Combination-Combination Projective Synchronization of Multiple Chaotic Systems Using Sliding Mode Control

Based on projective synchronization and combination synchronization model, a type of combination-combination projective synchronization is realized via nonsingular sliding mode control technique for multiple different chaotic systems. Concretely, on the basic of the adaptive laws and stability theory, the corresponding sliding mode control surfaces and controllers are designed to achieve the combination-combination projective synchronization between the combination of two chaotic systems as drive system and the combination of multiple chaotic systems as response systemwith disturbances. Some criteria and corollaries are derived for combination-combination projective synchronization of the multiple different chaotic systems. Finally, the numerical simulation results are presented to demonstrate the effectiveness and correctness of the synchronization scheme.


Introduction
With the development of the control area, chaotic synchronization has become an important and significance research direction in nonlinear science due to its potential practical application in many scientific and engineering fields such as electronic circuits [1], biological systems [2], and communication systems [3] during the recent years.From the viewpoint of control, synchronization of chaotic systems [4] is a hard task because of their nonlinear behavior and sensitivity to the initial values.It is due to the fact that for chaotic systems, with complex mathematical expression, stability analysis methods are difficult to be implemented.Synchronization of chaotic systems refers to a process wherein two or many chaotic systems adjust a given synchronization property of their motion to a common behavior due to a coupling or to a forcing.The main idea of chaos synchronization is to make the states of the response system follow the states of the drive system asymptotically.
Up to now, various kinds of synchronization have been intensively investigated and a lot of theoretical results have been obtained, such as complete synchronization [5,6], antisynchronization [7], phase synchronization [8], generalized synchronization [9][10][11][12], antiphase synchronization [13], lag synchronization [14], partial synchronization [15], projective synchronization [16][17][18][19], time scale synchronization [20], combination synchronization [21][22][23], and compound synchronization [24].Projective synchronization has attracted more and more attention due to the flexibility and adjustability of the proportionality factors, where the complexity of the system is enhanced and increases the secrecy property of the system.In this period, some projective synchronization types have been developed.Modified projective synchronization has been studied for a class of real nonlinear systems and a class of complex nonlinear systems [25][26][27].On another research frontier, the combination synchronization has been investigated for secure communication, which can double the number of variables of the chaotic systems to strengthen the contents and security of the transmitted information.For instance, a method has been designed to realize combination synchronization of three different chaotic complex systems, where the combination system of two drive systems can synchronize with one response system [28][29][30].And, a novel kind of compound synchronization among four chaotic systems is investigated to enhance the security of communication, where the compound system of three drive 2 Advances in Mathematical Physics systems can realize the synchronization with one response system [31,32].Although the combination synchronization of chaotic systems has more advantages and potential compared to the synchronization between one drive system and one response system, the study of combination synchronization has been mostly concerned with the synchronization of one response system and does not consider the influences of some uncertainties parameters for multiple response systems.
On the other hand, several control methods have been investigated to realize chaos synchronization such as feedback control method [33], active control method [34], back stepping method [35], adaptive control method [36][37][38], impulsive control method [39], coupling control method [40], and sliding mode control method [41].The concept and application of sliding mode control have become a popular research subject among many control theories.The difference between sliding mode control [42,43] and conventional control lies in the discontinuity of control.It is a nonlinear control, which is realized by switching functions.The structure of the controller is switched according to the degree of system state deviating from the sliding mode, so that the system runs according to the rule of sliding mode.By designing a switching surface and applying a discontinuous control law, the trajectories of dynamic systems can be forced to slide along the desired sliding surface.The advantage is that the controlled system with sliding mode control exhibits stability properties with respect to both internal parameter uncertainties and external disturbances, which has been applied to cope with the control problem of chaotic system [44].To the best of our knowledge, so far, the studies of chaos synchronization have little involvement with the synchronization scheme that combines projection synchronization and combination synchronization.The research on the combination-combination projective synchronization of multiple drive and multiple response chaotic systems with unknown disturbances is still an opportunity and challenge in the field of nonlinear science.
Motivated by the above discussions, a novel kind of combination-combination projective synchronization among multiple chaotic systems is investigated via the nonsingular sliding mode control, where the mixed system of two drive systems can realize the synchronization with multiple response systems.Moreover, the adaptive laws are proposed and sliding mode controllers are designed for the synchronization of uncertain chaotic systems with unknown parameters and external disturbances.Numerical simulation results are shown to verify the effectiveness and applicability of the combination-combination projective synchronization.
Compared with prior works [5,6,16], there are two advantages which make our paper more attractive and meaningful.At firstly, the paper extends projective synchronization of chaotic systems to combination-combination projective synchronization of chaotic systems, which make the dynamic behaviors and variables of the system more abundant and can greatly improve the security of communication.And importantly, the synchronization among multiple chaotic systems with unknown disturbances is discussed and realized by designing corresponding sliding mode surfaces and controllers, which have a better control performance than some existing results [5,6,16].Hence, the results of this paper may extend and improve existing results in some extent.
The organization of this work is organized as follows.Section 2 shows a scheme of modified combination-combination projective synchronization.In Section 3, sliding mode surface and controllers are designed and stability is analysed.In Section 4, the simulation example is realized to validate the effectiveness and the feasibility of the proposed control strategy.Finally, the concluding remarks are given in Section 5.

