Generalized chaos control and synchronization by nonlinear high-order approach

Available online at www.sciencedirect.com

Mathematics and Computers in Simulation 82 (2012) 2268–2281

Original article

Generalized chaos control and synchronization by nonlinear high-order approach Abdelkrim Boukabou ∗ , Naim Mekircha Department of Electronics, Jijel University, BP98 Ouled Aissa, Jijel 18000, Algeria Received 31 December 2011; received in revised form 7 July 2012; accepted 21 July 2012 Available online 28 July 2012

Abstract This paper investigates the generalized control and synchronization of chaotic dynamical systems. First, we show that it is possible to stabilize the unstable periodic orbits (UPOs) when we use a high-order derivation of the OGY control that is known as one of useful methods for controlling chaotic systems. Then we examine synchronization of identical chaotic systems coupled in a master/slave manner. A rigorous criterion based on the transverse stability is presented which, if satisfied, guarantees that synchronization is asymptotically stable. The Rössler attractor and Chen system are used as examples to demonstrate the effectiveness of the developed approach and the improvement over some existing results. © 2012 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Chaotic systems; High-order control; Synchronization; Transverse stability

1. Introduction Nowadays, chaos control and synchronization are important topics in the nonlinear control systems. Chaos control can be understood as the use of small perturbations to stabilize unstable periodic orbits (UPOs) embedded in chaotic systems via small control input. This concept was first initiated by Ott, Grebogi and Yorke known as OGY method [31]. However, the OGY method requires exact calculation of the UPO, which is often very hard in experiment. An alternative control method was proposed by Pyragas which states that, chaotic system can be stabilized by a feedback perturbation proportional to the difference between the present and the delayed state of the system [34]. However, it has been shown that the Pyragas method also has a limitation related to the odd number property [28,44]. Numerous research efforts are dedicated to overcome some limitations of these original methods. In fact, some improvements concerning the OGY method are reported in [2,7,12,19,35]. Based on the Pyragas method, several methods avoiding the odd number property are given in [1,17,20]. Boccaletti et al. [5] give a survey of the most relevant control methods. Recent progresses in controlling chaos can be found in [41,42]. On the other hand, since Fujisaka and Yamada’s 1983 paper on synchronized motion in coupled chaotic systems [13], many researchers have discussed the stability of this type of motion. Up to now, different methods and techniques are investigated on synchronization of chaotic systems. Here, we just mention the Pecora–Carroll synchronization [33], complete synchronization [47], phase and lag synchronizations [36,37], generalized synchronization [18,22,46], partial synchronization [49], predictive synchronization [39], and ∗

Corresponding author. Tel.: +213 34 472040; fax: +213 34 501189. E-mail address: [email protected] (A. Boukabou).

0378-4754/$36.00 © 2012 IMACS. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.matcom.2012.07.005

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adaptive–impulsive synchronization [45]. Boccaletti et al. [6] and Nijmeijer [29] give an overview of the proposed synchronization methods. Banerjee [3] and Stavroulakis [43] derived important results in secure communication using chaos synchronization. The discussion in this paper will center around the type of synchronization discussed by most of these authors. Namely, two or more identical chaotic systems, coupled in a master/slave manner, which exhibit motion that is chaotic and identical in time. Another interesting topic concerns the stability of the synchronized state to a mismatch of the parameter values between the two chaotic systems. Recent studies of this problem has led to the observation of new phenomena, such as riddled basins of attraction, attractor bubbling, on of intermittency and blow-out bifurcation [23,26,32,48]. The purpose of this paper is to generalize the high-order chaos control approach [7,8], and then to study the synchronization process of coupled controlled chaotic systems. First, the Poincaré section is employed to identify unstable periodic orbits embedded in the chaotic systems. Since a generic unstable periodic orbit is mapped on the Poincaré section by an ordered sequence of crossing points, the controller parameters are determined for the desired unstable periodic orbit evaluating the influence of small parametric variation on the unstable periodic orbit variation. Afterward, the transverse stability criterion is employed to synchronize identical chaotic attractors. We continue the study by examining the bifurcations through which low periodic orbits embedded in the synchronized chaotic state lose their transverse stability and produce the characteristic phenomena of local and global riddling, blow-out bifurcations, attractor bubbling, on–off intermittency, etc. As a potential application of the proposed control and synchronization strategy, we used it to study the control of unstable periodic orbits, and synchronization of coupled Rössler and Chen chaotic dynamical systems. This paper is organized as follows. After this introduction, we give in Section 2 the methodology of generalized control and synchronization by nonlinear high-order approach. In Section 3, we discuss chaos control and synchronization of two identical Rössler and two identical Chen dynamical systems, and numerical simulations are given to show this process. Conclusion is given in Section 4.

