Robust Synchronization of Hyperchaotic Systems with Uncertainties and External Disturbances

The robust synchronization of hyperchaotic systems with uncertainties and external disturbances is investigated. Based on the Lyapunov stability theory, the appropriate adaptive controllers and parameter update laws are designed to achieve the synchronization of uncertain hyperchaotic systems. The robust synchronization of two hyperchaotic Chen systems is taken as an example to verify the feasibility of the presented schemes. The size of the subcontroller gain’s influences on the convergence speed is discussed. Finally, numerical simulations are given to demonstrate the effectiveness of the proposed synchronization schemes.


Introduction
Since the method of synchronization between identical chaotic systems with different initial conditions was presented by Pecora and Carroll [1], chaos synchronization has attracted considerable attention because of its wide range of applications in many important research fields, such as secure communication, chemical reactions, artificial neural networks, and biological systems [2,3].The idea of synchronization is to use the output of the drive system to control the response system so that the output of the response system follows the one of the drive system asymptotically.Up to now, many types of synchronization phenomena have been reported, such as generalized synchronization [4,5], adaptive synchronization [6][7][8][9], projective synchronization [10][11][12], impulsive synchronization [13], lag synchronization [14,15], and function projective synchronization [16,17].And a wide variety of control approaches, such as backstepping design technique [18], fuzzy sliding mode control [19,20], adaptive control [21], optimal control [22], and  ∞ control [23], have been proposed to synchronize chaotic systems.
However, most of the reported schemes are mainly concerned with the synchronization of chaotic systems without uncertainties and external disturbances.This behaviour results in that the complexity of chaotic dynamics is limited while, in real world applications, there exist a mass of phenomena of the uncertainties and external disturbances in chaotic systems.In this regard, a number of researchers have paid their attention to the synchronization of chaotic systems with uncertainties and external disturbances [24][25][26][27][28][29][30].The corresponding works have solved the problem of synchronization of chaotic systems with uncertainties and external disturbances.However, these studies have not considered the synchronization of the hyperchaotic systems.
The chaos-hyperchaos transition occurs when the second Lyapunov exponent becomes positive [31].Compared with chaotic systems with one positive Lyapunov exponent, hyperchaotic systems are characterized by at least two positive Lyapunov exponents which indicates that they have more complex dynamics and much wider application.Moreover, the chaotic systems with higher-dimensional attractors have much more randomness and higher unpredictability of the corresponding system [32].So the hyperchaos may be more useful in some fields such as communication and encryption.Motivated by this, robust synchronization of uncertain hyperchaotic systems is investigated.Aghababa [33] proposed finite-time chaos control and synchronization of fractionalorder nonautonomous chaotic (hyperchaotic) systems.In [34,35], Fu considered robust adaptive modified function projective synchronization and robust adaptive antisynchronization of different hyperchaotic systems subject to external disturbances.Jawaada et al. [36] proposed robust active sliding mode antisynchronization of hyperchaotic systems with uncertainties and external disturbances.Li et al. [37] studied generalized function projective synchronization of two different hyperchaotic systems with unknown parameters.However, these studies have not considered the effects of both uncertainties and the different kinds of characteristics of external disturbances in the dynamics of the hyperchaotic systems.
In this paper, depending on the characteristics of external disturbance signals, appropriate adaptive controllers and parameter update laws are proposed for the robust synchronization of hyperchaotic systems with uncertainties and external disturbances.The uncertainties are bounded by a nonlinear state-dependent function, instead of a real constant.In the first adaptive controller, assuming that external disturbances are square integrable signals, the adaptivebased controllers and parameter update laws are designed to remove the effects of uncertainties.In the second proposed adaptive controller, the uncertainties and the external disturbances are not square integrable signals.The uncertainties and external disturbances are combined into an uncertain time-varying function with unknown bound.The robustness properties with respect to uncertainties and external disturbances are provided by the proposed controllers.
The organization of this paper is as follows.In Section 2, the robust synchronization of uncertain hyperchaotic Chen systems is formulated.According to the Lyapunov stability theory, the robust adaptive synchronization techniques and the error dynamical system's stability analysis are proposed in Section 3.Then, the simulation results in Section 4 are given to illustrate the effectiveness of the advised methods.Finally, the concluding remarks are given in Section 5.

