Hybrid projective combination–combination synchronization in non-identical hyperchaotic systems using adaptive control

In this paper, we investigate a hybrid projective combination–combination synchronization scheme among four non-identical hyperchaotic systems via adaptive control method. Based on Lyapunov stability theory, the considered approach identifies the unknown parameters and determines the asymptotic stability globally. It is observed that various synchronization techniques, for instance, chaos control problem, combination synchronization, projective synchronization, etc. turn into particular cases of combination–combination synchronization. The proposed scheme is applicable to secure communication and information processing. Finally, numerical simulations are performed to demonstrate the effectivity and correctness of the considered technique by using MATLAB.

A hyperchaotic (HC) system is characterized as a chaotic system having more than two positive Lyapunov exponents. Rossler [35] introduced the first classical HC system in the year 1979. Over the past few decades, a variety of classical HC systems have been emerged and investigated, for example, Lorenz system, Liu system, Chen system, Lu system, Cai system, Pehlivan system, Tigan system, Nikolov system, Sundarapandian system, Vaidyanathan system, etc.
Chaos synchronization among chaotic systems using adaptive control method (ACM) was introduced by Hubler [36] in 1989 for the first time. Synchronization of Rossler like system and Chua's circuit has been studied by using ACM in Ref. [37]. Also, adaptive control technique is studied to synchronize modified Chua's circuit system in Ref. [38]. Mainieri and Rehaceh firstly [39] formulated the notion of projective synchronization while synchronizing chaotic systems in 1999. In Ref. [1], projective synchronization and chaos in secure communication is studied. Further, in Ref. [40], complex projective synchronization between complex chaotic systems is discussed. Sun et al. [26] first proposed the C-C synchronization scheme while synchronizing four identical or non-identical chaotic systems via active control method in 2012. A detailed study of C-C synchronization between four non-identical chaotic systems with uncertain parameters using sliding mode control is described in Ref. [41]. Further in Ref. [42], C-C synchronization among four complex nonlinear chaotic systems is studied and also particular cases, for instance, projective synchronization and combination synchronization, are mentioned. Moreover, a generalized methodology of C-C synchronization between n-dimensional chaotic fractional order nonlinear dynamical systems is developed in Ref. [43]. Here, the scheme discussed in Ref. [26] is extended to C-C synchronization between four n-dimensional fractional order nonlinear chaotic systems. Also, complex C-C synchronization among four newly constructed complex chaotic systems is discussed in Ref. [44]. In Ref. [45], chaos analysis and C-C synchronization of new HC systems having no equilibria are studied. Furthermore, C-C synchronization is generalized to n-dimensional time-delay nonlinear chaotic systems using robust adaptive sliding mode control in Ref. [46]. Also, in Ref. [47], C-C anti-synchronization between four identical fractional order HC systems is discussed. Further, C-C phase synchronization between different fractional order complex nonlinear chaotic systems via active control is studied in Ref. [48]. Also, finite-time combination synchronization among three non-identical chaotic systems is discussed in Ref. [49]. Moreover, combination synchronization between three different order chaotic systems via active backstepping technique is studied in [50].
Considering the aforementioned studies about C-C synchronization, our paper aims to investigate a hybrid projective combination-combination synchronization (HPCCS) among four non-identical HC systems via ACM. Moreover, based on (LST), we formulate an adaptive control law (ACL) with an estimated parameter update law. Our approach has potentiality and is advantageous. If chaos synchronization can be controlled efficiently, then it may be easily possible to achieve significantly higher quality achievements for information processing and secure communications. The HPCCS scheme among four non-identical HC systems of integer order using ACM has not yet been explored till now. This shows the novelty of this presented research work.
Our studies are structured as follows: in Sect. 2, some necessary definitions and basic notations are presented to be used within this paper. Section 3 deals with the methodology of ACM. Section 4 contains the basic structured features of the considered HC systems. Section 5 investigates the HPCCS scheme via ACM strategy. Numerical simulations are executed to verify the obtained analytical results of the proposed scheme in Sect. 6 . Also, a comparative analysis has been discussed. Lastly, some conclusions are drawn in Sect. 7

Preliminaries
In this section, we initially describe the procedure to achieve C-C synchronization according to master-slave configuration among four non-identical integer order HC systems with two master systems and two slave systems which are requisites for the subsequent sections.

