Chaos generalized synchronization of coupled Mathieu-Van der Pol and coupled Duffing-Van der Pol systems using fractional order-derivative

Chaos, Solitons and Fractals 98 (2017) 88–100

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Chaos generalized synchronization of coupled Mathieu-Van der Pol and coupled Duffing-Van der Pol systems using fractional order-derivative Tene Alain Giresse a,b,∗, Kofane Timoleon Crépin c,d a

African Institute for Mathematical Sciences (AIMS), Cameroon Mesoscopic and Multilayer structures Laboratory, Department of Physics, Faculty of Science, University of Dschang, Cameroon Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I, P.O.Box 812, Yaounde, Cameroon d Centre d’Excellence Africain en Technologie de l’Information et de la Communication, University of Yaounde I, P.O.Box 812, Yaounde, Cameroon b c

a r t i c l e

i n f o

Article history: Received 18 August 2016 Revised 21 February 2017 Accepted 7 March 2017

Keywords: Chaos Synchronization GYC partial region stability theory Lyapunov function Adam–Bashforth–Moulton method

a b s t r a c t In the present paper, the synchronization by the Ge-Yao-Chen (GYC) partial region stability theory of chaotic Mathieu-Van der Pol and chaotic Duffing-Van der Pol systems with fractional order-derivative is proposed. Numerical simulations show that this synchronization technique is very effective and it turns out that the fractional order-derivative induces quick synchronization compared to integer orderderivative of these systems. In order to bring out the chaotic behavior of these systems either with fractional or with integer order-derivative, we simulate their phase portraits and the Lyapunov exponent. Moreover, we provide in this work an approximated solution to both systems to show that the solution of such a system can be represented as a simple power-series function. Furthermore, the representation of the error dynamics with respect to the time before and after the control action approves the effectiveness of the control method and proves the possibility of stabilization and controllability of chaotic systems with an appropriate. Furthermore, the synchronization of the fractional Mathieu-Van der Pol system using the fractional Duffing-Van der Pol system is simulated. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction The properties of non-linear systems have a great deal of interest, since they have many applications in various fields of sciences such as in biological, chemical and physical systems. The dynamics of such systems might be specified by various different means including ordinary differential equations which are the most known, partial differential equations, iterate maps and recently fractional differential equations. The last notion involves the derivative with memory and it is the generalization of ordinary and partial differential equations. Typical examples of systems that can be represented by these general types of equations are onset of coherent radiation in lasers and masers [1], self-excitations in electric circuits, selforganizations in chemical reactions [2], non-linear mechanics [3], etc. The occurrence of chaotic behavior in non-linear oscillators subjected to periodic forcing is widespread and well known. Exam-

∗

Corresponding author. E-mail addresses: [email protected], [email protected] (T.A. Giresse). http://dx.doi.org/10.1016/j.chaos.2017.03.012 0960-0779/© 2017 Elsevier Ltd. All rights reserved.

ples are the Duffing equation arising almost ubiquitously in models of mechanical oscillations [4], the Van der Pol equation describing, for example, the triode oscillation in electrical circuits [5] and the Mathieu equation which describes for example the motion of particles vibrating in an elliptic drum [6], which have been extensively studied. The parameter space of these equations are divided with great complexity into regions of different qualitative behavior, and the space of initial states is divided with similar complexity into the basin of attraction of competing attractors which may be steady, periodic or chaotic. The last notion describes erratic motions in non-linear dynamical systems. Nowadays, the chaos theory is used in several domains such as geophysics, meteorology, astronomy, economy, biology, etc. Moreover, the chaos theory is the best mechanism for signal design with potential application in telecommunication and coding systems [7]. A chaotic system is unpredictable but it is perfectly described by deterministic equations [8]. It is deterministic because, knowing the exact state of a system at some given time: the initial state can help define the state of the system at any time. Deterministic and unpredictability are two paradoxical notions but the link between them is determined by the sensitivity to initial conditions [9]. That means

