Adaptive Synchronization for Two Different Stochastic Chaotic Systems with Unknown Parameters via a Sliding Mode

and Applied Analysis 3 Definition 5 (see [26]). The trivial solution of the error system (7) is said to be almost surely exponentially stable, if for almost all sample paths of the solution e(t), we have lim sup t→∞ 1 t log ‖e (t)‖ < 0, (9) that is, the the drive system and response system are almost surely synchronization. Remark 6. If e(t) = 0, this means that x(t) = y(t), so u(t) = f(x) + F(x)θ + Δf(x) − g(y) − G(y)ψ − Δg(x) + (σ(x) − σ(y))?̇?(t), where ?̇?(t) = dw(t)/dt. This implies that u(t) directly depends on white Gaussian noises and it is an accessible causal signal; this means that the synchronization cannot be realized completely. The purpose of this paper is to consider the adaptive feedback synchronization problem for stochastic chaotic systems with unknown parameters and uncertainties. The main work of this paper consists of the following aspects. (i) Design an adaptive controller such that the asymptotical stability of the error system (7) can be achieved in mean squares that lim t→∞ E‖e(t)‖ 2 = 0. (ii) Design an adaptive control such that the error system (7) can be almost surely stable, that means the almost surely synchronization could be achieved between drive system and response system. Before proposing the main results, we introduce some lemmas which will be used in the following sections. Lemma 7 (see [20, 27]). The trivial solution of a stochastic differential equation as follows dx (t) = a (t, x) dt + b (t, x) dω (t) , (10) with a(t, x) and b(t, x) sufficiently differentiable maps, is globally asymptotically stable in probability if there exists a functionV(t, x)which is positive definite in the Lyapunov sense and satisfies LV (t, x) = t (t, x) + x (t, x) ⋅ a (t, x)


Introduction
In the past few years, chaotic synchronization has received particular interests [1][2][3] mainly due to its wide applications in secure communications, ecological systems, system identification, and so forth.During the past decades, many methods and experimental techniques have been presented to realize the synchronization of two identical chaotic systems [4][5][6][7][8][9][10], such as adaptive control [4,5], sliding mode control [6,7], nonlinear feedback control [8,9], and fuzzy system based control [10].Among all these methods, sliding mode control method has been used widely to treat the unknown parameters and uncertainties [11,12].For example, synchronization and finite synchronization between two different chaotic systems with uncertainties and unknown parameters via sliding mode method are discussed in [13,14], respectively.However, we have noted that in all of the above mentioned papers, the chaotic systems are deterministic differential equations without any random parameters or random excitation.
Recently, the stochastic modeling has played an important role in engineering application [15,16] and there are some works in the field of control and synchronization on stochastic neural networks [17][18][19][20][21][22][23][24][25].In accordance with the Lyapunov control theory, synchronization of stochastic delayed neural networks has been investigated in terms of linear matrix inequalities in [17].Reference [18] discussed the adaptive lag synchronization between stochastic neural networks with time delay and [19] discussed the lag synchronization between stochastic neural networks with unknown parameters using adaptive control method.Reference [20] considered the robust decentralized adaptive control for stochastic delayed Hopfield neural networks using sling mode control method and [21,22] discussed the almost surely exponential stability for stochastic neural networks.The almost surely synchronization between different stochastic chaotic systems is discussed in [23] using linear matrix equality technique.However, the parameters of the system need to be known, and the authors have not considered the chaotic system contained unknown parameters and uncertainties.
In [24], the authors designed an adaptive controller to make sure the synchronization error trajectories between two different stochastic Chua's systems enter a small zone around zero.The control of unstable periodic orbits of stochastic chaos is discussed in [25] using sliding mode method.As far as we know, there are no results on the asymptotical synchronization and almost surely synchronization for two different stochastic chaotic systems using adaptive sliding mode control method.
In this paper, we discussed the asymptotical synchronization and almost surely synchronization for two different stochastic chaotic systems with unknown parameters and uncertain terms using sliding mode method.The structure of this paper is outlined as follows.In Section 2, we introduce the model of chaotic systems with unknown parameters and uncertain terms and give several assumptions, definitions, and lemmas.Section 3 presents the main results of this paper; we design two adaptive sliding mode controllers to realize the synchronization.Numerical examples are given in Section 4 to show the effectiveness of our proposed results.Finally, some concluding remarks are made in Section 5.

