The Multi-Switching Sliding Mode Combination Synchronization of Fractional Order Non-Identical Chaotic System with Stochastic Disturbances and Unknown Parameters

: This paper deals with the issue of the multi-switching sliding mode combination synchronization (MSSMCS) of fractional order (FO) chaotic systems with different structures and unknown parameters under double stochastic disturbances (SD) utilizing the multi-switching synchronization method. The stochastic disturbances are considered as nonlinear uncertainties and external disturbances. Our theoretical part considers that the drive-response systems have the same or different dimensions. Firstly, a FO sliding surface is established in terms of the fractional calculus. Secondly, depending on the FO Lyapunov stability theory and the sliding mode control technique, the multi-switching adaptive controllers (MSAC) and some suitable multi-switching adaptive updating laws (MSAUL) are designed. They can ensure that the state variables of the drive systems are synchronized with the different state variables of the response systems. Simultaneously, the unknown parameters are assessed, and the upper bound values of stochastic disturbances are examined. Selecting the suitable scale matrices, the multi-switching projection synchronization, multi-switching complete synchronization, and multi-switching anti-synchronization will become special cases of MSSMCS. Finally, examples are displayed to certify the usefulness and validity of the scheme via MATLAB.


Introduction
Chaos is an inherent characteristic of nonlinear dynamic systems and a common phenomenon in real life. The chaotic phenomenon exhibited by the chaotic system is uncertain, unrepeatable, and unpredictable. Therefore, experts in mathematics and control have carried out a series of studies on the control and synchronization of chaotic systems. So far, some effective synchronization methods have been proposed, such as drive-response synchronization [1], projection synchronization [2,3], adaptive fuzzy synchronization [4][5][6], neural network synchronization [7,8], feedback synchronization [9], pulse synchronization [10,11], sliding mode synchronization [12,13], etc. People apply chaotic synchronization to secure communication, signal processing and life sciences, etc. Thus, chaos synchronization has gradually become a core research topic in the field of control science. Because chaotic systems are extremely sensitive to initial values, they are often subject to some SD. Whether it is the uncertainties within the system, such as parameter uncertainties, nonlinear uncertainties, or external disturbances, they cause an effect on the stability of the system.
In the beginning, these synchronization methods are used by people to research the synchronization of single-drive system and single-response system [14][15][16][17][18][19]. Gradually, some scholars have considered the influence of SD on this basis [20][21][22][23][24][25][26][27]. Since Runzi et al. [28] revealed the combination synchronization scheme, the synchronization of multi-drive and multi-response systems, multi-drive and single-response systems, single-drive and multi-response systems are suggested. Later, some new synchronization schemes appeared and have been developed by leaps and bounds, such as the combination-combination synchronization [29][30][31][32], compound synchronization [33][34][35] and double compound synchronization [36,37], etc. It will be a major breakthrough in chaos synchronization. A major advantage of these new synchronization schemes is that they can ensure the information security. However, with the increasing complexity of signal transmission, how to strengthen information security is a thought-provoking problem.
In recent years, a new multi-switching combination synchronization (MSCS) scheme was analyzed by Vincent U et al. [30], which means the state variables of the drive systems are synchronized with the different state variables of the response systems, breaking the conventional synchronization rules. Compared with the traditional synchronization or some extension of it, the MSCS scheme is very promising because it can provide greater security for secure communication. The topic of dual combination-combination multi-switching synchronization in terms of eight chaotic systems was solved in [38]. The global exponential MSCS was introduced in terms of three different chaotic systems [39]. Reference [40] solved the problem of MSCS between three different integer-order chaotic systems. Adopting adaptive control technology, reference [41] investigated the multi-switching combinationcombination synchronization of four integer-order chaotic systems which parameters are fully unknown. An further work of [41] has been developed by [42], which is indicated as an integer-order