Multi-Switching Combination Synchronization of Three Fractional-Order Delayed Systems

: Multi-switching combination synchronization of three fractional-order delayed systems is investigated. This is a generalization of previous multi-switching combination synchronization of fractional-order systems by introducing time-delays. Based on the stability theory of linear fractional-order systems with multiple time-delays, we propose appropriate controllers to obtain multi-switching combination synchronization of three non-identical fractional-order delayed systems. In addition, the results of our numerical simulations show that they are in accordance with the theoretical analysis.


Introduction
Fractional calculus has attracted researchers from various fields due to fractional dimensions widely existing in nature and engineering fields [1][2][3]. Compared to the integer-order dynamical systems, the fractional-order counterparts can exhibit more complex dynamical behaviors. Some of the researches on integer-order dynamical systems can be generalized to fractional-order dynamical systems. Fractional-order dynamical systems have been widely investigated, such as synchronization [4], identification [5], stabilization [6] and approximate entropy analysis [7,8]. Time-delay is a frequently encountered phenomenon in real applications, such as physical, communication, economical, pneumatic and biological systems [9]. Introducing time-delay into a system can enrich its dynamic characteristics and describe a real-life phenomenon more precisely. Thus, the fractional-order delayed differential equation (FDDE) is becoming a hot topic for scientists and engineers, and it has many theoretical and practical applications [10]. Nowadays, the chaotic behavior and synchronization of FDDE attract intensive research interests. In [11], Bhalekar et al. introduced the fractional-order delayed Liu system. The fractional-order delayed financial system was presented in [12], and hybrid projective synchronization between the aforementioned two systems was achieved in [13]. The fractional-order delayed Chen system was proposed in [14], while its adaptive synchronization was investigated in [15]. The fractional-order delayed porous media was proposed in [16]. In [17], a fractional-order delayed Newton-Leipnik system was taken as an example to present intermittent synchronizing delayed fractional nonlinear system.
Due to its wide applications in secure communication, synchronization of fractional-order delayed chaotic systems are extensively investigated [18][19][20][21]. However, all the above-mentioned and other synchronization schemes are in traditional drive-response ways, which only have unique drive and response systems. Recently, Luo et al. [22] extended the traditional drive-response synchronization to combination synchronization, which has two drive systems and one response system. Compared to the drive-response synchronization, combination synchronization has stronger

Preliminaries
Fractional calculus is a generalization of integration and differentiation to non-integer order fundamental operator a D r t , which is defined as There are several different definitions for the fractional-order differential operator [34]. Because the Caputo definition is easy to understand and is frequently used in the literature, we apply this definition in this paper, which is where 1 < r < n.
. Let φ(t) ∈ R be a continuous and derivable function. Then, for any time instant t ≥ t 0 Given the following n-dimensional linear fractional-order system with multiple time-delays [36]: where q i ∈ (0, 1) is the fractional-derivative order, y i (t) is the state, and τ ij > 0 is the time-delay, the initial value Performing Laplace transform on the system in Equation (6) yields where Y(s) = (Y 1 (s), Y 2 (s), ..., Y n (s)) T is the Laplace transform of y(t) = (y 1 (t), y 2 (t), ..., y n (t)) T , b(s) = (b 1 (s), b 2 (s), ..., b n (s)) T is the remaining non-linear part, the characteristic matrix of the system in Equation (6) is Theorem 1 ([36]). If all the roots of the characteristic equation det(∆(s)) = 0 have negative real parts, then the zero solution of the system in Equation (6) is Lyapunov globally asymptotically stable.

Multi-Switching Combination Synchronization Scheme
Multi-switching combination synchronization among three non-identical fractional-order delayed systems is investigated in this section.
Define the error state as e klm = f k z k − g l x l − h m y m (k, l, m = 1, ..., n). Then, we have the error state vector where e(t) is the vector form of e klm , F = diag{ f 1 , f 2 , . . . , f n } ∈ R n×n , G = diag{g 1 , g 2 , . . . , g n } ∈ R n×n and H = diag{h 1 , h 2 , . . . , h n } ∈ R n×n are real scaling matrix. Accordingly,

Remark 1.
If k = l = m, the systems in Equations (9) and (10) and the system in Equation (11) are defined to be combination synchronization [22].