Problem Statement and Mathematic Preliminaries
In this paper, we consider the following stochastic systems with uncertain parameters in the following form: or, in a compact form: d = ( () +  ()  + Δ ()) d +  () d () .
Remark 1.Note that condition (A1) is very weak.We do not impose the usual conditions such as Lipschitz condition and differentiability on the unknown uncertainties functions.It can be discontinuous or even impulsive functions.Since the trajectories of chaotic systems are always bounded, hence, condition (A1) can be easily satisfied.
Remark 2. The condition (A2) is the linear growth condition in fact, it is easy to see this condition is equivalent to the condition in [23].
Let () = () − (), then with subtracting (3) from (1) the error dynamics is obtained as follows: It is clear that the synchronization problem can be transformed to be the equivalent problem of stabilizing the error system (7).
Definition 4 (see [19]).The error system (7) is said to be globally stable in mean squares if for any given initial condition such that lim where [⋅] is the mathematical expectation.
Abstract and Applied Analysis 3 Definition 5 (see [26]).The trivial solution of the error system ( 7) is said to be almost surely exponentially stable, if for almost all sample paths of the solution (), we have lim sup that is, the the drive system and response system are almost surely synchronization.
The purpose of this paper is to consider the adaptive feedback synchronization problem for stochastic chaotic systems with unknown parameters and uncertainties.The main work of this paper consists of the following aspects.(i) Design an adaptive controller such that the asymptotical stability of the error system (7) can be achieved in mean squares that lim  → ∞ ‖()‖ 2 = 0. (ii) Design an adaptive control such that the error system (7) can be almost surely stable, that means the almost surely synchronization could be achieved between drive system and response system.
Before proposing the main results, we introduce some lemmas which will be used in the following sections.

Main Results
To design the adaptive feedback controller to realize the synchronization for stochastic chaotic systems with unknown parameters and uncertainties, we use the sliding mode control method.In this section, the nonsingular terminal sliding mode is chosen as where   () ∈ , () = [ 1 (),  2 (), . . .,   ()] T and   > 0 are constants.

Design of an Adaptive Controller to Realize Asymptotical
Synchronization in Mean Squares.In this section, we are going to design an adaptive controller with updating laws such that the state trajectories will move to the sliding surface in mean squares.To ensure the occurrence of the sliding motion, an adaptive sliding mode controller is proposed as where θ, ψ, α , β , k are the estimations for , ,   ,   ,   , respectively.  > 0 is the switching gain and a constant,  = 1, 2, . . ., .
The proposed control input in (15) with the updating laws in (16) will guarantee the reaching condition lim  → ∞ ‖()‖ 2 = 0 and ensure the occurrence of the sliding motion, which is proved in the following theorem.Theorem 9. Suppose that the assumption conditions (A1) and (A2) hold; consider the error dynamics (7); this system is controlled by () in (15) with updating laws in (16), then the error system trajectories will converge to the sliding surface () = 0 in mean squares.
Taking mathematical expectation on both sides of (20), in view of (18) and the definition of (), we obtain Based on the LaSalle invariance principle of stochastic differential equation, which was developed in [28,29], we have () → 0 when  → ∞, which in turn illustrates that lim  → ∞ ‖()‖ 2 = 0.This complete the proof.
Remark 11.Since the control law (15) contains the sign function as a hard switcher, the undesirable chattering phenomenon occurs.According to Lemma 2 and Remark 2 in [13], we can replace the sign(  ) function by tanh(  ),  > 0.
In fact, the unknown parameters ,  in (7) cannot identify with θ, ψ.We offer the following theorem.
Theorem 14. Suppose that the assumption condition (A2) holds.Then under the controller (15) with updating laws (16), the response system and the drive system are asymptotical synchronized in mean squares.Moreover, if   (),   () are linearly independent of the synchronization manifold, then Proof.It is easy to get the following error system: Define the following Lyapunov function candidate Since d  () =   d  (), by Itô's differential rule, the stochastic derivative of () along trajectories of error system (7) can be obtained as follows: where the weak infinitesimal operator L is given by From the update laws ( 16), we can always choose the appropriate initial values of α0 and β0 to make α > 0 and β > 0. Since α   () sign(  ()) ≥ 0 and β   () sign(  ()) ≥ 0. Using the facts ∑  =1     ()  () =  T  T ()(), ∑  =1     ()  () =  T  T ()() and the updating laws in (18), with the same procedure of the proof of Theorem 9, we also arrive at ‖()‖ 2 → 0.
Remark 15.Certainly, under the controller without (α  + β ) sign   () term, the response system can also synchronize the drive system in mean squares.However, under the controller (15) with this term, it is more effective to realize the synchronization with this term.If   () >   (), this term will help to increase   (), and if   () <   (), this term will help to decrease   ().Hence, this term (α  + β ) sign   () can enhance the synchronization speed.