Remark 2.2
The constant matrices P, Q, R and S are called the scaling matrices. In addition, P, Q, R, S can be extended as matrices of functions of state variables x 1 , x 2 , y 1 and y 2 .

Remark 2.3
If R = 0 or S = 0, then C-C synchronization problem turns out to be the combination synchronization problem.

Remark 2.4
The combination synchronization problem will be converted into the chaos control problem for P = Q = R = 0 or P = Q = S = 0.
Remark 2.6 Definition 2.1 displays that the combination of master systems and slave systems can be expanded to three or more chaotic systems. Further, master systems and slave systems of the C-C synchronization scheme can be identical or non-identical.

Methodology of adaptive control method
Consider the chaotic (or HC) master system and the corresponding chaotic (or HC) slave system aṡ where is the controller vector , f 1 , g 1 : R n → R n are non-linear continuous vector functions, f 2 : R n → R n×m 1 , g 2 : R n → R n×m 2 are matrix. η ∈ R m 1 , and β ∈ R m 2 are non-linear parameter vectors. Suppose error is described as where e(t) = (e i ; i = 1, 2, . . . , n) T is the error function and . represents the vector norm. It follows thatė Using (6) and (7), the error dynamics becomeṡ Now, we design appropriate control function u and parameter update laws accordingly to ensure that master and slave systems with unknown parameters approach to desired synchronization state.
For that the controller is chosen as whereη(t),β(t) are the estimated values of unknown parameter vectors, respectively, and k is an arbitrarily chosen positive number known as gain constant.
The parameter update laws are described aṡ Choosing the Lyapunov function as The derivative of V is written aṡ which establishes thatV is negative definite.
In accordance with LST, the error dynamical system is globally and asymptotically stable in the neighbourhood of the equilibrium points.
Wei et al. [54] found the HC system: where x 41 , y 41 , z 41 , w 41 are the state variables for the system (11) and for the system parameters (11) depicts HC behaviour as shown in Ref. [54]. In addition, Fig. 1 and Fig. 2 display 2-D and 3-D phase portraits respectively of the aforementioned four hyperchaotic systerms.
In the next section, we shall present the HPCCS scheme using ACM.

Illustrative example
In this section, we discuss the ACL and the parameter estimation update law to stabilize the HC systems. For convenience, the systems (8) and (9) are treated as the master systems and the systems (10) and (11) are treated as the slave systems with controllers may be written aṡ where u 11 , u 12 , u 13 , u 14 , u 21 , u 22 , u 23 , u 24 are the adaptive control inputs to be designed using ACM.
Putting the values of controllers (17) in error dynamics (16), we geṫ We define parameter estimation error as follows: From Eqs. (18) and (19), we obtain Next, we consider the candidate for Lyapunov function given by 11 + e 2 12 + e 2 13 + e 2 14 +ã 1 2 +ã 2 2 +ã 3 2 +ã 4 2 +b 1 2 +b 2 2 +b 3 2 +b 4 2 +c 1 2 +c 2 2 +c 3 2 +c 4 2 +d 1 2 +d 2 2 +d 3 2 +d 4 2 +h 3 2 +h 4 2 ], which is a positive definite function. Differentiating V along the trajectories of (21), we obtaiṅ In view of (22), we define the parameter estimation laws as follows: This shows thatV < 0 which ensures thatV is negative definite in R 22 . Thus, according to LST, we deduce that the HPCCS error e(t) → 0 asymptotically as t → ∞ for all initial conditions e(0) ∈ R 4 . Hence, the proof is complete. Case 1. Let us assume the scaling matrix α with α 1 = α 2 = α 3 = 1. In this case, we achieve complete C-C synchronization with master systems (8)-(9) and slave systems (12)- (13). The trajectories of master and slave systems with the control inputs are depicted in Fig. 3a-d. Further, initial states for error system are obtained as (e 11 , e 12 , e 13 , e 14 ) = (1.2, −4.9, 9.75, 2). The convergence of synchronization error e(t) to zero as t approaches infinity as displayed in Fig. 3e shows that the HPCCS among two master systems and two slave systems has been achieved via ACM. Moreover, Fig. 3f displays that the estimated parameters converge to their respective original values asymptotically with time. Thus, the proposed HPCCS scheme between master and slave systems is verified computationally.