T.A. Giresse, K.T. Crépin / Chaos, Solitons and Fractals 98 (2017) 88–100

two almost initial conditions can lead to very different states of the system. The impossibility to predict the evolution of deterministic system is a real characteristic of chaotic systems. Due to their unpredictability character, chaotic phenomena are very difficult to control. However, scientists have shown that a chaotic motion can be controlled under certain conditions. Even so, our main interest in this work is the study of the chaotic or hyperchaotic systems described by fractional order-derivative systems. It has been shown recently that, chaotic behavior can appear with fractional order differentiation [10]. Over the unpredictability of such a system, its instability character makes its controllability a big challenge. However, because of its several potential applications, many researchers are nowadays interested and have even succeeded to control chaotic motions either theoretically or experimentally [11]. Hence, the synchronization of chaotic systems has gained increasing attention after the pioneer work of Pecora et al. [12] in 1990. That is, many types of chaos synchronization have been proposed such as the phase synchronization developed by Pecora in [12]. The complete synchronization has also been developed by Liu et al. in [13], the adaptive synchronization by Shihua et al. [11]. Furthermore, Zheng-Ming et al. [14] developed the generalized synchronization using the GYC (Ge-Yao-Chen) partial region stability theory for the case of Mathieu-Van der Pol and Duffing-Van der Pol systems. Added to this, we have also the projective synchronization [15], hybrid synchronization [16], etc. Generally, these types of synchronization have been carried out using several synchronization schemes such as: linear and non-linear feedback synchronization [17], adaptive control [11], time delay feedback approach [18], etc. In their work, Hongmin et al. [19] have shown that for the order of derivative α = 0.9, the system behaves chaotically. In the same idea, Ghaderi et al. [20] studied the control and synchronization of chaotic Coullet system using fractional order-derivative. They used active control for synchronization and the simulations show the effectiveness of the method. Very recently, Kumar et al. [21] introduced a Mathieu-Van der Pol system with fractional order-derivative. They came out with some remarkable conclusions: the synchronization of Mathieu-Van der Pol chaotic system of fractional order-derivative by linear feedback can be achieved and then, the stability of the system is possible under certain conditions. However, they did not take into consideration the more general case where we have different order-derivatives in the system which implies α 1 = α 2 = α 3 = α 4 . In this paper, the coupled Duffing-Van der Pol and the coupled Mathieu-Van der Pol chaotic systems with fractional orderderivative are studied taking into consideration this generality. The synchronization is approached by the GYC partial region stability theory. This implies by theorem that if V is a positive definite function on the partial region with opposite sign to that of its derivative and the function V itself permits an infinitesimal upper limit, then the undisturbed motion is asymptotically stable on the partial region [14]. Compared to other techniques of synchronization which include the backstepping method, the adaptive design method, the linear and non-linear feedback method, sampled-data feedback synchronization, time-delay feedback, etc. [22–25], this techniques is very appropriate due to the fact that it introduces less simulation error in the system, also from the fact that using this theory we are able to construct the Lyapunov function as a simple linear-function from where the control parameters are easily designed. This synchronization approach is more general since it is applicable for the autonomous and non-autonomous systems, for perturbed and unperturbed systems and finally for linear and non-linear systems. The upper drawbacks can be overcome by using this method. This drawback can only be calculated in the case of finite evolution time in computer simulation. However, infinite evolution time is needed by definition of Lyapunov exponent [26]. It has been applied to several chaotic systems and the simulation

89

results demonstrate the effectiveness and feasibility of the method. These systems include the fly-ball governor with and without system structure perturbation [26], Lorenz system [26],etc. These are the motivations for choosing to apply this synchronization technique to the fractional Mathieu-Van der Pol and fractional DuffingVan der Pol systems in this work. The present paper is structured as follows: In Section 2, the coupled Mathieu-Van der Pol and the coupled Duffing-Van der Pol systems with fractional order-derivative are introduced. In Section 3, we provide an approximated solution to the MathieuVan der Pol and Duffing-Van der Pol fractional systems using the Variational Iterative Method. In Section 4, we introduce the stability analysis, the synchronization scheme and the fractional Lyapunov exponent is presented. Section 5 presents the numerical results. Finally, the conclusion is provided in Section 6.

2. Coupled Mathieu-Van der Pol and the coupled Duffing-Van der Pol systems with fractional order-derivative The coupled Mathieu-Van der Pol system [21] with fractional order-derivative is given as follows:

⎧ α1 D x t = x2 ⎪ ⎨ ∗α2 1 ( ) D∗ x2 (t ) = −(a + bx3 )x1 − (a + bx3 )x31 − cx2 + dx3 α3 ⎪D∗α x3 (t ) = x4   ⎩ D∗ 4 x4 (t ) = −ex3 + f 1 − x23 x4 + gx1 ,

(2.1)

with initial values x1 (0), x2 (0), x3 (0), x4 (0) and a, b, c, d, e, f, g the parameters of the system. For all the simulations in this paper, we considered (a, b, c, d, e, f, g)=(10, 3, 0.4, 70, 1, 5,0.1) and the initial conditions (x10 , x20 , x30 , x40 )=(0.1, -0.5, 0.1, -0.5). Let us now consider α 1 = α 2 = α 3 = α 4 all real numbers taken in (0,1). Fig. 1 gives the phase portrait of the system plotted in Python with different values of α so that, α1 = 0.98, α2 = 0.99, α3 = 0.999, α4 = 0.99. Analogically, the Duffing-Van der Pol equation with fractional order derivative is given by:

⎧ α1 D z = z2 ⎪ ⎨ ∗α2 1

D∗ z2 = −z1 − z13 − hz2 + iz3 Dα3 z = z4 ⎪   ⎩ ∗α4 3 D∗ z4 = − jz3 + k 1 − z32 z4 + lz1 ,

(2.2)

where h, i, j, k, l are also the parameters of the system with the values (h, i, j, k, l)=(0.0 0 06, 0.67, 1,5, 0.05) and the initial conditions (z10 , z20 , z30 , z40 )=(2, 2.4, 5, 6). Let us now consider the fractional order-derivative where α 1 = α 2 = α 3 = α 4 are all real numbers taken in (0,1), As in the previous case the system also exhibits chaotic behavior which is shown in Fig. 2 depicting the phase portrait of the system for different order of derivative, α1 = 0.98, α2 = 0.99, α3 = 0.999, α4 = 0.99. α D∗ i , 0 < αi < 1 stands for the Caputo derivative defined so that, for n − 1 < α < n, n ∈ N, α ∈ R and a function f(t) such that Dα ∗ f (t ) exists, one has

Dα∗ f (t ) =

1 (n − α )

 a

t

(t − s )−1−α+n f (n) (s )ds,

(2.3)

where f(n) (s) denotes the ordinary derivative of order n of the function f(s). From Figs. 1 and 2, we observed that, as well as the MathieuVan der Pol and the Duffing-Van der Pol using integer-order derivative, the fractional order derivative of these systems also depicts a chaotic behavior. This is the case for the values of α i in the range of [0.9-1[. These curves are the so called attractor from their character to keep a particle or an object in the same region no matter where the particle starts its motion.