2. Nonlinear high-order control and synchronization principles Consider the two nonlinear systems modeled by a set of ordinary nonlinear differential equations as follows: ˙ 1 (t) = f (X1 (t), α) X

(1)

˙ 2 (t) = g(X1 (t), X2 (t)) X

(2)

where f, g : RN → RN are continuous functions, X1 , X2 ∈ RN are the state variables and α ∈ R is a control parameter. The system given by Eq. (1) will be called the master system and the system given by Eq. (2) will be called the slave system.

2.1. Control principle The chaos control method may be understood as a two-stage technique. In the first step, the learning stage, the unstable periodic orbits are identified and control parameters are evaluated. After that, there is the control stage where the desired unstable periodic orbit is stabilized. The learning stage uses, in a large sense, the Poincaré section (PS) properties. For this, we determine the influence of control parameter on the original unstable periodic orbit. Secondly, we determine the variation that should be applied to the control parameter in order to force the system to rejoin the desired unstable periodic orbit. After information about the PS has been gathered, the system is kept to remain on the desired orbit by perturbing the appropriate parameter. Similar to the original OGY control method, we wish to make only small controlling perturbations to the system. We do not envision creating new orbits with different properties from the already existing orbits. Let us consider the chaotic system (1), where x is a N dimensional vector and α a control parameter, and start by setting α in order to have a stable periodic orbit of period τ . A N − 1 dimensional PS intercepts the orbit at a point which repeats after a time τ. Let X be an unstable periodic orbit of the chaotic system (1) represented by a point at PS.

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We assume that α is able to vary in a small neighborhood of X with feedback rules. Therefore, the following control law is directly derived from the PS: δα(t) =

∂α | × δX1 (t), ∂X1 X1 =X

(3)

where ∂α/∂X1 |X1 =X determines the influence of small parametric variation on unstable periodic orbit variation and δX1 (t) represents the difference between actual and desired values, i.e. δX1 (t) = X1 (t) − X. The task in the control stage is to apply an appropriate control such that the controlled master system tracks the target lim δX1 (t) = 0,

(4)

t→∞

where || · || is the Euclidean norm. It seems more sensitive, from a practical point of view, to introduce some delay between the computation of the control law and the effective modification of the control parameter. Then by Boukabou and Mansouri improvement of the OGY method [7], this is realized by adding to the control law (3) a term depending on the previous value of the control parameter α weighted with a parameter ξ . Thus, the proposed high-order control law is applied to the master as follows: δα(t) =

∂α | × δX1 (t) + ξδα(t − 1). ∂X1 X1 =X

(5)

The parameter ξ of Eq. (5) measures how the distance between actual and desired trajectory evolves. Contraction or expansion of ξ result in perturbing the dynamics more or less robustly to stabilize the desired UPO. The introduced adaptive weighting procedure assures the effectiveness of the method (perturbation is larger or smaller whenever it has to be). The controlled master system is then given by ˙ 1 (t) = f ((X1 (t), α + δα(t)) . X

(6)

In terms of control performance, once the control is activated, the controlled master system must be maintained at its new trajectory along its evolution. Lemma 1. Suppose that the master system (1) is autonomous. Let J(X) = ∂α/∂X be the system Jacobian on the Poincaré section and assume that J(X) is uniformly bounded by a constant matrix J0 , namely J(X) ≤ J0 for all X(t) and all t : t0 ≤ t ≤ ∞ . Then, there is a control gain ξ such that the orbit of system (6) is driven to approach a periodic orbit as t → ∞ . Proof. According to the fundamental theory of differential equations, the solution of the linearized system of (6) is  t X1 (t) = e(t−t0 )[J(X1 )+ξ] X(t0 ) + e(t−s)[J(X1 )+ξ] X1 (s)ds t0

which satisfies X1 (t) ≤ e

(t−t0 )[J0 +ξ]

 X(t0 ) + 

t

e(t−s)[J0 +ξ] X1 (s)ds

t0

where X(t0 ) is the initial condition and X1 (t) is chaotic, and, hence is uniformly bounded in the phase space so that the integral term above converges. Since the real parts of all eigenvalues of [J0 + ξ] can be made negative by a suitable choice of gain ξ, the first term on the right-hand of the above inequality tends to zero as t → ∞ . This implies that the controlled orbit is always bounded.  Remark 1. It follows from the extended Poincaré–Bendixon theorem [14] that the controlled orbit approaches a limit set in the phase plane. In particular, for a chaotic system, it converges to a periodic orbit.

