Synchronization of Fractional-Order Complex Chaotic Systems Based on Observers

By designing a state observer, a new type of synchronization named complex modified projective synchronization is investigated in a class of nonlinear fractional-order complex chaotic systems. Combining stability results of the fractional-order systems and the pole placement method, this paper proves the stability of fractional-order error systems and realizes complex modified projective synchronization. This method is so effective that it can be applied in engineering. Additionally, the proposed synchronization strategy is suitable for all fractional-order chaotic systems, including fractional-order hyper-chaotic systems. Finally, two numerical examples are studied to show the correctness of this new synchronization strategy.


Introduction
The fractional-order complex chaotic systems (FOCCS), as a special kind of nonlinear systems, combine advantages of fractional-order real systems and integer-order complex chaotic systems, and thus have more complex and richer behavior. Furthermore, a broader application of FOCCS has been developed in cryptography and signal processing. Therefore, many scholars have devoted a lot of effort to study FOCCS and have obtained lots of useful results on the dynamic behavior, stabilization, control, and synchronization of FOCCS in recent years. As shown in [1], Gao and Yu employed numerical simulation to study chaotic characteristics of a fractional-order complex duffing oscillator. The chaotic behavior of fractional-order logistic equations with complex variables was discussed in detail in [2]. Subsequently, a large number of FOCCS, including the fractional-order complex Lorenz system [3], complex Chen system [4], complex T system [5], complex Lü system [6], and the fractional-order hyper-chaotic complex Lü system [7], have been found one after another. In the meantime, lots of meritorious results on chaos synchronization of FOCCS have been reported, and various regimes of synchronization have been presented, such as complete synchronization (CS) [3,8,9], anti-synchronization (AS) [6,10], hybrid projective synchronization [11,12], combination synchronization [13,14], combination-combination synchronization [15], etc. For other recent works on this subject, please refer to the previous literature [16][17][18][19][20][21][22][23][24][25].
Complex modified projective synchronization (CMPS) is a new type of complex synchronization based on complex chaotic systems that was proposed almost simultaneously in 2013 by Mahmoud et al. [26] and Zhang et al. [27]. CMPS means that state variables of the master system where q ∈ (m − 1, m), m = [q] + 1, [q] is the integer part of q, Γ represents the Gamma function, and D q * indicates the q-order Caputo differential operator.
In this paper, we always suppose that q is a positive number less than 1 since the fractional-order q often lies in (0, 1) in engineering. For the sake of our synchronization result, we introduced the following stability results for linear fractional differential equations. Given the autonomous system where the state variable y ∈ R n and the initial condition y(0) = y 0 , system (1) has the following results.
where arg(λ l (B)) stands for the argument of the eigenvalue λ l of B. For this case, the component of the state decay converges to 0 as t −q .

Problem Description and Synchronization Scheme
FOCCS are generally described by a set of nonlinear differential equations. Generally speaking, a FOCCS can be divided into two major parts: one is linear and the other is nonlinear. Therefore, we study the following FOCCS: where the state vector z = (z 1 , z 2 , . . . , z n ) T = z r + jz i ∈ C n , z r is the real part of z and z i is the imaginary part of z. Θ ∈ R n×n and Ψ ∈ R n×m are real matrices, Ω ∈ R n×1 (or Ω ∈ C n×1 ), and f = ( f 1 , f 2 , . . . , f m ) T ( f i stands for complex nonlinear function) are column vectors.
In order to realize CMPS, we take system (2) as the master system. Thus, suppose that system (2) has the following output where K ∈ R n×m is a gain matrix. Given an invertible constant matrix Φ = Φ r + jΦ i ∈ C n×n , its inverse matrix can be expressed as Φ −1 . Thus, design the observer of FOCCS (2) as the slave system and define the output in the following form: In the sequel, define the synchronization error function as where Φ is called a complex scaling matrix. Thus, based on systems (2) and (4), the definition of CMPS can be stated as the following.
Next, we investigate the process of CMPS based on a nonlinear state observer. From the error equation (6), we obtain that Thus, taking into account the system (2) and the observer (4), we have

Substituting (5) into the above equation, one can conclude that
Separating real and imaginary parts, we have two real systems as follows: and where Θ − ΨK is the linear time invariant matrix. For the sake of making systems (7) and (8) controllable, we choose the appropriate gain matrix K to satisfy |arg(λ l (Θ − ΨK))| > qπ/2, (l = 1, 2, . . . , n). Thus, on the basis of Lemma 1, we can obtain e r → 0 and e i → 0 as t → ∞, that is, e = e r + je i → 0 as t → ∞. Hence, system (2) and the observer (4) can realize CMPS.

Remark 3.
The eigenvalues of matrix Θ − ΨK are independent of the complex scaling matrix Φ, so the complex scaling matrix Φ does not affect the controllability of the error systems (7) and (8). Therefore, the proposed method can arbitrarily adjust the complex scaling matrix in the synchronization process without worrying about the robustness of other synchronization methods.

Remark 5.
In this paper, we only study the CMPS of FOCCS theoretically but do not study hardware implementation. Recently, there are many papers considering the implementation of the fractional-order operator and fractional-order synchronization scheme [40][41][42][43], which provide good research ideas for the implementation of the CMPS proposed in this paper. Therefore, we will further investigate the hardware implementation of the CMPS of FOCCS in future work.

Numerical Simulations
Next, we respectively study CMPS of the following two pairs of examples to show our proposed theory.

CMPS of the Fractional-Order Complex Lü Systems
The following fractional-order complex Lü system is considered as the master system, which is denoted as where β, γ, δ are real constants, z 1 = m 1 + jm 2 , z 2 = m 3 + jm 4 , and z 3 = m 5 are state variables.

Conclusions
This article studies the observer-based CMPS of FOCCS in detail. On the basis of the assumed output, the authors construct nonlinear state observers to realize CMPS of a large class of FOCCS. In this new synchronization scheme, it is not necessary to calculate the conditional Lyapunov exponents, and it is so effective that it can be applied in engineering. Additionally, the proposed CMPS scheme is suitable for all FOCCS, including fractional-order complex hyperchaotic systems. We respectively achieve CMPS of fractional-order complex chaotic systems: complex Lü systems, and hyper-chaos complex Lü systems. The corresponding simulation results show the correctness of this new synchronization strategy. Since CMPS has a wide application in many fields, we will consider the following two aspects in our future work: one is to extend the obtained results of this paper to other systems including impulsive systems and hybrid systems, and the other is to investigate the hardware implementation of CMPS.

Author Contributions:
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript, writing-review and editing, Z.L., T.X. and C.J.