time-delay chaotic system. Shafiq M et al. [43] proposed a robust adaptive multi-switching technology to solve the issue of anti-synchronization for unknown hyperchaotic systems under SD. The authors of references [44,45] considered the multi-switching synchronization of the single-drive and single-response system which the parameters are unknown. Their innovations lie in the orders of the drive-response systems are different in [44] and the dimensions are different in [45]. On the contrary, Chen et al. [46] considered the synchronization of multiple chaotic systems with unknown parameters and disturbances. It's a pity that the case of multi-switching is not considered. Although reference [47] took multi-switching scheme into account in terms of multiple chaotic systems, it does not consider the influence of unknown parameters and SD.
As we all know, sliding mode control is a useful tool for the fractional (or integer) order systems due to its strong robustness for the SD. Therefore, some scholars prefer to use sliding mode control technology to research the uncertain chaotic systems, such as, in [48], Modiri et al. have considered the fractional order uncertain chaotic systems and designed a fractional-order adaptive terminal sliding mode controller to estimate the upper bounds of stochastic disturbances. Sun et al. [49] have researched the synchronization of fractionalorder chaotic systems with non-identical orders, unknown parameters and disturbances via sliding mode control. The issue of synchronization of a class of chaotic systems with disturbances and unknown parameters are focused where the disturbances are supposed as bounds [50]. In addition, some scholars proposes the fractional-order sliding mode control based on the disturbance observer to study the fractional order chaotic systems with SD [51][52][53][54][55]. It is a good idea to study the multi-switch synchronization of fractional-order chaotic systems under stochastic disturbance based on the sliding mode control.
Hence, in order to address this limitation, we plan to solve the multi-switching sliding mode combination synchronization (MSSMCS) of fractional order (FO) chaotic systems with different (or same) dimensions under double stochastic disturbances (SD). Meanwhile, the parameters of two drive systems and one response system are fully unknown. The double SD are considered as nonlinear uncertainties and external disturbances. In the light of Lyapunov stability theory and sliding mode control technique, we introduce two multiswitching adaptive controllers (MSAC) and some multi-switching adaptive updating laws (MSAUL) to realize the multi-switching synchronization of D-R systems and assess the unknown parameters. Numerical simulations via MATLAB demonstrate the multi-switching controllers we conducted have good robustness and anti-interference performance. There are three innovations points in this article. (a) On the basis of reference [45], the fractionalorder chaotic system is considered. The combination synchronization has been extended to two drive systems and one response system. (b) The multi-switching sliding mode combination synchronization (MSSMCS) of fractional order (FO) chaotic systems with different dimensions under double stochastic disturbances (SD) is investigated for the first time. (c) Several existing synchronization schemes (projection synchronization, complete synchronization and anti-synchronization) are obtained as special cases of MSSMCS.
The rest of the paper is described as follows. In Section 2, some definitions and lemmas are introduced. In Section 3, the problem statements are given. In Section 4, the MSAC and MSAUL are designed for D-R systems with the same dimension. In Section 5, the MSAC and MSAUL are designed for D-R systems with different dimensions. In Section 6, the numerical simulations conducted that our method is effective and dependable. In Section 7, there is a conclusion.

Preliminaries
The fractional calculus is an ancient and "fresh" concept. As early as the establishment of integer calculus, some scholars began to consider its meaning. Up till the present moment, there are some commonly used types of fractional derivatives, such as the Riemann-Liouville (R-L), Caputo and Grunwald-Letnikov (G-L) derivative, etc. Among these definitions, Caputo's derivative definition is the most generally utilized. Definition 1 ([56]). The mathematical expression of the fractional integral of the function f (t) is following where Γ(α) indicates the Gamma function.

Remark 1.
It is worth noting that x(t), y(t) ∈ R m , z(t) ∈ R n , m = n or m = n; and the parameters of the D-R systems are unknown. We useθ,β,θ to represent the estimation of parameters θ, β, ϑ.
then the drive systems (6), (7) and response system (8) can be reach combination synchronization.