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2.2. Synchronization principle Let X1 (t, X1 (0)) and X2 (t, X2 (0)) be solutions to the master system (1) and to the slave system (2), respectively. In this framework, complete synchronization is defined as the identity between the trajectories of the slave system ˙  (t) = g(X1 (t), X (t)) for the same chaotic master system. X2 (t) and of one replica X2 (t) of it X 2 2 Let us define the error dynamical system as follows ˙ 1 (t) − X ˙ 2 (t), e˙ (t) = X

(7)

Then, the master system and the slave system are said synchronized if and only if: lim e(t) = 0.

(8)

t→∞

In other words, the slave system forgets its initial conditions, though evolving on a chaotic attractor. Hence, the synchronization objective is to force X2 (t) → X1 (t) as t → ∞ . Lemma 2. If the slave system tracks the master system as t→ ∞ and if the control law δα(t) → 0, then the slave system synchronizes with master system no matter how they are initialized. Moreover, the synchronized master and slave systems are stabilized on the unstable periodic orbit. Proof. If X2 (t) tracks X1 (t) as t → ∞ , then we have both ||X1 (t) − X2 (t)||→0 and ||e(t)||→0 (so that e˙ (t) → 0) as t → ∞ . It thus follows from (6) that δα(t) → 0 . Since X1 (t) = X by high order control implying X − X2 (t) → 0, then the slave system tracks the desired unstable periodic orbit.  The next theorem demands weaker conditions, but provides weaker results in the sense that it guarantees the transverse stability of the synchronized chaotic systems. Lemma 3. For the error dynamical system (7), if there are two positive definite and symmetric constant matrices P and Q such that the Riccati polynomial matrix AT P + PA + PBQ−1 BT P + PB + Q

(9)

is either zero or (semi)-negative definite, then when ||e(t)|| is small enough it will always approach zero as t → ∞ . Proof. We begin by determining the behavior of the error system e(t) in Eq. (7) by dividing the corresponding Jacobian into a time independent part, A, and an explicitly time dependent part, B(e ; t), J(e(t)) = A + B(e; t) Consider the Lyapunov function of the form  t V (e, t) = eT (t)Pe(t) + eT (t)Qe(t)dt t0

Then, since zero is an equilibrium point of the error dynamical system, we have a Taylor expansion e˙ (t) = Ae(t) + B(e(t) − e(t0 )) + O(e) where O(e) are higher order terms in e(t) . Thus, V˙ (e, t) = =

e˙ T (t)Pe(t) + eT (t)P e˙ (t) + eT (t)Qe(t) − eT (t0 )Qe(t0 ) [Ae(t) + B(e(t) − e(t0 )) + O(e)]T Pe(t) +eT (t)P [Ae(t) + B(e(t) − e(t0 )) + O(e)]

+eT (t)Qe(t) − eT (t0 )Qe(t0 )   = eT (t) AT P + PA + PBQ−1 BT P + PB + Q e(t)  T  1/2  Q e(t0 ) + Q−1/2 BT Pe(t) − Q1/2 e(t0 ) + Q−1/2 BT Pe(t) +OT (e)Pe(t) + eT (t)PO(e) <

0, ∀ small e(t).