Combination-Combination Projective Synchronization of Multiple Chaotic Systems with Disturbances
2.1.Problem Formulation.In this section, the combinationcombination projective synchronization problem of multiple chaotic systems is proposed, which has two drive systems that synchronize multiple response systems.The precise definitions of combination-combination projective synchronization scheme and some definitions are introduced.Without loss of generality, consider the following  chaotic systems with disturbances; the first system is drive system and is described by where The corresponding  − 1 systems for system (1) with control inputs are response systems and can be written as follows: where . .,   (  ()) are the continuous nonlinear functions.  = [ 1 ,  2 , . . .,   ]  is a coefficient matrix,  1 ,  2 , . . .,   are an ×1 real vector of system parameters, respectively.  () = [ 1 (),  2 (), . . .,   ()]  is the disturbance of the system (2), which is an ×1 vector.The control input is The second drive system is given where  1 () = [ 11 (),  12 (), . . .,  1 ()]  is the state vector of the chaotic drive system (3).

The Theory of Combination-Combination Projective Synchronization.
In the section, we firstly design the scheme of combination-combination projective synchronization in our drive-response synchronization scheme with two drive systems and multiple response system.It is assumed that the system  1 () is the drive system, and the other  − 1 systems  2 (), . . .,   () are response systems.The system  1 () is the drive system, and the other  − 1 systems  2 (), . . .,   () are response systems.Therefore, the following definition 1 will be given.

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Remark 7.There are many different scaling factors in a scaling matrix , so all the dynamical states of the combinationcombination projective synchronization can be adjusted of the different states independently.
As we all known, the main idea of synchronization control is to change the combination synchronization of chaotic systems problem into the combination synchronization error stability problem of the chaotic system.
The sliding motion exists with the error system trajectories moving on sliding surface and staying on it forever, if and only if After choosing the corresponding sliding surface, we design the control law to drive the error system trajectories to go onto the sliding surface.Therefore, to ensure the existence of the sliding motion, the corresponding combination controller V −1 () =  −1 () +  −1 () is designed to ensure the existence of the sliding motion.The combination controller can be proposed as follows: where is the constant gain that needs to be given ( = 2, 3, . . ., ).
where  −1 () + and  −1 () − are right-hand limit and left-hand limit of the  −1 (), respectively.Substituting V −1 () =  −1 () +  −1 () from (11) into the above equation ( 8), the error system (7) can be further given as So according to formulas ( 8), (9), and ( 13), we have the following result: Obviously, the external disturbances are generally uncertain, so the control law  −1 () can be simply designed in the following form: The proposed combination controller in (11) and the control law in (15) will guarantee the occurrence of the sliding motion, which is proved in the following Theorem 8.
Theorem 8. Consider the error system ( 5) is controlled with the combination controller in (11) and the combination law in (15).If the following condition ( 16) is satisfied, then the errors of the system (5) will go toward the sliding surface and will reach the sliding surface, which means that the combinationcombination projective synchronization of multiple chaotic systems is realized.

Numerical Simulations
In order to validate the efficiency and effectiveness of the proposed theory, the complex Lorenz system, the complex Chen system, the complex L ü system, and R ö ssler system with disturbance are proposed and corresponding controllers are designed.

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The first drive system is the Lorenz system (24) and corresponding response system is the Chen system (25).
The second drive system is the L ü system (26) and corresponding response system is the R ö ssler system (27).
Remark 10.The sliding control method is included in the discontinuous control methods as a special case.Hence, the sliding control method can be also applied in an array of coupled neural networks.The design of sliding control law is also a difficult point for the discontinuous control methods.The sliding mode control laws sometimes are too complex to realize in the real control.How to simplify the sliding mode control laws to reach the better effect is an interesting yet challenging problem during our future study.
Remark 11.It can be easily obtained that the drive systems and response systems can be the combination of nonidentical chaotic systems or identical chaotic systems.

Conclusions
In this study, a novel modified combination-combination projective synchronization between the combination of two chaotic systems as the drive system and the combination of multiple chaotic systems as the response system with unknown parameters and disturbances are proposed.Furthermore, combined to the adaptive laws, the adaptive combination controllers and sliding mode manifold have been designed, and its convergence has been gained analytically.Finally, the simulations analyses have been given and have shown that the proposed combination controller works well for synchronizing the combination of drive systems and the combination of response systems.The proposed method may be more advantageous and have more potential than the traditional control method to complete intelligent synchronization.How to realize combination-combination projective synchronization in actual practice is our next research topic.

Remark 3 .Remark 4 .
where  1 ,   ,  1 and   are known positive constants ( = 2, 3, 4 ⋅ ⋅ ⋅ , ).If the scaling matrix  = , then the combinationcombination projective synchronization problem will be reduced to combination-combination complete synchronization, where I is a  ×  identity matrix.If the scaling matrix  = −, then the combination-combination projective synchronization problem will be reduced to combination-combination anticomplete synchronization.