Design of an Adaptive Controller to Realize Almost Surely
Synchronization.In this section, we are going to design an adaptive controller with update laws such that the state trajectories will move to the sliding surface almost surely.We first introduce the following assumptions for the unknown parameters.
(A3) The unknown parameters vectors  and  are norm bounded with known bounds, that is, where  and  are two known positive constants.(A4) Assume (A1) and (A2) hold, and   ,   , and   are known positive constants.
To tackle the uncertainties and unknown parameters, appropriate adaptive laws are defined as follows: where () = [ 1  1 (),  2  2 (), . . .,     ()] T , and θ0 , ψ0 are the initial values of the update parameters, respectively.Based on the control input in (29) with the updating laws in (30) to guarantee the reaching condition lim  → ∞ () = 0 almost surely holds and to ensure the occurrence of the sliding motion, a theorem is proposed and proved.
Theorem 16.Suppose that the assumption conditions (A3) and (A4) hold; consider the error dynamics (7); this system is controlled by () in (29) with updating laws in (30), then the error system trajectories will converge to the sliding surface () = 0 almost surely.
Proof.Select a positive definite function as a Lyapunov function candidate in the form of Since d  () =   d  (), by Itô's differential rule, the stochastic derivative of () along trajectories of error system (7) can be obtained as follows: where the weak infinitesimal operator L is given by Using the facts ∑  =1     ()  () =  T  T ()(), ∑  =1     ()  () =  T  T ()() and the updating laws in (30), one has Since ) and similarly we have ‖ψ − ‖ 2 ≤ 2(‖ψ‖ 2 +  2 ), so we can conclude that Then from Lemma 8, we can obtain: lim  → ∞ () = 0 almost surely.This completes the proof.
Remark 17.From the proof, it is easy to see that the positive numbers in (A4) also could be unknown; we just modify the estimate parameters in the controller.To simplify, we discussed the problem under condition (A4).

Remark 19.
With the similar method in Theorem 13 and Theorem 14, we can also discuss the problem of the identification between the unknown parameters ,  in (7) and θ, ψ in (30).

Numerical Simulations
In this section, we will show that the proposed adaptive controllers are efficient and that the theoretical results are correct.Numerical simulations are performed using MATLAB software.The well-known stochastic chaos between Lorenz system and Chen systems is synchronized using the adaptive controller (15) in the first example.The synchronization between Chen system and Lu system is shown in the second example using the adaptive controller (29).The Lorenz, Chen, and Lu systems are given by the following differential equations, respectively, In all the cases, the uncertainties Δ(), Δ(), and () and the noise intensity function are given as follows, respectively, In all simulations, we choose the initial value of the adaptive parameters vectors θ0 = [5, 5, 5] T , ψ0 = [3, 3, 3] T , α0 = β0 = k0 = 2, the constants  1 = 10 and  = 0.01.

Example 1: Synchronization between Lorenz Systems and
Chen Systems.The nonlinear part of master and slave systems can be rewritten in the form of ( 2) and (4) as follows: ) .
(38) Consequently, three sliding surfaces are chosen as The stochastic Lorenz and Chen systems are started with the initial conditions as follows:  0 = [1, 1.5, 2] T and  0 = [2, 2.5, 3] T .The synchronization of the Lorenz and Chen systems without control input is shown in Figure 1 and the error simulation under the control input is shown in Figure 2. As one can see the synchronization errors converge to zero in mean squares.The control input is shown in Figure 3 and the sliding mode surface is shown in Figure 4.The updated vector parameters of α, β, θ, and ψ are shown in Figures 5, 6, 7, and 8 and k are depicted in Figures 9, 10, and 11, respectively.Obviously, all of updated parameters approach some constants.

Example 2: Synchronization between Chen Systems and Lu
Systems.The nonlinear part of master and slave systems can be rewritten in the form of ( 2) and (4) as follows: Consequently, the same sliding surfaces are chosen as in Example 1.To simplify, we choose the noise intensity function () = [0.4 2 + 0.3    and the sliding mode surface is shown in Figure 15.The updated vector parameters of θ and ψ are depicted in Figures 16 and 17, respectively.Obviously, all of updated parameters approach some constants.Remark 20.As it is observed in Figures 5,6,16, and 17, the limits of unknown parameter vectors θ and ψ are not equal to the vectors  and  in (36).This point is consistent with the results of Theorem 13, Theorem 14, and Remark 19.

Conclusion
In this paper, adaptive sliding mode controllers are designed to realize the asymptotical synchronization in mean squares and the almost surely synchronization for two different stochastic chaotic systems with unknown parameters and uncertain terms, respectively.The designed controllers' robustness and efficiency are proved between two different pairs of stochastic chaos systems (Lorenz-Chen and Chen-Lu) with unknown parameters and uncertainties.

Figure 1 :
Figure 1: The trajectories of the error system without control input in Example 1.

Figure 2 :
Figure 2: Time responses of error system under control input in Example 1.

Figure 3 :Figure 4 :
Figure 3: Time response of the control input in Example 1.

Figure 5 :
Figure 5: The trajectories of the adaptive laws of the parameter θ in Example 1.

Figure 6 :Figure 7 :
Figure 6: Time responses of the adaptive update laws ψ in Example 1.

Figure 8 :
Figure 8: The trajectories of the adaptive laws  in Example 1.

Figure 12 :Figure 13 :
Figure 12: The trajectories of the error system without control input in Example 2.

Figure 14 :Figure 15 :
Figure 14: Time responses of the control input in Example 2.

Figure 16 :Figure 17 :
Figure 16: The trajectories of the adaptive laws of the parameter θ in Example 2.