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This completes the proof.  Remark 2. If additionally the following inequality holds ˙e(t) < 0, then the synchronization is monotic, i.e. after each perturbation the distance between the actual trajectory and the chaotic error system is a decreasing function as t → ∞ . In the case of monotic synchronization of all initial values in the neighborhood of e(t) = 0, we refer to the chaotic error system as monotically stable. 2.3. Synchronization transition We conclude the investigation of two coupled chaotic systems by studying the nature of the synchronization transition. Note that the synchronization is achieved for all initial conditions in the neighborhood of the error system (7) for −λ1 > ||P−1 B(e ; t)P|| where λ1 is the largest transverse Lyapunov exponent and P is a prespecified matrix. At −λ1 = ||P−1 B(e ; t)P|| we have a bifurcation where the error system (7) looses its asymptotic stability but it is still stable. For a specific value of matrix P where −λ1 < ||P−1 B(e ; t)P||, the error system (7) has a locally riddled basin [9,15,32,23] where all transverse Lyapunov exponents are negative. However, there are still initial conditions dense in the error system for which one of the transverse exponents is positive. For smaller values of ||P−1 B(e ; t)P||, the error system (7) undergoes a blow-out bifurcation, i.e., transition between chaos and hyperchaos. After this bifurcation occurs the phenomenon of on–off intermittency (chaos-hyperchaos intermittency) in which a typical phase space trajectory spends some of the time in the neighborhood of the error system and occasionally burst away from it. In this case, the largest transverse Lyapunov exponent is positive but small. For more smaller values of ||P−1 B(e ; t)P||, the largest transverse Lyapunov exponent is sufficiently large. In this case the error system becomes a repelling chaotic saddle. 3. Numerical simulations In this section, we apply the proposed nonlinear high-order control and synchronization approach to the Rössler and Chen chaotic systems. 3.1. Rössler system Let us consider the Rössler system [38] as an example: ⎧ ⎪ ⎨ x˙ = −y − z, y˙ = x + ay, ⎪ ⎩ z˙ = b + z(x − c).

(10)

The parameters values are a = 0.398, b = 2 and c = 4 . These values correspond to a chaotic behavior. The parameter c is used as the control parameter. The chaotic behavior of the Rössler system is shown in Fig. 1. An examination of Eq. (10) results in the following equations of A and B : ⎡ ⎡ ⎤ ⎤ 0 −1 −1 0 0 0 ⎢ ⎢ ⎥ ⎥ 0 ⎦, B = ⎣0 0 0⎦ A = ⎣1 a 0 0 z 0 x −c For this example, Brown and Rulkov show in [9] that transverse stability is guaranteed in the neighborhood of the fixed point, period-1 and period-2 by numerical calculations of the invariant manifolds. Moreover, Yanchuk et al. [48] reported the influence of a parameter mismatch by extending the approach proposed initially by Johnson et al. [21], and describe the interesting effect of a shift of the synchronization manifold which can be observed numerically when the considered parameters are included into the system in some special way. To apply the chaos control algorithm to the Rössler master system, we have to determine the Poincaré section. This section is described by one dimensional map and corresponds to the set of points where attractor is at its maximum. That is x = max(x) as shown in Fig. 2. The value of the first state variable of the unstable periodic orbit is represented at the Poincaré section by the point x = 4.39.

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Fig. 1. (a) (x, y, and z) projection of the phase space portrait for the transient Rössler attractor. (b) Time evolution of the state variables x, y and z.

current maximum of state x

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previous maximum of state x Fig. 2. Poincaré section of the current maximum of x versus the previous maximum of the same state variable with c = 4 . The original unstable periodic orbit is then obtained at the intersection of the map with the diagonal line.

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current maximum of state x

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1.5 1.5

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3.5 4.5 previous maximum of state x

5.5

Fig. 3. Influence of the small parametric variation on unstable periodic orbit variation for c = 4 and c = 4.1 .

To determine influence of the small parametric variation on unstable periodic orbit variation, we generate a Poincaré section for c = 4.1 as shown in Fig. 3. For parameter c = 4.1, we obtain x = 4.65. Based on Eq. (5), the control law is applied to variable x of the Rössler master system as follows δc(t)

∂c 4.1 − 4 |x=x × δx(t) + ξδc(t − 1) = × δx(t) + ξδc(t − 1), ∂x 4.65 − 4.39 = 0.38δx + ξδc(t − 1).

=

(11)

We let ξ = 0.1 . Thus, the control law becomes δc(t) = 0.38δx(t) + 0.1δc(t − 1). The Rössler master system is then given by ⎧ x˙ 1 = −y1 − z1 , ⎪ ⎪ ⎨ y˙ 1 = x1 + ay1 , ⎪ ⎪ ⎩ z˙ 1 = b + z1 (x1 − (c + δc)).