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T.A. Giresse, K.T. Crépin / Chaos, Solitons and Fractals 98 (2017) 88–100

Fig. 1. Phase portrait for chaotic Mathieu-Van der Pol system with fractional order-derivative (α1 = 0.98, α2 = 0.99, α3 = 0.999, α4 = 0.99).

Fig. 2. Phase portrait for chaotic Duffing-Van der Pol system with fractional order-derivative (α1 = 0.98, α2 = 0.99, α3 = 0.999, α4 = 0.99).

3. Approximated solutions of Mathieu-Van der Pol and Duffing-Van der Pol systems with fractional order-derivative 3.1. Case of Mathieu-Van der Pol system Non-linear phenomena play a crucial role in applied mathematics and physics. The Mathieu-Van der Pol fractional system given by Eq. (2.1) is a perfect non-linear system. Therefore, providing an analytical solution to this system is a real big challenge. However, the Variational Iterative Method (VIM) and the Adomian Decomposition Method (ADM), are among others some powerful tools to provide an approximation of solution to such a system. But we recall that, the VIM is the simplest and accurate method to provide solution either to linear or non-linear system. Then, the solution of the following system:

⎧ α1 D x t = x2 ⎪ ⎨ ∗α2 1 ( ) D∗ x2 (t ) = −(a + bx3 )x1 − (a + bx3 )x31 − cx2 + dx3 Dα3 x (t ) = x4 ⎪   ⎩ ∗α4 3 D∗ x4 (t ) = ex3 + f 1 − x23 x4 + gx1 ,

(3.1)

where we have:

⎧ ⎪ ⎪ f 1 ( τ , x1 ( τ ), x2 ( τ ), x3 ( τ ), x4 ( τ ) ) = x2 ⎪ ⎨ f2 (τ , x1 (τ ), x2 (τ ), x3 (τ ), x4 (τ ) ) = −(a + bx3 )x1 − (a + bx3 )x3 − cx2 + dx3 1 f τ , x ( τ ) , x ( τ ) , x ( τ ) , x ( τ ) = x ( ) ⎪ 3 1 2 3 4 4 ⎪ ⎪   ⎩ f4 (τ , x1 (τ ), x2 (τ ), x3 (τ ), x4 (τ ) ) = ex3 + f 1 − x23 x4 + gx1 (3.3) By using system (3.3) and the definition of Caputo derivative and keeping the same initial conditions of our system, with α1 = α2 = α3 = α4 = α , we come out with the solution up to 3rd order of system (3.1) given by: x01 = 0.1 x11

(t ) = 0.1 − 0.5t

x21

(t ) = 0.1 − 3.17t 2 +

x31 (t ) = x21 (t ) − 0.5t − 0.09t 2 − 5.35t 3 − −

103 4 1 6 7 5 t + t − t 800 100 160

t 3−2α 2(4 − 2α )

. . .

can be built as follows:

  t  ⎧ k+1 x1 (t ) = xk1 (t ) − 0 Dα∗ 1 xk (τ ) − f1 τ , xk1 (τ ), xk2 (τ ), xk3 (τ ), xk4 (τ ) dτ ⎪ ⎪   ⎪ ⎨xk+1 (t ) = xk (t ) − t Dα2 xk (τ ) − f τ , xk (τ ), xk (τ ), xk (τ ), xk (τ )dτ ∗ 2 2 0 2 2 3 4 2  1   ⎪xk3+1 (t ) = xk3 (t ) − 0t Dα∗ 3 xk3 (τ ) − f3 τ , xk1 (τ ), xk2 (τ ), xk3 (τ ), xk4 (τ ) dτ ⎪ ⎪    ⎩ k+1 t x4 (t ) = xk4 (t ) − 0 Dα∗ 4 xk4 (τ ) − f4 τ , xk1 (τ ), xk2 (τ ), xk3 (τ ), xk4 (τ ) dτ , (3.2)

t 2−α 2(3 − α )

x02 = −0.5 x12

(t ) = −0.5 + 6.34t

x22 (t ) = −0.5 + 12.67t − 16.04t 2 −

103 3 3 5 7 4 t 2−α t + t − t − 6.34 200 20 80 2(3 − α )

x32 (t ) = x22 (t ) + 6.16t + 43.2t 3 + 0.05t 4 + 3.66t 5 + 0.0025t 6 + 45.21t 7

T.A. Giresse, K.T. Crépin / Chaos, Solitons and Fractals 98 (2017) 88–100

+ 13.64t 9 + − −

t 3−α (3 − α )

64.55t 2−α 4.5t 6−α 6.34t 3−2α 8.4t 5−α − + + (3 − α ) (6 − α ) (7 − α ) (4 − 2α )



5.45 156.11t 4 10.32t 6 12.7t 2 − + + 3−α 5−α 7−α 9−α



t 5−2α

((3 − α ) )

2

−

25.62t 2 1.54 13.44t 4 − + 5 − 2α 7 − 2α 9 − 2α



t 7−3α



63 − 48α + 9α 2 ((3 − α ) )

3







12.6 − 21.6t 2 + α −4.2 + 9.25t 2



91

−

6.65t 3−2α 13.82t 7−3α − (4 − 2α ) (7 − 3α )((3 − α ) )3

−

t 3 −α (3 − α )

+

 



95.78t 2 28.8 79.63t 4 − + 3−α 5−α 7−α



t 5−2α



35 − 24α + 4α 4 ((3 − α ) )

2



× 241.91 − 287.34t 2 + α −69.12 + 114.93t 2

. . .



