Effective synchronization of a class of Chua's chaotic systems using an exponential feedback coupling

In this work a robust exponential function based controller is designed to synchronize effectively a given class of Chua's chaotic systems. The stability of the drive-response systems framework is proved through the Lyapunov stability theory. Computer simulations are given to illustrate and verify the method.

For applications such as telecommunications, where the transmission of messages is not possible unless transmitter and receiver are synchronized [10][11][12], the investigation of new chaotic systems as 1 arXiv:1312.7124v1 [nlin.CD] 26 Dec 2013 well as the most effective means of synchronization is always of great importance. Thus, mathematical models [28,29], mechanical systems [30], electronic circuits [10,13] are continually built. One of the best known electronic circuits is the Chua's oscillator [33,34]. Although Chua's circuit is one of the simplest circuits in the literature, it has various complex chaotic dynamics properties which has made it a topic of extensive study [31][32][33][34]. A modified version of the circuit has also been topic of attention [31,32,35]. Its theoretical analysis and numerical simulations agree very well with experimental results.
Recently, some authors proposed a nonlinear controller in order to force synchronization with the purpose of saving energy [22,23]. The nonlinear controllers used are based on bounded nonlinear functions [22,23]. In this work we apply the exponential function based nonlinear controller to achieve synchronization between the drive-response systems when disturbances are present. Our controller has certain properties which makes it more advantageous to use it, such as: (1) it is easy to implement in practice; (2) it needs no adaptation algorithm, hence its electrical circuit remains simple;(3) it is faster than the synchronization based on fixed feedback gain which is usually used.
This work is organized as follows: In section 2, the problem is formulated and the assumptions are given. Section 3 presents the main results. We use Lyapunov stability theory to study the robustness of our proposed controller. We show that with this controller the drive and response systems are practically synchronized -the errors between the master system and the slave system do not tend to zero but to a limit value. In this case, it is shown that the derivative of Lyapunov function is contained in a closed domain to which the error between master and slave system converge.
Since the error is sufficiently small, using the principle of the "ultimate boundedness property", we arrive to the conclusion that the system is globally stable because the derivative of the Lyapunov function is negative definite. Ultimate boundedness is in particular compatible with local instability about zero and implies global stability. This was demonstrated by Z. Ding and G. Cheng in ref. [36].
They proposed a new criterion of globally uniformly ultimate boundedness for discrete-time nonlinear systems which helps to relax the condition of stability based on Lyapunov function. The same ideas were successfully applied by De La Sen and S. Alonso [37], while in ref. The conclusions are given in section 5.
In this paper, we study the master-slave synchronization of a class of Chua's chaotic systems, represented in Figure 1 and described by the equations that follow.
The master system is given by: where τ is a dimensionless time, x i (t), i = 1, 2, 3 are the state variables, v (τ ) is an external force and α, β, γ and R are positive constant parameters of the system. The function f (x 1 (τ )) represents the nonlinearity of the system and d (τ ) the disturbances. The function f (x 1 (τ )) defines Chua's circuit, which is given by f ( [34]. The latter represents the behavior of a tunnel diode [34]. For an autonomous system v (τ ) is constant and d(τ ) = 0.
The slave system is given by: where y i (τ ) , i = 1, 2, 3 is the slave state variables and U (τ ) the feedback coupling.
Here we present a scheme to solve the synchronization problem for system (1). That is to say, if the uncertain system (1) is regarded as the drive system, a suitable response system should be constructed to synchronize it with the help of the driving signal x. In order to do so, we assume that: (i) There is a bounded region U ⊂ R 3 containing the whole basin of the drive system (1) such that no orbit of system (1) ever leaves it.
(ii) The disturbance d(τ ) is bounded by an unknown positive constant D, namely where · denotes the euclidian norm of a vector.
(iii) All chaotic systems are supposed to be confined to a limited domain, hence it exists a positive constant L such that We shall now try to synchronize the systems described in (1) and (2) designing an appropriate control U (τ ) in system (2) such that where r is a sufficiently small positive constant.
Let us define the state errors between the transmitter and the receiver systems as and the feedback coupling as: where ϕ and k are positive fixed constants.
Introducing the definition of the systems (1), (2) and equation (6) into (2), the dynamics of the error states (5) becomes: The problem is now reduced to demonstrating that with the chosen control law U (τ ), the error states e i , i = 1, 2, 3 in (7) are at most a sufficiently small positive constant r, which will prove the proposition.

Main results
If we consider the master-slave chaotic systems (1) and (2) with all the aforementioned assumptions (3) and with the exponential function based feedback coupling given by the relation (6), we shall show that the overall system will be practically synchronized, i.e., y i (τ ) − x i (τ ) ≤ r, where r is a sufficiently small positive constant for large enough τ .
In order to do so, let us consider the following Lyapunov function: Differentiating the function V with respect to time yieldṡ Expanding the exponential function as follows where θ(e 1 ) and ζ(e 1 ) constitute the rest of the expansion in order greater than n for odd part and for even part of the development respectively, and substituting by the maximum value of the disturbance, D, it follows thatV Hence, we haveV Here we use r as an upper bound for the error in each axis. Then we see that the derivative of the Lyapunov function (11) is lower than that in equation (12), which in turn is smaller than the one given by equation (13). Thus expression (13) is maximized and the radius of the close domain to which the error is attracted is determined. Defining one obtainsV Eq. (15) is in principle a form of the ultimate boundedness property in the sense that if the error is sufficiently small, then the system is globally stable because the upper-bound is negative [36]. From (15), it follows that if therefore,V (τ ) < 0, hence V (τ ) decreases, which implies that e 1 decreases. It then follows from standard invariance arguments as in Ref. [23] that asymptotically for increasing time the error satisfies the following bound So if φ is sufficiently small, the bound for the synchronization error will also be sufficiently small.
Therefore, the synchronization state error would be contained within a neighborhood of the origin, as we wanted to prove. In addition as V (τ ) decreases, then there exists a continuous and strictly increasing function and a finite integer η, such that ) where e(τ ) = (e 1 (τ ), e 2 (τ ), e 3 (τ )) .
Thus, the Lyapunov function respect the Theorem 3.1 in [36] and then Eq. (7) is globally uniformly ultimate bounded near the origin.

Simulation results and discussion
The controller's parameters are ϕ = 10 and k = 3. The controller circuit was realized through the Considering the case without disturbances, if we compare the proposed scheme with the one for which the controller is given by the following relation, where ζ is a positive constant chosen equal to ϕ, it appears that, as one can visually appreciate on the graphs of Figs (5) and (6), the exponential function based nonlinear controller is faster than the linear controller with fixed gain. In this paper the synchronization between two different delayed chaotic systems is studied via a simple -exponential function based -nonlinear controller. Although different initial conditions and disturbances make synchronization more difficulty, a simple exponential function based nonlinear controller is designed which facilitates the task. This is proven through the Lyapunov stability theory, it is shown that both master-slave systems should be practically synchronized. It is important to note that the proposed scheme improves the linear controller with fixed gain usually used. To show the effectiveness of the proposed strategy, some numerical simulations are given, they show the efficiency of the proposed strategy in front of the linear fixed gain based controller. The electronic circuit of the used controller is also given followed by some simulations.