(12)

(13)

Simulation results of the control procedure are illustrated in Fig. 4. We observe how the trajectory approaches the unstable periodic orbit where a small pulse is activated automatically so that, at sufficient amplitude determined by Eq. (12), eventually the system orbit converges to the desired unstable periodic orbit. Once controlled Rössler master system is obtained, we construct a Rössler slave system which exhibits a generalized kind of synchronization motion with the master system by making a nonlinear transformation among the variable y2 . Thus, the Rössler slave system will be given by ⎧ x˙ 2 = −y2 − z2 , ⎪ ⎪ ⎨ y˙ 2 = x2 + ay1 , (14) ⎪ ⎪ ⎩ z˙ 2 = b + z2 (x2 − c). Notice that the slave system consists of a copy of the master system without control (δc = 0) where the variable y1 is considered as the drive signal.

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Fig. 4. (a) Projection of the phase space portrait for the controlled Rössler master system. (b) Time evolution of the state variables before and after control. Dashed line indicates the instant at which control task begins. The initial values of the Rössler master system are x1 (0) = 2, y1 (0) = 2 and z1 (0) = 3 .

Introducing the error variables e1 = x1 − x2 , e2 = y1 − y2 and e3 = z1 − z2 , then, we obtain the error dynamics ⎧ e˙ 1 = −e2 − e3 , ⎪ ⎪ ⎨ e˙ 2 = e1 , (15) ⎪ ⎪ ⎩ e˙ 3 = z1 x1 − z2 x2 − ce3 − δcz1 . The synchronization problem is essentially to stabilize the error dynamics (15), i.e. ⎧ ⎪ ⎪ −e2 − e3 = 0, ⎨ e1 = 0, ⎪ ⎪ ⎩ ⎧ z1 x1 − z2 x2 − ce3 − δcz1 = 0, ⎪ e2 + e3 = 0, ⎪ ⎨ e1 = 0, ⇒ ⎪ ⎪ ⎩ z1 e1 + (x2 − c)e3 − δcz1 = 0,

(16)

Since the parameter c and the variables x2 , z1 are different from zero and δc → 0, it follows that the error variables e1 , e2 and e3 asymptotically converge to zero. In other words, the slave system synchronizes with the master system no matter how they are initialized. The initial values of the slave system are x2 (0) = −1, y2 (0) = 4 and z2 (0) = 7 . Simulation

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Fig. 5. Results of numerical simulations for synchronizing coupled Rössler master and slave systems. The Rössler master and slave systems are synchronized to the original unstable periodic orbit. The initial values of the slave system are x2 (0) = −1, y2 (0) = 4 and z2 (0) = 7 .

e1

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0 −10 −20 0

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Fig. 6. Time evolution of the synchronization error variables e1 , e2 and e3 .

results for the synchronized Rössler master and slave systems, and for the error variables are shown in Figs. 5 and 6, respectively. 3.2. Chen system Chen’s system was observed and reported in [11]. In contrast with the Lorenz butterfly system, Chen’s system is topologically more complex but without changing the smooth quadratic function. This chaotic system show some new phenomena including homoclinic bifurcation, and coexistence of two stable limit cycles and one chaotic attractor as well as some periodic solutions emerging from Hopf bifurcation but ending in a homoclinic bifurcation [10]. The mathematical model of the chaotic Chen’s system can be described by the following state equations: ⎧ ⎪ ⎨ x˙ = a(y − x), y˙ = −(a + z)x + c(x + y), (17) ⎪ ⎩ z˙ = xy − bz.

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Fig. 7. (a) (x, y, and z) projection of the phase space portrait for the transient Chen system numerically simulated with a = 35, b = 3, c = 28 . (b) Time evolution of the state variables x, y and z.

The chaotic behavior of the Chen system is illustrated in Fig. 7. The plot of the Poincaré section for the Chen system is shown in Fig. 8 for parameter c = 28 and z = max(z) . Poincaré section of the Chen system indicates the existence of many different period-1 orbits. Our emphasis is focused on the determination of the original unstable periodic orbit. In this case, we obtain z = 27.29. Additionally, we generate a Poincaré section for c = 28.1 as shown in Fig. 9. For parameter c = 28.1, we obtain z = 27.52. The control law is applied to variable z of the Chen master system as follows: ∂c 28.1 − 28 |z=z × δz(t) + ξδc(t − 1) = × δz + ξδc(t − 1), ∂z 27.52 − 27.29 = 0.43δz(t) + 0.1δc(t − 1).

δc(t) =

The Chen master system is under control law described by ⎧ x˙ 1 = a(y1 − x1 ), ⎪ ⎪ ⎨ y˙ 1 = −(a + z1 )x1 + (c + δc)(x1 + y1 ), ⎪ ⎪ ⎩ z˙ 1 = x1 y1 − bz1 . Simulation results of the control procedure are shown in Fig. 10.