.. .

x03 = 0.1 x13

(t ) = 0.1 − 0.5t

x23 (t ) = 0.1 + 1.29t 2 +

t 2−α 2(3 − α )

z30 = 5

66253 4 t 3−2α 128753 5 x33 (t ) = x23 (t ) − 0.5t − 2.08t 3 − t + t − 120 0 0 0 0 80 0 0 0 0 2(4 − 2α )

z31

z32 (t ) = z31 + 6t − 362.45t 2 +

. . .

(t ) = −0.5 − 2.57t

+

66253 3 128753 4 t 2−α x24 (t ) = −0.5 − 5.15t − 6.25t 2 − t + t + 2.57 30 0 0 0 0 160 0 0 0 (3 − α )

+ 0.2t 8 − 0.7t 9 − 0.55



5t 5−2α

((5 − 2α ) )2

3−α

5t + (3 − α )

t 3−α

(4 − α )

−

z40 = 6

3.2t 7−3α

(7 − 3α )((3 − α ) )

3

0.13 0.2t 4 1.28t 1.74t 2 0.05t 3 − + + + − 5 − 2α 6 − 2α 7 − 2α 8 − 2α 9 − 2α



2.59 0.51t 0.6t 2 6.62t 3 0.28t 4 + + + + 3−α 4−α 5−α 6−α 7−α

1.03t 6 0.28 + − 8−α 9−α



z41 z42

(t ) = 6 − 724.9t (t ) = z41 − 724.9t + 44396.94t 2 − 72850t 3 + 32622t 4 +

− 3.18 ∗ 107 t 5 − 1.22 ∗ 107 t 6 + 8.16 ∗ 108 t 7 − 1.2 ∗ 109 t 8 88799.78t 3−α 1449.8t 2−α − 4.76 ∗ 108 t 9 + − (3 − α ) (4 − α ) +

3.2. Case of Duffing-Van der Pol system

−

As in the previous case, the Duffing-Van der Pol given by (2.2) is also a perfect non-linear system. Using the same procedure, we have the approximated solution of the Duffing-Van der Pol fractional system which is given as follows:

−

z11 (t ) = 2 − 2.4t

(t ) =

13.31t 3−α 2.4t 3−2α − (4 − α ) (4 − 2α )

z20 = 2.4

(t ) = 2.4 − 6.65t 6.65t 2−α

(3 − α )

z23 (t ) = z22 (t ) − 12.3t + 6.65t 2 + 10.35t 3 + 2.88t 4 − 13.96t 5 + 5.25t 7 −

69.12t 4−α 83.04t 5−α − (5 − α ) (6 − α )

+

−

t 5−2α

((3 − α ) )2

43710 52192.2t 1.55 ∗ 106 t 2 2.62 ∗ 106 t 3 + + + 5 − 2α 6 − 2α 7 − 2α 8 − 2α

1.17 ∗ 106 t 4 9 − 2α



+

t 3 −α

(3 − α )

17757.6 10320t 2 1.93 ∗ 106 t 3 86988t + − − 3−α 4−α 5−α 6−α

3.17 ∗ 108 t 5 9.6 ∗ 107 t 4 1.42 ∗ 108 t 6 − + 7−α 8−α 9−α



.. .

.. .

z22 (t ) = z21 + 6.65t + 13.59t 2 − 11.59t 3 + 3.46t 4 +

(7 − 3α )((3 − α ) )

3

× −

2

z13 (t ) = z12 (t ) + 2.4t − 6.65t 2 + 4.53t 3 − 2.88t 4 + 0.69t 5

z21

26096.4t 7−3α



2.4t 2−α + 2.4t − 3.32t + (3 − α )

+−

437100t 4−α 782928t 5−α 724.9t 3−2α − − (5 − α ) (6 − α ) (4 − 2α )

× −

z10 = 2

z11

724.9t 2−α (3 − α )

z43 (t ) = z42 (t ) − 148.9t + 17397.6t 2 − 347806.1t 3 − 876600t 4

. . .

z12

6t 3−2α 1449.8t 3−α − (4 − α ) (4 − 2α )

.. .

x34 (t ) = x24 (t ) − 2.61t − 12.7t 2 − 10.36t 3 + 1.35t 4 + 3.2t 5 + 7.1t 6 + 7.2t 7

+

6t 2−α

(3 − α )

z33 (t ) = z12 (t ) + 6t − 724.9t 2 − 14799t 3 − 18212.5t 4 + 6524.4t 5

x04 = −0.5 x14

(t ) = 5 − 6t

The approximated solution provided in this Section leads us to conclude that one can approximate the solution of such a system by simple power-series function which can be manipulable easily. These results reveal that the proposed method is very effective, simple and leads to accurate, approximately convergent solutions of non-linear equations. These approximated solutions preserve the non-linearity and give an additional insight into the dynamic of the simulated system.