Remark 2.
If A = (a vk ) n×m , B = (b wj ) n×m , C = (c li ) n×n , C = 0, we can redefined the error system (9) as where p, i, j, k, l, w, v ∈ (1, 2, · · · , n). The subscript (p) represents p th error component of e; the (ijk) represents i th components of z, j th components of y, and k th components of x, respectively; the superscript (lwv) represents l th row of matrix C, w th row of matrix B, and v th row of matrix A, respectively ( l, w, v means the switching mode). Suppose that l = w = v, then the error variables are expressed in the form of Definition 3; if l = w = v, definition 3 will no longer apply.

Definition 4.
We redefine the error state in Definition 3 as

The Synchronization of Multi-Switching FO Chaotic System with Same Dimension
In Section 4, the MSSMCS of FO chaotic systems with same dimension is formulated. It means that the dimension of D-R systems (6)-(8) satisfies m = n. Thus, the scaling matrices A, B, C are given as diagonal matrices. Firstly, we know that even if a chaotic system is slightly disturbed, its state orbits will change drastically over time. Therefore, it is crucial to suppose them as bounded. Then, we designed appropriate multi-switching adaptive controllers (MSAC) and some multi-switching adaptive updating laws (MSAUL) to realize the synchronization of the D-R systems, which are proved in Theorem 2.
When we choose m = n and the diagonal matrices A = diag(a 11 , a 22 , · · · , a nn ), B = diag(b 11 , b 22 , · · · , b nn ), C = diag(c 11 , c 22 , · · · , c nn ), the error system (11) can be described as where i, j, k, p ∈ (1, 2, · · · , n), i = j = k or i = j = k or i = j = k or i = j = k. The subscript (p) represents p th error component of e; the superscript (ijk) represents i th components of z, j th components of y, and k th components of x.

Assumption 1.
Assume the external disturbances d k , D j , µ i , uncertain nonlinear vectors ∆ f k , ∆g j , ∆h i all have a bounded norm. Namely, there are suitable positive constants (ijk) r p , (ijk) q p that satisfy where p = 1, 2, · · · , n, i = j = k or i = j = k or i = j = k or i = j = k.

Remark 8.
The positive constants (ijk) r p , (ijk) q p are unknown. (ijk)r p , (ijk)q p are used to represent the estimation of parameters (ijk) r p , (ijk) q p .
Proof. Adopting the Lyapunov function as: Taking the α derivative Then substituting the Equation (17) and the MSAUL (18) into Equation (20), we obtain Using the fact is negative semi-definite. Based on Barbalat's lemma [58], lim t→∞ 0 D α t V(t, x(t)) = 0 is obtained. We have lim t→∞ s = 0. Then, the trajectory of the error system is driven onto the predefined sliding surface, i.e., we can say that the MSSMCS of the drive systems (6), (7) and response system (8) is accomplished in terms of m = n.

