Parameter identiﬁcation and the multi-switching sliding mode combination synchronization of fractional order non-identical chaotic system under stochastic disturbances

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 (S-D) utilizing the multi-switching synchronization method. The stochastic disturbances are considered as nonlinear uncertainties and external disturbances. Our theoretical part is divided into two cases, namely, the dimension of the drive-response system are diﬀerent (or same). Firstly, a FO sliding surface was established in term of fractional calculus. Secondly, depended on the FO Lyapunov stability theory, the adaptive control technology and sliding mode control technique, the multi-switching adaptive controllers (MSAC) and some suitable multi-switching adaptive updating laws (M-SAUL) are designed, so 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 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 demonstrated 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 the fields of mathematics and control have carried out a series of researches on the control and synchronization of chaotic systems. So far, some effective synchronization control methods have been proposed, such as drive-response synchronization [1], projection synchronization [2,3], adaptive fuzzy control [4][5][6], neural network synchronization [7,8], feedback synchronization [9], pulse synchronization [10,11], sliding mode control [12,13] etc. People apply chaotic synchronization to secure communication, signal processing and life sciences, etc which has attracted great attention. 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 [29][30][31][32], compound synchronization [33][34][35] and double compound synchronization [36,37], etc. It can be said to be a major breakthrough in the field of chaos synchronization. A major advantage of these new synchronization schemes is that they can ensure the security of information. However, as the transmission of signals become more and more complex, how to strengthen the security of information is a thought-provoking question.
In recent years, a new multi-switching combination synchronization (MSCS) scheme was analyzed by Vincent U et al [38] 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 the secure communication according to chaotic synchronization. The topic of dual combination-combination multi-switching synchronization in terms of eight chaotic systems was solved in [39]. The global exponential MSCS was introduced in terms of three different chaotic systems [40]. Reference [41] solved the problem of M-SCS between three different integer-order chaotic systems. Adopting adaptive control technology, reference [42] investigated the multi-switching combination-combination synchronization of four integer-order chaotic systems which parameters are fully unknown. An further work of [42] has been developed by [43], which is indicated as integer-order time-delay chaotic system. Shafiq M et al [44] proposed a robust adaptive multi-switching technology to solve the issue of anti-synchronization for unknown hyper-chaotic systems under SD. The authors of references [45] and [46] considered the multi-switching synchronization of the single-drive and single-response system which the parameters are unknown. Their innovations lie in the order of the drive-response system is different in [45] and the dimension is different in [46]. On the contrary, Chen et al [47] 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 [48] take multi-switching scheme into account in terms of multiple chaotic systems, it does not consider the influence of unknown parameters and SD.
Hence, 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) dimension 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 disturbance. In the light of Lyapunov stability theory, adaptive control technology and sliding mode control technique, we introduce two multi-switching adaptive controllers (MSAC) and multi-switching adaptive updating laws (MSAUL) to realise the multiswitching synchronization of D-R systems and assess the unknown parameters. There are two innovations points in this article. 1. Considering the combination D-R systems with unknown parameters and double SD, the designed MSAC can make the state variables of drive systems are synchronized with the different state variables of the response system. Simultaneously, the designed MSAUL can identify the unknown parameters accurately and bounded estimates are made for uncertainties and external disturbances. 2. When the dimension of D-R systems are different (or same), scale matrices are chose as non-diagonal (or diagonal) matrices. If we adopt suitable scale matrices, the multiswitching projection synchronization, multi-switching complete synchronization, and multi-switching anti-synchronization will become the special cases of MSSMCS. Finally, numerical simulations via MATLAB demonstrate the multi-switching controllers we conducted have good robustness and anti-interference performance and this method we investigated can make the multi-switching error systems quickly converges to the equilibrium point. The rest of the paper is described as follows. In Section 2, some definitions, lemmas that need to be used are introduced. In Section 3, the problem statement are given. In Section 4, the MSAC and MSAUL are designed about D-R systems with the same dimension . In Section 5, the MSAC and MSAUL are designed about D-R systems with the different dimension. 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. [49] The mathematical expression of the fractional integral of the function f (t) is following where Γ(α) indicates the Gamma function.
[18] When x(t) ∈ R n has continuous first derivative, then where α ∈ (0, 1) and Q ∈ R n × R n indicates a positive definite matrix.
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 parameter θ, β, ϑ.
then the drive systems (4), (5) and response system (6) can be reach combination synchronization.
Remark 2. We can redefined the error system (7) as where i, j, k, p ∈ (1, 2, · · · , n) represent p th error component of e, i th components of z, j th components of y, and k th components of x respectively, l, w, v represent 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 then the drive systems (4), (5) and the response systems (6) realize the MSCS, where Remark 3. The matrices A ∈ R n×m , B ∈ R n×m , C ∈ R n×n , C = 0 are hailed as the scaling matrices. Furthermore, they can be either constant matrices or the functions with regard to state variables x, y, z. 4 The synchronization of multi-switching FO chaotic system with same dimensions In Section 4, the MSSMCS of FO chaotic systems with same dimension is formulated which means the dimension of D-R systems (4), (5) and (6) 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 orbit 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 system which are proved in Theorem 1.
Assumption 1. Assumed that 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.
Remark 7. The positive constants (ijk) r p , (ijk) q p are unknown. (ijk)r p , (ijk)q p are used to represent the estimation of parameter (ijk) r p , (ijk) q p .
Proof. Adopting the Lyapunov function as: Taking the α derivative Then substituting the Eq. (15) and the MSAUL (16) into Eq. (18), we obtain Thus Which establishes that the 0 D α t V (t, x(t)) is negative definite. Thus, it follows from the FO Lyapunov stability theory that lim t→∞ (ijk) e p = 0 is achieved, i.e. we can say that the MSSMCS of the drive systems (4), (5) and response system (6)is accomplished in terms of m = n.