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T.A. Giresse, K.T. Crépin / Chaos, Solitons and Fractals 98 (2017) 88–100

4. Stability analysis and synchronization scheme

point corresponding to system (2.2) is given by:

⎡

4.1. Case of the coupled Mathieu-Van der Pol system with fractional order-derivative In order to analyze the stability of the system, let us consider P (x∗1 , x∗2 , x∗3 , x∗4 ) as an equilibrium point. Then, using the property of α Caputo derivative where D∗ i c = 0 ∀ i = 1, 2, · · · , 4 with c a constant, we have:

⎧ α1 ∗ D x = 0 ⇒ x∗2= 0 ⎪ ⎨ ∗α2 1∗







∗ ∗ D∗ x2 = 0 ⇒ − a + bx∗3 x∗1 − a + bx∗3 x∗3 1 − cx2 + dx3 = 0 α3 ∗ ∗ D x = 0 ⇒ x = 0 ⎪ 4   ∗ ⎩ ∗α4 3∗ ∗ D∗ x4 = 0 ⇒ −ex∗3 + f 1 − x∗2 3 x4 + gx1 = 0.

(4.1) After simplifying Eq. (4.1), we get:



x∗3 = ge x∗1 bg ∗4 x − ax∗3 1 + e 1

bg ∗2 x e 1

+ (a −

bg e

0

g

1

0

0

−c

d − bx∗1 − bx∗3 1

0

⎤ ⎥ ⎥ ⎥ ⎦

0

0

1

0

−e − 2 f x∗3 x∗4

f (1 − x∗2 3 ).

(4.3) At P1 (x∗1 , x∗2 , x∗3 , x∗4 ) = P1 (0, 0, 0, 0 ), we have:

⎡

0

⎢ −a J ( 0, 0, 0, 0 ) = ⎣ 0 g

1 −c 0 0

0 d 0 −e

0 0 1 f

0 i 0 −j

0 0 1 k

⎤ ⎥ ⎦

(4.6)

The associated characteristic polynomial is the following: P (λ ) = λ4 − 5λ3 + 1.9976λ2 − 4.9994λ + 0.95. The resolution leads to: λ1 = −0 √ .1729, λ2 = 4.809, λ3 = 0.18 + 1.05i, λ4 = 0.18 − 1.05i with i = −1. We realize that λ2 = 4.809 is positive. Therefore, Q(0, 0, 0, 0) is an unstable equilibrium point. The study of the stability shows that, in both cases, we can never have natural stability of such a system even if the system presents equilibrium points. However, it is clear that such a system is controllable under certain conditions. That is what we are going to study in the next section. This confirms the fact that fractionalorder differential equations are, at least, as stable as their integer order counterpart. 4.3. Lyapunov exponent

J x1 , x2 , x3 , x4

⎢ −a + bx∗  − 3a + bx∗ x∗2 ⎢ 3 3 1 =⎢ ⎣ 0

1 −h 0 0

(4.2)

)x∗1 = 0.

Using Maxima to solve (4.2), we get the following values: x∗1 = 0 or x∗1 = −33.312, corresponding to two equilibrium points P1 (x∗1 , x∗2 , x∗3 , x∗4 ) = P1 (0, 0, 0, 0 ) and P2 (x∗1 , x∗2 , x∗3 , x∗4 ) = P2 (−33.312, 0, −3.3312, 0 ). The Jacobian matrix at the equilibrium point associated to system (2.1) is given by:  ∗ ∗ ∗ ∗

⎡

⎢

J ( 0, 0, 0, 0 ) = ⎣

0 −1 0 l

⎤ ⎥ ⎦

(4.4)

The characteristic polynomial of this matrix is given by:

P (λ ) = λ4 − 4.6λ3 + 9λ2 − 49.6λ + 3 Using Maxima, we get the following eigenvalues: λ1 = 0.0611, λ2 = 4.884, λ3 = −0.147 + 3.2i, λ4 = −0.147 − 3.2i. We realize that λ1 and λ2 are positive, which means that P1 (0, 0, 0, 0) is an unstable equilibrium point. Doing the same analysis for P2 (−33.312, 0, −3.3312, 0 ), we come out with the following characteristic polynomial: P (λ ) = λ4 + 50.88λ3 + 42.2λ2 + 1061.26λ − 11086.1, where after solving using Maxima we get λ√ 1 = 50.54, λ2 = −4.6, λ3 = 2.45 − 6.46i, λ4 = 2.45 + 6.46i with i = −1 similarly, λ1 = 50.54 is positive. Therefore, P2 (−33.312, 0, −3.3312, 0 ) is also an unstable equilibrium point. This is in agreement with the results found by Kumar in [21].

The Lyapunov exponent is a dynamical quantity characterizing the rate of separation of infinitesimally closed trajectories. For any given Lyapunov exponent, the following statements hold:  if λ < 0 then, slightly separated, the orbit attracts to a stable point or stable periodic orbit. That means trajectories converge,  if λ > 0 then, divergence of trajectories and the evolution of the system is sensitive to initial condition. In this case, the system presents a chaotic behavior. The synchronization occurs when all the Lyapunov exponents are negative. In this case, the Lyapunov exponent can be understood as the measure of sensitivity or dependence of a system to initial conditions. Obviously, how quickly two nearby state diverge. Let us consider x ∈ Rn and f a vector function such that f : Rn −→ Rn . Then, for an orbit starting with initial condition x0 the Lyapunov exponent in Rn is defined by:

λ(x0 ) = nlim −→∞

n−1 1 log | f (xk )|. n

(4.7)

k=0

where f (xk ) stands for the Jacobean matrix at x=x0 . But the most interesting in this part is the Lyapunov exponent for fractional order-derivative system. For this purpose, let’s . be an arbitrary norm in Rn then, we have the following definition for fractional Lyapunov exponent. Definition 4.3.1 (ref. [27]). Let f : Rn −→ Rn be an arbitrary function, we have:

λα ( f ) = lim sup t −→∞

1 N logα f (t ) tα

(4.8)

The theorem below provides a practical formulation of λα (f). 4.2. Case of the coupled Duffing-Van der Pol system with fractional order-derivative

Theorem 4.3.2 ([27]). Let f : Rn+ −→ Rn be an arbitrary function. The following conditions are satisfied:

As in the previous case, to study the stability of the system, we consider Q (z1∗ , z2∗ , z3∗ , z4∗ ) as an equilibrium point. Then:

 λα (f) > 0 if and only if λ(f) > 0. This implies that:

D∗ z2 = 0 ⇒ −z1 − z1∗3 − hz2∗ + iz3∗ = 0 Dα3 z∗ = 0 ⇒ z4∗ = 0  ⎪  ⎩ ∗α4 3∗ D∗ z4 = 0 ⇒ − jz3∗ + k 1 − z3∗2 z4∗ + lz1∗ = 0.

 When limt −→∞ supt α f (t ) is finite then, λα (f) < 0. In this case,

⎧ α1 ∗ D z = 0 ⇒ z2∗ = 0 ⎪ ⎨ ∗α2 1∗ ∗

1 α λα ( f ) = lim sup α logNα f (t ) , t −→∞

(4.5)

After solving system (4.5) we find only one possible equilibrium point which is Q(0, 0, 0, 0). The Jacobean matrix at this equilibrium

λα ( f ) =

t

1

(1 − α ) limt −→∞ supt α f (t )

,

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93

Fig. 3. Fractional Lyapunov exponent for the Mathieu-Van der Pol and Duffing-Van der Pol systems with q = 0.95.

 λα ( f ) = 0 if and only if λ(f) ≤ 0 and limt −→∞ sup t α f (t ) is infinite. Applying this theorem to the Fractional Mathieu-Van Der Pol and Duffing-Van Der Pol systems, the largest Lyapunov exponent for both systems has been simulated and the behavior is shown on Fig. 3: It is observed from these graphs that, the systems move quickly from non chaotic to hyper-chaotic character confirming the results found previously. This is the real proof that the fractional Mathieu-Van der Pol and Duffing-Van der Pol systems behave chaotically as shown by their phase portraits. 4.4. Synchronization scheme Generally, Chaos synchronization implies making two chaotic systems with different initial conditions oscillate in the same way by using an active control parameter. Let us consider the following coupled chaotic systems.

Dα∗ i x(t ) = f (t, x ), Dα∗ i y(t ) = g(t, y ) + u(t, x, y ),

(4.9)

(4.10)

αi

where D∗ , denotes the Caputo derivative of order αi , i = 1 , 2 , · · · , n . x = [x1 , x2 , x3 · · · , xn ]T ∈ Rn and y= T n [y1 , y2 , y3 · · · , yn ] ∈ R , denote the master and the slave (or response) equations, respectively. f and g are two vector functions so that f : Rn → Rn , consequently, g : Rn → Rn . It is important to notice that f and g are continuous non-linear functions that fulfill the Lipschitz condition. u(t, x, y) denotes the control parameter vector and is defined such that u = [u1 , u2 , x3 · · · , un ]T ∈ Rn . Let’s define the error vector function as follows: e(t ) = [e1 (t ), e2 (t ), e3 (t ) · · · , en (t )]T ∈ Rn , with ei (t ) = xi (t ) − yi (t ). The synchronization can be achieved when

lim e(t ) = lim x(t ) − y(t ) → 0,

t →∞

t →∞

where . denotes the Euclidean norm. The error dynamics corresponding to Eqs. (4.9) and (4.10) is defined as follows:

Dα∗ i ei (t ) = fi (t, x1 , x2 , · · · , xn ) − gi (t, y1 , y2 , · · · , yn ) − ui ,

(4.11)

But the error dynamical system to be controllable need to be a linear system with control input. That is, we redefine the

control function such that the non-linear term in ei (t ) ∀ i = 1, 2, 3, · · · , n cancels. Furthermore, synchronization implies also stabilization of the system. So, from the partial region stability theory [28], one can construct the Lyapunov function from where the controller can be designed. In fact, if we consider a function V (x ) : Rn → Rn such that, the following conditions:  V(x) is a positive definite function (V(x) ≥ 0 with equality if and only if x = 0),  The derivative of V(x) is a negative semi-definite function α ( d dtVα(x ) ≤ 0) with additional property like boundedness, are fulfilled. Then, V(x) is a good Lyapunov function candidate. In these conditions, the system is stable in the sense of Lyapunov. The concrete examples are studied in Section 5, where the master and the slave equations are both Mathieu-Van der Pol fractional orderderivative system and also for Duffing-Van der Pol fractional orderderivative system. The study of the stability of these systems leads to the conclusion that, taking into consideration the fractional derivative can lead to more stability of the system. On the other hand, the synchronization scheme proposed in this Section shows that, compared to other methods, the synchronization by the GYC partial region stability theory is very appropriated. Moreover, from the phase portrait representation and the lyapunov exponent, we conclude that, these systems effectively exhibit chaotic behavior.