The Synchronization of Multi-Switching FO System with Different Dimensions
In Section 5, the MSSMCS of FO chaotic systems with different dimensions is formulated. It means that the dimensions of D-R systems (6)-(8) satisfy m = n. Thus, the scaling matrices A, B, C are given as non-diagonal matrices. We designed appropriate multi-switching adaptive controllers (MSAC) and some multi-switching adaptive updating laws (MSAUL) to realize the synchronization of the D-R systems which are proved in Theorem 3.
When we choose m = n and the non-diagonal matrices A = (a vk ) n×m , B = (b wj ) n×m , C = (c li ) n×n , C = 0, the form of the error system can be explained as (11), namely: where the meaning of p, i, j, k, l, w, v can be seen (10), where p = 1, 2, · · · , n.
According to the definition of the error vector (11), we get the FO error system as The errors of unknown parameters θ, β, ϑ have been defined in (14). For convenience, we define error of unknown constants (lwv) Thus, the sliding mode surface is designed as (lwv) s p = λ{ (lwv) e p }. We can get the following multi-switching adaptive controller (MSAC) (25) and multi-switching adaptive updating laws (MSAUL) (28): where λ, k 1 are constants. Substituting (25) into Equation (24), we obtain Therefore, a column vector representing the general form of the error system (26), whose elements are chosen arbitrarily form (lwv) e (i.e., when l = 2, w = 3, v = 1, an error mode is generated (231) e), can be obtained the following form: The MSAUL with regard to unknown parameters θ, β, ϑ, (lwv) ρ p , (lwv) p are selected as where m 1 , m 2 , ϕ 1 , ϕ 2 , ϕ 3 , w are positive constants and a column vector representing the general form of sliding mode surface (ijk) s p = λ{ (ijk) e p } can be obtained as s = λe, p = 1, 2, · · · , n.
Proof. Adopting the Lyapunov function as: Taking the α derivative Substituting the (27) and MSAUL (28) into Equation (30), then, the rest of the proof process is similar to Theorem 2. Finally, one can obtain 0 D α x(t)) = 0 is obtained. We have lim t→∞ s = 0. Then, the trajectory of the error system is driven onto the predefined sliding surface, i.e., we can say that the MSSMCS of the drive systems (6), (7) and response system (8) is accomplished in terms of m = n.
The following corollaries are successfully analyzed from Theorem 3 and their proofs are omitted here. By the way, the Theorem 2 has the same theory, we are not going to describe it. Corollary 1. If the matrices A = 0, B = 0, C = 0, then the drive system (6) achieves the MSSMCS with the response system (8) providing the following controller, In addition to the adaptive updating laws, If the matrices A = 0, B = 0, C = 0, then the drive system (7) achieves the MSSMCS with the response system (8) providing the following controller, In addition to the adaptive updating laws,  (8) is asymptotically stable provided the following controller, In addition to the adaptive updating laws,

Numerical Simulation
This section mainly demonstrates the reliability and validity of the suggested multiswitching sliding mode combination synchronization scheme. For the D-R systems (6)- (8) with the same dimensions, we selected two error states to elaborate the method, namely, i = j = k and i = j = k. For the D-R systems (6)-(8) with different dimensions, we selected l = w = v and l = w = v. In each case, we give the specific forms of controllers and parameter adapting laws via the specific FO hyper-chaotic or chaotic systems.

Switch-2
It follows from Switch-2 (35) that the FO error dynamic systems are expressed as: It follows from the forms of MSAC (15) and MSAUL (18) that the controllers are designed as follows: and the parameters updating laws are designed as follows: In this two numerical simulations, we adopt the matrices A, B, C as identity matrices.
q 36 shown in sub-pictures (a,b).

Conclusions
In this article, we investigated the multi-switching sliding mode combination synchronization (MSSMCS) of fractional order (FO) non-identical chaotic systems with unknown parameters and double stochastic disturbances (SD). In the theoretical parts, the FO chaotic systems with the different (or same) dimensions are considered. Our idea for this topic is that with the help of the Lyapunov theory and sliding mode control technique, we put forward a fractional order sliding surface, multi-switching adaptive controllers (MSAC) and multi-switching adaptive updating laws (MSAUL) that can achieve the state variables of the drive systems are synchronized with the different state variables of the response systems. Meanwhile, the unknown parameters are identified and upper bound values of stochastic disturbances are examined accurately. What's more, the combination drive systems and single response system we chose are very general. The different description of the scale matrices can make the multi-switching projection synchronization, multi-switching complete synchronization, multi-switching anti-synchronization, etc., become the special cases of MSSMCS. Motivated by the numerical simulation results, it is clear that the different error variables quickly converge to the equilibrium point. Therefore, the multi-switching adaptive controllers (MSAC) are effective and robust. Next, we will concentrate on the fractional order multi-switching synchronization of time-delay systems under multiple stochastic disturbances, and the parameters of system are still unknown.

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Data Availability Statement:
The data used to support the findings of this study are available from the corresponding author upon request.