Remark 2.
If the scaling matrix F = 0, G = 0 or H = 0, the multi-switching combination synchronization mentioned above is simplified to multi-switching hybrid projective synchronization.
From the systems in Equations (9)-(11), we have the error system as follows To achieve multi-switching combination synchronization among the above systems, a non-linear controller is constructed: Substituting the systems in Equations (9)-(11) and (15) into the system in Equation (14), we have Thus, the multi-switching combination synchronization between the systems in Equations (9) and (10) and the system in Equation (11) is changed into the analysis of the asymptotical stability of the system in Equation (16).
In light of Corollary 1, a sufficient condition to achieve multi-switching combination synchronization between the systems in Equations (9) and (10) and the system in Equation (11) is obtained as follows. (9) and (10) and the system in Equation (11) can be achieved if there exists a matrix K = diag{k 1 , k 2 , . . . , k n } in the system in Equation (16) such that k i < −1/ sin(απ/2), (i = 1, 2, . . . , n).

Proposition 1. Multi-switching combination synchronization between the systems in Equations
Proof. For the system in Equation (16) Performing Laplace transform on the system in Equation (16) yields where has a root s = wi = |w| (cos(π/2) + i sin(±π/2)). Thus, From the above equation, we can get Hence, Since k i < −1/ sin(απ/2), α ∈ (0, 1), the discriminant of the roots satisfies Then, Equation (22) has no real solutions, and Equation (18) has no purely imaginary roots. In light of Corollary 1, the zero solution of the system in Equation (16) is globally asymptotically stable, i.e., multi-switching combination synchronization is obtained between the systems in Equations (9) and (10) and the system in Equation (11).

Numerical Examples
Numerical simulations were carried out to illustrate the above proposed multi-switching combination synchronization scheme. We used the same systems as in [32] with time-delays, which are fractional-order delayed Lorenz, Chen, and Rössler systems, and the numerical simulations were carried out in MATLAB.
The fractional-order delayed Lorenz system [37] was considered as the first drive system The system in Equation (24) exhibits a chaotic attractor, as illustrated in Figure 1. The system in Equation (24) can be rewritten as where The fractional-order delayed Chen system [14] is the second drive system The system in Equation (27) displays a chaotic attractor, as shown in Figure 2. We rewrite the system in Equation (27) as D α y(t) = y(t) + y(t − τ) + B(y(t), y(t − τ)), where The fractional-order delayed Rössler system is the response system, given by where U 1 , U 2 and U 3 are determined afterwards. Without the controllers, the system in Equation (30) exhibits a chaotic attractor, as illustrated in Figure 3. The system in Equation (30) is rewritten as where For the systems in Equations (24), (27) and (30)   We randomly pick two cases and In the following, we analyze these two cases in detail.

Case 1
From the systems in Equations (24), (27) and (30), we have the error dynamical system such that Substituting the systems in Equations (24), (27) and (30) into the system in Equation (35) yields Here, we obtain the following results. (24) and (27) and the system in Equation (30) can be achieved with the following controllers

Theorem 2. Multi-switching combination synchronization between the systems in Equations
Supposing F = 0 and G = 0 or H = 0, we have the following results.

Case 2
From the systems in Equations (24), (27) and (30), we have Substituting the systems in Equations (24), (27), and (30) into the system in Equation (42) yields: Here, we have the following similar results. (24) and (27) and the system in Equation (30) can be achieved with the following controllers
The feedback gain matrix K is an important factor to affect the convergence of the error systems. With the increase of the absolute value of k i , the convergence time will be shortened. Thus, we carried out one more simulation with K = diag{−40, −40, −40}. Figures 8 and 9 illustrate the synchronization errors for Cases 1 and 2, respectively. By comparing Figures 4 and 8, as well as Figures 6 and 9, it is easy to see that convergence times are shortened obviously.

Conclusions
We extended previous work [32] to investigate multi-switching combination synchronization among three non-identical fractional-order delayed systems by introducing time-delays. Based on the stability theory for linear fractional-order systems with multiple time-delays, we designed appropriate controllers to obtain multi-switching combination synchronization among three non-identical fractional-order delayed systems. The simulations are in accordance with the theoretical analysis.
On the one hand, when applying multi-switching combination synchronization of fractional-order delayed chaotic systems in secure communications, fractional-order and time-delay can enrich systems' dynamics. On the other hand, the origin information can be separated into two parts and embedded different parts in separate drive systems via combination synchronization scheme. Besides, because the switched states are unpredictable, this synchronization scheme can increase the security of the transmitted information in secure communication. Thus, the communication security will be enhanced, which makes multi-switching combination synchronization of fractional-order delayed chaotic systems able to find better applications in security communication.