(𝑦)−𝐹_𝑟(𝑀(𝑥)𝑥), and 𝐷₁(𝑥,𝑦)∈𝑅^𝑚×𝑚. So, the MFPS between drive system (2.1) and response system (2.2) is transformed into the following problem: choose a suitable vector function 𝐶₂(𝑥,𝑦) such that system (2.6) is asymptotically converged to zero.
In what follows we present the stability theorem for nonlinear fractional-order systems of commensurate order [22–25]. Consider the following nonlinear commensurate fractional-order autonomous system 𝐷^𝑞𝑥=𝑓(𝑥), (2.7) the fixed points of system (2.7) is asymptotically stable if all eigenvalues (𝜆) of the Jacobian matrix 𝐴=𝜕𝑓/𝜕𝑥 evaluated at the fixed points satisfy |arg𝜆|>0.5𝜋𝑞. Where 0<𝑞<1, 𝑥∈𝑅^𝑛, 𝑓∶𝑅^𝑛→𝑅^𝑛 are continuous nonlinear functions, and the fixed points of this nonlinear commensurate fractional-order system are calculated by solving equation 𝑓(𝑥)=0.
Now, the following theorem is given based on the above discussion in order to achieve the MFPS between the drive system (2.1) and the response system (2.2).

Theorem 2.3. Choose the control vector 𝐶₂(𝑥,𝑦)=𝐷₂(𝑥,𝑦)𝑒, and if 𝐷₁(𝑥,𝑦)+𝐷₂(𝑥,𝑦) satisfy the following conditions:(1)𝑑_𝑖𝑗=−𝑑_𝑗𝑖(𝑖≠𝑗),(2)𝑑_𝑖𝑖≤0 (all 𝑑_𝑖𝑖 are not equal to zero),then the modified function projective synchronization (MFPS) between (2.1) and (2.2) can be achieved. Where 𝐷₂(𝑥,𝑦)∈𝑅^𝑚×𝑚, and 𝑑_𝑖𝑗(𝑖,𝑗=1,2,…𝑚,forall𝑑_𝑖𝑗∈𝑅) are the matrix element of matrix 𝐷₁(𝑥,𝑦)+𝐷₂(𝑥,𝑦).

Proof. Using 𝐶₂(𝑥,𝑦)=𝐷₂(𝑥,y)𝑒, so fractional-order system (2.6) can be rewritten as follows: 𝑑^𝑞𝑟𝑒𝑑𝑡^𝑞𝑟=𝐷₁𝑥,𝑦+𝐷₂𝑥,𝑦𝑒. (2.8)
Suppose 𝜆 is one of the eigenvalues of matrix 𝐷₁(𝑥,𝑦)+𝐷₂(𝑥,𝑦) and the corresponding non-zero eigenvector is 𝜓, that is, 𝐷₁𝑥,𝑦+𝐷₂𝑥,𝑦𝜓=𝜆𝜓. (2.9)
Take conjugate transpose (H) on both sides of (2.9), we yield 𝐷₁𝑥,𝑦+𝐷₂𝑥,𝑦𝜓^𝑇=𝜆𝜓^𝐻. (2.10)
Equation (2.9) multiplied left by 𝜓^𝐻 plus (2.10) multiplied right by 𝜓, we derive that 𝜓^𝐻𝐷₁𝑥,𝑦+𝐷₂𝑥,𝑦+𝐷₁𝑥,𝑦+𝐷₂𝑥,𝑦^𝐻𝜓=𝜓^𝐻𝜓𝜆+𝜆. (2.11)
So, 𝜆+𝜓𝜆=^𝐻𝐷₁𝑥,𝑦+𝐷₂𝑥,𝑦+𝐷₁𝑥,𝑦+𝐷₂𝑥,𝑦^𝐻𝜓𝜓^𝐻𝜓. (2.12)
Because 𝑑_𝑖𝑗=−𝑑_𝑗𝑖(𝑖≠𝑗,forall𝑑_𝑖𝑗∈𝑅) in matrix 𝐷₁(𝑥,𝑦)+𝐷₂(𝑥,𝑦), so 𝜆+𝜓𝜆=^{𝐻2𝑑110⋯002𝑑22⋯000⋯00002𝑑𝑚𝑚}𝜓𝜓^𝐻𝜓. (2.13)
Because 𝑑_𝑖𝑖≤0(forall𝑑_𝑖𝑖∈𝑅), and all 𝑑_𝑖𝑖 are not equal to zero. So, 𝜆+𝜆≤0. (2.14)
From (2.14), we have arg𝜆𝐷₁𝑥,𝑦+𝐷₂𝑥,𝑦≥0.5𝜋>0.5𝑞_𝑟𝜋. (2.15)
According to the stability theorem for nonlinear fractional-order systems of commensurate order [22–25], system (2.8) is asymptotically stable. That is lim_𝑡→+∞‖𝑒‖=0. (2.16)
Therefore, lim_𝑡→+∞‖𝑒‖=lim_𝑡→+∞𝑦−𝑀(𝑥)𝑥=0. (2.17)
This indicates that the modified function projective synchronization between drive system (2.1) and response system (2.2) will be obtained. The proof is completed.