Problem Formulation
Throughout the paper, for a vector  ∈   , ‖‖ denotes the Euclidean vector norm and ‖‖ 2  :=   , with the weighting matrix .Make ‖‖ 2 := ‖‖ 2 = =   .Furthermore, Consider the nonlinear systems in the form of where  = [ ]  which will be designed in the following section.The errors between system (1) and system (2) are denoted as   =   −   ( = 1, 2, . . ., ).And the error system is where The aim of the robust synchronization is to find an adaptive controller  = [ 1 ,  2 , . . .,   ]  , such that the states of the nonlinear system (2) can follow the ones of the nonlinear system (1) asymptotically.Now, we adopt the adaptive control theory to achieve the robust synchronization of hyperchaotic Chen systems.The hyperchaotic Chen system [38] is given by where , , , and  are state variables and , , , , and  are the real constants.
In order to observe the synchronization behavior of the hyperchaotic Chen systems, assume that there are two identical hyperchaotic Chen systems where the drive system with subscript 1 drives the response system having identical equations denoted by subscript 2. For system (4), the drive system is and the response system is where  1 , Subtract ( 5) from ( 6), error dynamical system can be represented in a matrix form as where ) . ( The aim of the robust synchronization is to design a controller vector  = [ 1 ,  2 ,  3 ,  4 ]  such that the drive system (5) and the response system (6) are synchronized in the sense that lim  → ∞ ‖()‖ = 0.

Robust Synchronization of Two Hyperchaotic Chen Systems
In this section, depending on the characteristics of disturbance vector ℎ in (6), two adaptive controllers are designed to provide the robustness properties with uncertainties.Two robust adaptive techniques are developed based on a control input in the form as where  = diag( Theorem 2. With Assumption 1, the robust synchronization between the drive system (5) and the response system (6) is realized by controller (9) with and the parameter update laws where  > 0 denotes the subcontroller gain;  and  are positive constants.
Proof.Choose a Lyapunov function as With the controller (9), the time derivative of ( 12) along the error trajectory of ( 7) is Substituting subcontrollers  1 and  2 from (10) implies that Then, taking into account the update laws (11) and Assumption 1, one can obtain Now, the inequality ( 15) is integrated from  = 0 to  = , one can receive for all 0 ≤  < ∞.The inequality (16) implies that () is square-integrable.Besides, the external disturbance ℎ is bounded which means that there exists a  > 0 such that ‖ℎ‖ ≤ .Due to inequality (15), V can be bounded as V ≤ − min ‖‖ 2 + 2 +, where  min is the minimum eigenvalue of .Choose  min > ( 2 +)/ 2 for any small  > 0. There exists  =  min −( 2 + )/ 2 > 0 such that V ≤ −‖‖ 2 < 0 for all ‖‖ > .Thus, there is  > 0 such that ‖‖ ≤  for all  ≥ .This implies that the synchronization error is uniformly ultimately bounded [39] and all the closed-loop signals are bounded.Because of the boundedness of all variables, Ė () in the error dynamical system ( 7) is also bounded.Therefore, according to Barbalat's lemma [39], one can conclude that lim  → ∞ ‖()‖ = 0.The robust synchronization of hyperchaotic systems ( 5) and ( 6) with uncertainties and external disturbances is achieved.The proof is completed.
Remark 3. Choosing a smaller  > 0 causes the faster convergence speed of system (7).In fact, there exists a trade-off between the value of subcontroller gain  and the magnitude of control input .
Remark 4. To satisfy the smoothness of controller ( 9) without violating the convergence property of synchronization error, the exponential term in subcontroller  1 formed by  > 0 and  > 0.
Proof.Choose a Lyapunov function as where m =  − m denotes the estimation error.The time derivative of ( 18) along ( 7) can be received: Replacing  1 and  2 , one can get Taking into account Assumption 5 and the parameter update laws (11), one can obtain Substituting the update law (17) into inequality (21), Following a procedure, similar to the proof of Theorem 2, this implies that the goal of robust synchronization is achieved despite the systems ( 5) and ( 6) with time-varying parameters, the unstructured uncertainties, and external disturbances.The proof is completed.
the response system (6) with  = 1 and  = 0.2 are shown in Figures 2 and 3.The comparison of Figures 2 and 3 shows that the smaller the subcontroller gain  > 0, the faster the speed of achieving synchronization.The states of the error dynamical system (7) with  = 1,  = 0.2, and  = 0.04 in Figure 4 show that the smaller of the subcontroller gain  > 0 provides the faster convergence speed of synchronization errors. in Figure 5.The states of the hyperchaotic Chen systems (5) and ( 6) are plotted in Figure 6.The convergence of synchronization errors are achieved, as depicted in Figure 7.

Conclusion
In this paper, the robust synchronization of two hyperchaotic Chen systems with uncertainties and external disturbances is discussed.Depending on the different characteristics of disturbance signals ℎ ∈  2 [0, ∞) or ℎ ∉  2 [0, ∞), two robust adaptive controllers are proposed.The appropriate adaptive controllers and the suitable parameter update laws are designed to remove the influence of uncertainties and external disturbances, so that robust synchronization of uncertain hyperchaotic Chen systems can be realized.The stability analysis is presented based on the Lyapunov stability theory.The smaller of the subcontroller gain causes the faster convergence speed of the system (7).At last, the efficiency and applicability of the introduced controllers and the parameter update laws are demonstrated using some numerical simulations.