Case 2.
When the scaling matrix α is taken as α 1 = α 2 = α 3 = α 4 = −1, then we achieve C-C antisynchronization between the master systems (8)-(9) and the slave systems (12)- (13).   13.75, −4). The convergence of synchronization error e(t) to zero as t approaches infinity as depicted in Fig. 5e ensures that the HPCCS among two master systems and two slave systems is achieved via ACM. Further, Fig. 5f exhibits that the estimated parameters converge to their respective original values asymptotically with time. Thus, the proposed HPCCS scheme between master and slave systems is ensured computationally.
A comparison analysis between the proposed HPCCS scheme and the earlier published work. In Ref. [26], author used nonlinear active control to investigate the C-C synchronization between four identical Lu systems and non-identical chaotic systems where it is noticed that the synchronization is obtained at t = 5 (approx.) and t = 4.8 (approx.), respectively. Also, author adopted sliding mode control approach in Ref. [41] to achieve C-C synchronization among four different systems with uncertain parameters, here it has been found that synchronization error is converging to zero at t = 2.4 (approx.). Further, in Ref. [42], author studied C-C synchronization among four complex nonlinear chaotic systems where it has been recorded that the error synchronization is realized at t = 5 (approx.). Also in Ref. [45], author applied ACM to achieve C-C synchronization between four identical HC systems where it noted that the synchronization state is attained at t = 5.1 (approx.). Moreover, in Ref. [46], author utilized robust adaptive sliding mode control to generalize the C-C synchronization of n-dimensional chaotic systems with time delay in which the synchronization is obtained at t = 4.9 (approx.). Also in Ref. [48], author discussed phase synchronization between different complex chaotic systems of fractional order, by this procedure, the C-C synchronization has been carried out at t = 4.5 (approx.). Furthermore, in Ref. [55], author proposed ACM to study complex modified hybrid function projective synchronization between different complex chaotic systems with unknown complex parameters, where it has been seen that the desired synchronization is obtained at t = 4.5 (approx.). Apart from the above-described studies, we have investigated the HPCCS scheme among four non-identical HC systems using ACM in which it has been found that the synchronization occurs at t = 1.2 (approx.) as shown in Fig. 6. Hence, in comparison to the above discussed techniques, the synchronization time in our investigated scheme is much lesser which in turn shows the vitality and effectivity of the considered methodology.

Conclusion
In this paper, we have investigated the HPCCS scheme among four non-identical HC systems via ACM. The analytical expressions of control inputs and the parameter update laws are obtained in accordance with Lyapunov stability theory. Numerical simulations are demonstrated to check the effectivity and correctness of the theoretical results analysed by using MATLAB. Remarkably, the experimental outcomes and theoretical results are both in excellent compatibility. It is noticed that numerous synchronization schemes, viz., chaos control problem, combination synchronization, projective synchronization, etc. become the particular cases of C-C synchronization. In our study, the synchronization error takes less time to converge to zero which implies that our proposed scheme is more preferred over earlier published work. The considered synchronization scheme may be used in secure communications and information processing with several applications in biological, social and physical systems.
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