5
The synchronization of multi-switching FO system with different dimensions In Section 5, the MSSMCS of FO chaotic systems with different dimension is formulated which means the dimension of D-R systems (4), (5) and (6) satisfies 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 system which are proved in Theorem 2.
When we choose m = n and the non-diagonal matrices A ∈ R n×m , B ∈ R n×m , C ∈ R n×n , C = 0, the form of the error system (9) can be explained as: Assumption 2. Assumed that 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 where p = 1, 2, · · · , n.
According to the definition of the error vector (22), we get the FO error system as The errors of unknown parameters θ, β, ϑ have been defined in (12). For convenience, we define error of unknown constants (lwv) Thus, the sliding mode surface is designed as (lwv) s p = (lwv) λ p ( (lwv) e p ). We can get the following multi-switching adaptive controller (MSAC) (25) multi-switching adaptive updating laws (MSAUL) (28): where (lwv) λ p is a constant. Substituting (25) into Eq. (24), we obtain Thus, it follows from (lwv) s p = (lwv) λ p ( (lwv) e p ) and Eq. (26) that we obtain the following summation result.
Proof. Adopting the Lyapunov function as: Taking the α derivative Then substituting the (27) and MSAUL (28) into Eq. (30), we obtain Since Thus Which establishes that the 0 D α t V (t, x(t)) is negative definite. Thus, it follows from the FO Lyapunov stability theory that lim t→∞ (lwv) e p = 0 is achieved. On the other hand, we can say that the MSSMCS of the drive systems (4), (5) and response system (6) is accomplished in terms of m = n.
The following corollaries are successfully analyzed from Theorem 2 and their proofs are omitted here. By the way, the Theorem 1 has the same theory, we are not going to describe. Corollary 1. If the matrices A = 0, B = 0, C = 0, then the drive systems (4) achieve the MSSMCS with the response systems (6) provided the following controller, And the adaptive updating laws, where the explanation of γ, ̟ can be seen (28).
Corollary 2. If the matrices A = 0, B = 0, C = 0, then the drive systems (5) achieve the MSSMCS with the response systems (6) provided the following controller, . And the adaptive updating laws, where the explanation of ξ, ̟ can be seen (28).  (6) is asymptotically stable provided the following controller, , And the adaptive updating laws, where the explanation of ̟ can be seen (28).

Numerical simulation
This section mainly demonstrates the reliability and validity of the suggested multi-switching sliding mode combination synchronization scheme. For the D-R systems (4), (5) and (6) with same dimension, we selected two error states to elaborate the method, namely, i = j = k and i = j = k. For the D-R systems (4), (5) and (6) with different dimensions, we selected l = w = v and l = w = v. In each case, we have given the specific form of controller and parameter adapting rate via the specific FO hyper-chaotic or chaotic systems. .

Switch-1
It follows from switch-1 (37) that the FO error dynamic system is expressed as: It follows from the form of MSAC (13) and MSAUL (16) that the controller is designed as follows: and the parameters updating laws are designed as follows:

Switch-2
It follows from switch-2 (38) that the FO error dynamic system is expressed as: It follows from the form of MSAC (13) and MSAUL (16) that the controller is designed as follows: and the parameters updating laws are designed as follows: results, it can reveal that the drive systems (4), (5) and response system (6) achieve MSSMCS. Therefore, the multi-switching adaptive controllers (MSAC) (13) and some suitable multi-switching adaptive updating laws (MSAUL) (16) are effective.

Conclusion
In this article, we investigated the multi-switching sliding mode combination synchronization (MSSMCS) of fractional order (FO) non-identical chaotic systems with unknown parameters under double stochastic disturbances (SD). In the theoretical parts, the FO chaotic systems with the different (or same )dimension are considered. Our idea for this topic is that with the help of the Lyapunov theory, adaptive control technology and sliding mode control technique, we put forward the fractional order sliding surface, multi-switching adaptive controllers (MSAC) and multi-switching adaptive updating laws (MSAUL) which can achieve the 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 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 converges to the equilibrium point. Therefore, the multi-switching adaptive controllers (MSAC) are effective and have strong robustness. 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.

Declarations
Funding: This work is partly supported by Sichuan Youth Science and Technology Foundation (Grant No. 2019YJ0456), Fund of Sichuan University of Science and Engineering (Grant No. 2020RC26, 2020RC42).
Competing interests: The authors declare that they have no competing interests.
Availability of data and material: The data used to support the findings of this study are available from the corresponding author upon request.