(18)

(19)

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current maximum of state z

60

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previous maximum of state z Fig. 8. Poincaré section of the current maximum of z versus the previous maximum of the same state variable with c = 28 .

current maximum of state z

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50 c = 28 c = 28.1

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20 20

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previous maximum of state z Fig. 9. Influence of the small parametric variation on unstable periodic orbit variation for c = 28 and c = 28.1 .

In order to synchronize between the Chen master and slave systems, we introduce the signal y1 in the variable y2 of the slave system as follows ⎧ x˙ 2 = a(y2 − x2 ), ⎪ ⎪ ⎨ y˙ 2 = −(a + z2 )x2 + c(x2 + y1 ), (20) ⎪ ⎪ ⎩ z˙ 2 = x2 y2 − bz2 . The dynamics of the error variables e1 , e2 and e3 are given by ⎧ e˙ 1 = −a(e1 − e2 ), ⎪ ⎪ ⎨ e˙ 2 = −ae1 − z1 x1 + z2 x2 + ce1 + δc(x1 + y1 ), ⎪ ⎪ ⎩ e˙ 3 = x1 y1 − x2 y2 − be3 .

(21)

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Fig. 10. (a) Projection of the phase space portrait for the controlled Chen system. (b) Time evolution of the state variables before and after control. Dashed line indicates the instant at which control task begins. The initial values of the Chen master system are x1 (0) = −2, y1 (0) = 2 and z1 (0) = 10 .

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Fig. 11. Results of numerical simulations for synchronizing coupled Chen master and slave systems. The initial values of the slave system are x2 (0) = 1, y2 (0) = 3 and z2 (0) = 17 .

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e1

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time (s)

Fig. 12. Time evolution of the synchronization error variables e1 , e2 and e3 .

Demanding that all of the equations of system (21) are zeros, we get the following ⎧ −a(e1 − e2 ) = 0, ⎪ ⎪ ⎨ (c − a)e1 − z1 x1 + z1 x2 − z1 x2 + z2 x2 + δc(x1 + y1 ) = 0, ⎪ ⎪ ⎩ ⎧ x1 y1 − x1 y2 + x1 y2 − x2 y2 − be3 = 0, −a(e1 − e2 ) = 0, ⎪ ⎪ ⎨ (c − a − z1 )e1 − x2 e3 + δc(x1 + y1 ) = 0, ⇒ ⎪ ⎪ ⎩ y2 e1 + x1 e2 − be3 = 0.

(22)

Since the parameters a, b, and c and the variables x1 , x2 , y1 , y2 , and z1 are different from zero and δc → 0, it follows that the error variables e1 , e2 and e3 asymptotically converge to zero. Simulation results for the synchronized Chen master and slave systems, and for the error variables are shown in Figs. 11 and 12, respectively. 4. Conclusion In this paper, modification based on Poincaré section to design a nonlinear high-order controller is investigated in order to control and synchronize two identical chaotic systems. Numerical simulations are also given to validate the proposed approach. For two different initial conditions, the slave system is forced to track the master system and the states of the two systems become ultimately the same and converge toward the original unstable periodic orbit. Since the Lyapunov exponents are not required for the calculation, the nonlinear controller is effective and convenient to synchronize two chaotic dynamical systems. The simulation results show that the states of two identical Rössler and Chen systems are perfectly synchronized. Note that this control approach demands special attention when we deal with the stabilization problem of unstable periodic orbits. Same as in the original OGY and TDFC methods, the proposed technique suggests that the mathematical model has to be known beforehand in order to achieve a successful stabilization. However, for chaotic system with parameter uncertainty or without a precise model, in such situation, the stabilization needs improvements by using intelligent approaches such as fuzzy logic and neural networks. This task will be subject of future works. References [1] A. Ahlborn, U. Parlitz, Stabilizing unstable steady states using multiple delay feedback control, Physical Review Letters 93 (2004) 264101. [2] F.T. Arecchi, S. Boccaletti, M. Ciofini, R. Meucci, C. Grebogi, The control of chaos: theoretical schemes and experimental realizations, International Journal of Bifurcation and Chaos 8 (1998) 1643–1655.

A. Boukabou, N. Mekircha / Mathematics and Computers in Simulation 82 (2012) 2268–2281

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