Remark 2.4. Theorem 2.3 indicates that the condition of the MFPS between drive system (2.1) and response system (2.2) are |arg𝜆[𝐷₁(𝑥,𝑦)+𝐷₂(𝑥,𝑦)]|>0.5𝑞_𝑟𝜋. So, in practical applications, we can easily choose the matrix 𝐷₂(𝑥,𝑦) according to the matrix 𝐷₁(𝑥,𝑦). Moreover, in order to reserve all the nonlinear terms in response system or error system, the controller in our work may be complex than the controller reported by [16, 17]. But, all the nonlinear terms in response system or error system are absorbed in [16, 17].

Remark 2.5. Perhaps our result can be extended to the modified function projective synchronization of complex networks of fractional order chaotic systems [26–28] and the complex fractional-order multi scroll chaotic systems [29–31]. But, the modified function projective synchronization for complex networks and complex fractional-order multi-scroll chaotic systems would be much more complex. Further work on this issue is an ongoing research topic in our group.

3. Applications

In this section, to illustrate the effectiveness of the proposed MFPS scheme for different dimension fractional-order chaotic systems. Two groups of examples are considered and their numerical simulations are performed.

3.1. The MFPS between 3-Dimensional Fractional-Order Lorenz System and 4-Dimensional Fractional-Order Hyperchaotic System

The fractional-order Lorenz [3] system is described as follows: 𝐷^𝑞𝑟𝑦₁=10𝑦₂−𝑦₁𝐷^𝑞𝑟𝑦₂=28𝑦₁−𝑦₂−𝑦₁𝑦₃𝐷^𝑞𝑟𝑦₃=𝑦₁𝑦₂−8𝑦₃3. (3.1) The fractional-order Lorenz system exhibits chaotic behavior [3] for 𝑞_𝑟≥0.993. The chaotic attractor for 𝑞_𝑟=0.995 is shown in Figure 1.

(a)

(b)

Figure 1 

Chaotic attractors of the fractional-order Lorenz system (3.1) for 𝑞𝑟=0.995.

Recently, Pan et al. constructed a hyperchaotic system [17]. Its corresponded fractional-order system is described as follows: 𝐷^𝑞𝑑𝑥₁=10𝑥₂−𝑥₁+𝑥₄𝐷^𝑞𝑑𝑥₂=28𝑥₁−𝑥₁𝑥₃𝐷^𝑞𝑑𝑥₃=𝑥₁𝑥₂−8𝑥₃3𝐷^𝑞𝑑𝑥₄=−𝑥₁𝑥₃+1.3𝑥₄. (3.2) The hyperchaotic attractor of system (3.2) for 𝑞_𝑑=0.95 is shown in Figure 2.

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Figure 2 

Hyperchaotic attractors of the fractional-order system (3.2) for 𝑞𝑑=0.95.

Consider the fractional-order hyperchaotic system (3.2) with fractional-order 𝑞_𝑑=0.95 as drive system, and the fractional-order Loren system with fractional-order 𝑞_𝑟=0.995 as response system. According to the above mentioned, we can obtain 𝐹_𝑟𝑦−𝐹_𝑟(𝑀(𝑥)𝑥)=𝐷₁𝑥,𝑦𝑒=−1010028−𝑦₃−1−∑⁴_𝑗=1𝑀_1𝑗𝑥_𝑗𝑦∑₂⁴_𝑗=1𝑀_1𝑗𝑥_𝑗−83𝑒. (3.3)

Now, we can choose 𝐷₂𝑥,𝑦=00−𝑦₂−38+𝑦₃00000. (3.4)

So, 𝐷₁𝑥,𝑦+𝐷₂𝑥,𝑦=−1010−𝑦₂−10−1−∑⁴_𝑗=1𝑀_1𝑗𝑥_𝑗𝑦∑₂⁴_𝑗=1𝑀_1𝑗𝑥_𝑗−83. (3.5)

According to the above theorem, the MFPS between the 3-dimensional fractional-order Lorenz system (3.1) and the 4-dimensional fractional-order hyperchaotic system (3.2) can be achieved. For example, choose 𝑀(𝑥)=^{1+𝑥221+𝑥232𝑥}₁^1𝑥_221𝑥40.5. The corresponding numerical result is shown in Figure 3, in which the initial conditions are 𝑥(0)=(2,1,2,1)^𝑇, and 𝑦(0)=(18,13,13.5)^𝑇, respectively.

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Figure 3 

The modified function projective synchronization errors between the fractional-order Lorenz system (3.1) and the fractional-order hyperchaotic system (3.2).

3.2. The MFPS between 4-Dimensional Fractional-Order Hyperchaotic Lǔ System and 3-Dimensional Fractional-Order Arneodo Chaotic System

In 2002, Lü and Chen reported a new chaotic system [32], which be called Lü chaotic system. The Lü chaotic system is different from the Lorenz and Chen system. Based on Lü chaotic system, the hyperchaotic Lü chaotic system and the fractional-order hyperchaotic Lü system have been constructed recently. The fractional-order hyperchaotic Lü system [16] is described by the following 𝐷^𝑞𝑟𝑦₁=36𝑦₂−𝑦₁+𝑦₄𝐷^𝑞𝑟𝑦₂=20𝑦₂−𝑦₁𝑦₃𝐷^𝑞𝑟𝑦₃=𝑦₁𝑦₂−3𝑦₃𝐷^𝑞𝑟𝑦₄=𝑦₁𝑦₃−𝑦₄. (3.6) The hyperchaotic attractor of system (3.6) for 𝑞_𝑟=0.96 is shown in Figure 4.

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Figure 4 

Hyperchaotic attractors of the fractional-order hyperchaotic Lǔ system (3.6) for 𝑞𝑟=0.96.

The fractional order Arneodo chaotic system [16] is defined as follows: 𝐷^𝑞𝑑𝑥₁=𝑥₂𝐷^𝑞𝑑𝑥₂=𝑥₃𝐷^𝑞𝑑𝑥₃=5.5𝑥₁−3.5𝑥₂−𝑥₃−𝑥³₁. (3.7) The chaotic attractor of system (3.7) for 𝑞_𝑑=0.998 is shown in Figure 5.

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Figure 5 

Chaotic attractors of the fractional-order Arneodo chaotic system (3.7) for 𝑞𝑑=0.998.

Consider the fractional-order Arneodo chaotic system (3.7) with fractional-order 𝑞_𝑑=0.998 as drive system, and the fractional-order hyperchaotic Lǔ system (3.6) with fractional-order 𝑞_𝑟=0.96 as response system. According to the above mentioned, we can yield 𝐹_𝑟𝑦−𝐹_𝑟(𝑀(𝑥)𝑥)=𝐷₁𝑥,𝑦𝑒=−363601−𝑦₃20−∑³_𝑗=1𝑀_1𝑗𝑥_𝑗0𝑦∑₂³_𝑗=1𝑀_1𝑗𝑥_𝑗𝑦−30₃0∑³_𝑗=1𝑀_1𝑗𝑥_𝑗−1𝑒. (3.8)

Now, we can choose 𝐷₂𝑥,𝑦=00−𝑦₂0−36+𝑦₃−2100000−∑³_𝑗=1𝑀_1𝑗𝑥_𝑗−1−𝑦₃000. (3.9)

So, 𝐷₁𝑥,𝑦+𝐷₂𝑥,𝑦=−3636−𝑦₂1−36−1−∑³_𝑗=1𝑀_1𝑗𝑥_𝑗0𝑦∑₂³_𝑗=1𝑀_1𝑗𝑥_𝑗−3−∑³_𝑗=1𝑀_1𝑗𝑥_𝑗−10∑³_𝑗=1𝑀_1𝑗𝑥_𝑗−1. (3.10)

According to above theorem, the MFPS between the 4-dimensional fractional-order hyperchaotic Lǔ system (3.6) and the 3-dimensional fractional-order Arneodo chaotic system (3.7) can be achieved. For example, choose 𝑀(𝑥)=^{1+𝑥20001+𝑥30}_{000.5+𝑥111−𝑥11}. The corresponding numerical result is shown in Figure 6, in which the initial conditions are 𝑥(0)=(2,2,2)^𝑇, and 𝑦(0)=(11,10,11,2)^𝑇, respectively.

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Figure 6 

The modified function projective synchronization errors between the fractional-order system (3.6) and the following fractional-order system:

4. Conclusions

In this paper, based on the stability theory of the fractional-order system and the tracking control, a modified function projective synchronization scheme for different dimension fractional-order chaotic systems is addressed. The derived method in the present paper shows that the modified function projective synchronization between drive system and response system with different dimensions can be achieved. The modified function projective synchronization between 3-dimensional fractional-order Lorenz system and 4-dimensional fractional-order hyperchaotic system, and the modified function projective synchronization between the 4-dimensional fractional-order hyperchaotic Lǔ system, and the 3-dimensional fractional-order Arneodo chaotic system, are chosen to illustrate the proposed method. Numerical experiments shows that the present method works very well, which can be used for other chaotic systems.

Acknowledgments

The authors are very grateful to the anonymous reviewers for their valuable comments and suggestions, which have greatly improved the presentation of this paper. This work is supported by the Foundation of Science and Technology project of Chongqing Education Commission under Grant KJ110525, and by the National Natural Science Foundation of China (61104150).