5. Numerical simulations In this Section, the purpose is to simulate the synchronization of the Mathieu-Van der Pol and Duffing-Van der Pol systems with fractional order-derivative, the time evolution of the error dynamics after computing it for each system and the phase portrait of this error. For this reason, we are called to solve systems of differential equation with fractional order-derivative numerically. The fractional Adams-Bashforth-Moulton method is used to approximate the solution. It is an efficient method for linear or non-linear fractional order differential equations and it is very accurate and clearly developed by Haci et al. [29]. All the simulations have been done using the free software Python.

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Fig. 4. Phase portrait for chaotic Mathieu-Van der Pol error dynamical system with fractional order derivative (α = 0.98).

5.1. The Mathieu-Van der Pol system with fractional order-derivative Let us consider as the unidirectional master and slave (or response) systems, the Mathieu-Van der Pol equation using fractional order-derivative both defined as follows:

⎧ α1 D x (t ) = x2 ⎪ ⎨ ∗α2 1 D∗ x2 (t ) = −(a + bx3 )x1 − (a + bx3 )x31 − cx2 + dx3 Master equation: α3 ⎪D∗ x3 (t ) = x4   ⎩ Dα∗ 4 x4 (t ) = −ex3 + f 1 − x23 x4 + gx1 ,

In order to have Dα∗ V as a negative semi-definite function, let’s choose our control parameters as follows:

⎧ u1 = x2 − y2 + e1 ⎪ ⎪ ⎪ ⎨u2 = −[(a + bx3 )x1 − (a + by3 )y1 ] − [(a + bx3 )x31 − (a + by3 )y31 ] − c ( x2 − y2 ) + d ( x3 − y3 ) + e2 ⎪ ⎪ u3 = x4 − y4 + e3 ⎪     ⎩ u4 = −e(x3 − y3 ) + f 1 − x23 x4 − f 1 − y23 y4 + g(x1 − y1 ) + e4 .

(5.6)

(5.1) And

⎧ α1 D y (t ) = y2 + u1 ⎪ ⎨ ∗α2 1 D∗ y2 (t ) = −(a + by3 )y1 − (a + by3 )y31 − cy2 + dy3 + u2 Slave equation: α3 ⎪D∗ y3 (t ) = y4 + u3   ⎩ Dα∗ 4 y4 (t ) = −ey3 + f 1 − y23 y4 + gy1 + u4 , (5.2) where u1 , u2 , u3 , u4 are control parameters to be designed. Dα∗ i ∀ i = 1, · · · , 4 stands for Caputo derivative. a, b, c, d, e, f are the parameters defined by a=10, b=3, c=0.4, d=70, e=1, f=5, g=0.1 under the initial conditions defined by x10 = −0.5, x20 = 0.1, x30 = −0.5, x40 = 0.1 for the master system and y10 = 0.3, y20 = −0.1 ,y30 = 0.3, y40 = −0.1 for the slave system. The goal is to be able to synchronize the system introducing a control. That is, the generalized synchronization as developed in Section 4.4 of the error dynamics is given by ei (t ) = xi (t ) − yi (t ). The synchronization is achieved when the error tends to zero as the time increases such that we have yi −→ xi ∀ i = 1, 2, · · · , n.

Having ei (t ) = xi (t ) − yi (t ) ∀ i = 1, 2, · · · , n, we choose to make this error always happens in the first quadrant by taking ei (t ) = xi (t ) − yi (t ) + 100 ∀ i = 1, 2, · · · , n. Using this, we derive the following error dynamical system using Eqs. (5.1) and (5.2): ⎧ α1 D∗ e1 (t ) = x2 − y2 − u1 ⎪ ⎪ ⎪Dα∗ 2 e2 (t ) = −[(a + bx3 )x1 − (a + by3 )y1 ] − [(a + bx3 )x3 − (a + by3 )y3 ] ⎨ 1 1 − c ( x2 − y2 ) + d ( x3 − y3 ) − u2 α3 ⎪ ⎪ ⎪     ⎩D∗α4 e3 (t ) = x4 − y4 − u3 D∗ e4 (t ) = −e(x3 − y3 ) + [ f 1 − x23 x4 − f 1 − y23 y4 ] + g(x1 − y1 ) − u4 .

(5.3) By the GYC partial region stability theory clearly stated in [14], the Lyapunov function can be chosen as a simple linear positive semi-definite function. In this case, the good candidate is:

Then, from system (5.5), we have: Dα ∗ V = −e1 − e2 − e3 − e4 < 0 since ei

∀ i = 1, 2, · · · , n is positive definite-function in the first quadrant. Fig. 4 below depicts the phase portrait of the error dynamics for α = 0.98. We can easily remark from this picture that, as in the case of integer-order derivative, the error dynamics also behaves chaotically for fractional order-derivative. This behavior is obtained just before the control action (control-off). Figs. 5–8 depict the synchronization of the Mathieu-Van der Pol system using active control (control-on). In these pictures, Figs. 5a, 6a, 7a and 8a represent the synchronization of the system with fractional orderderivative (α = 0.95) while Figs. 5b, 6b, 7b and 8b represent the synchronization of the system using integer order-derivative. The zoomed parts in these pictures show clearly how the synchronization starts. Observing carefully these pictures, we realize that, for the fractional order-derivative of the system, the synchronization starts earlier compared to the synchronization of the system with integer order-derivative where we have α = 1. We conclude that using fractional order-derivative is better for chaos synchronization than using integer order-derivative. This implies that, as predicted in Section 4, the system tends to be more stable as the order of derivative decreases. But this should not exceed some order otherwise it will not be more possible to observe chaos in the system. The same conclusions are observed when analyzing the DuffingVan der Pol system by the same procedure. Let us now consider the case where the generalized synchronization error function is defined in the following form:

(5.4)

V = e1 + e2 + e3 + e4 . For the reason of simplification, we assume this assumption, we get

α1 = α2 = α3 = α4 = α . Using

ei =

1 3 x − yi + 10 0 0 0. 3 i



Dα∗ V = (x2 − y2 − u1 ) + − [(a + bx3 )x1 − (a + by3 )y1 ] − [(a + bx3 )x31



−(a + by3 )y31 ] − c (x2 − y2 ) + d (x3 − y3 ) − u2 + x4 − y4 − u3









− e(x3 − y3 ) + [ f 1 − x23 x4 − f 1 − y23 y4 ] + g(x1 − y1 ) − u4 .

(5.5)

where the 10,0 0 0 added is to make the error always happens in the first quadrant. Taking its Caputo derivative and applying the α α α property D∗ i ( f g) = f D∗ i (g) + gD∗ i ( f ), we get:

T.A. Giresse, K.T. Crépin / Chaos, Solitons and Fractals 98 (2017) 88–100

Fig. 5. synchronization of chaotic Mathieu-Van der Pol system with fractional order-derivative compared to integer order-derivative for (x1 and y1 ) via active control.

Fig. 6. synchronization of chaotic Mathieu-Van der Pol system with fractional order-derivative compared to integer order-derivative for (x2 and y2 ) via active control.

Fig. 7. synchronization of chaotic Mathieu-Van der Pol system with fractional order-derivative compared to integer order-derivative for (x3 and y3 ) via active control.

95

96

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Fig. 8. synchronization of chaotic Mathieu-Van der Pol system with fractional order-derivative compared to integer order-derivative for (x4 and y4 ) via active control.

α

α

α

D∗ i ei = x2i D∗ i xi − D∗ i yi . Using systems (5.1) and (5.2) we get the following error dynamical system: ⎧ α D∗ 1 e1 (t ) = x21 x2 − y2 − u1 ⎪ ⎪ ⎪ ⎪ α ⎪ ⎪D∗ 2 e2 (t ) = −[(a + bx3 )x1 x22 − (a + by3 )y1 ] − [(a + bx3 )x31 x22 − (a + by3 )y31 ] ⎨ − c (x32 − y2 ) + d (x3 x22 − y3 ) − u2 ⎪ α3 ⎪ ⎪ D∗ e3 (t ) = x23 x4 − y4 − u3 ⎪ ⎪ ⎪     ⎩ α4 D∗ e4 (t ) = −e(x3 x24 − y3 ) + [ f 1 − x23 x34 − f 1 − y23 y4 ] + g(x1 x24 − y1 ) − u4 .

(5.7)

Der Pol system are strongly correlated since we can realize that as the order of the derivative increase the error goes quickly to zero this confirm the results found in the previous section for which the time of synchronization depend on the order of the derivative of the system.

5.2. The Duffing-Van der Pol system with fractional order-derivative In this section, we consider as master (or drive) and slave (or response) systems, the Duffing-Van der Pol equation with fractional order derivative. That is, we define them by:

⎧ α1 D z = z2 ⎪ ⎨ ∗α2 1

By the same procedure as in the previous case the control parameters are chosen in the following form:

⎧ u1 = x21 x2 − y2 + e1 ⎪ ⎪ ⎪ 2 3 2 3 ⎪ ⎪ ⎨u2 = −[(a + bx3 )x1 x2 − (a + by3 )y1 ] − [(a + bx3 )x1 x2 − (a + by3 )y1 ] − c (x32 − y2 ) + d (x22 x3 − y3 ) + e2 ⎪ ⎪ 2 ⎪ u ⎪ 3 = x3 x4 − y4 + e3 ⎪     ⎩ u4 = −e(x3 x24 − y3 ) + f 1 − x23 x34 − f 1 − y23 y4 + g(x1 x24 − y1 ) + e4 .

D∗ z2 = −z1 − z13 − hz2 + iz3 Dα3 z = z4 ⎪   ⎩ ∗α4 3 D∗ z4 = − jz3 + k 1 − z32 z4 + lz1 ,

Master system

and

(5.8) Fig. 9 represents the phase portrait of the error dynamics and we can easily see that the system presents chaotic behavior. Figs. 10–13 represent the time evolution of the error function before and after the control action. It is clear that applying our control kindly designed we are able to stabilize the system from the fact that the error goes to zero. From these graphs plotted for different values of the derivative, it is observed that the error dynamics and the order of the derivative α of the fractional Mathieu Van

(5.9)

⎧ α1 D y = y2 + u1 ⎪ ⎨ ∗α2 1 3

Slave system

D∗ y2 = −y1 − y1 − hy2 + iy3 + u2 α3

D y = y4 + u3  ⎪ ⎩ ∗α4 3

D∗ y4 = − jy3 + k 1 −

y23



(5.10)

y4 + ly1 + u4 ,

α

where of course D∗ i stands for Caputo derivative. u1 , u2 , u3 , u4 are control parameters to be designed. h, i, j, k, l are given perimeters defined in Section 4.2. Using the same procedure as in the previous Section 5.1, we derive the error dynamics ei = zi − yi ∀ i =

Fig. 9. Phase portrait of the generalized synchronization error function of chaotic Mathieu-Van der Pol system with fractional order derivative just before the control action (ei = 13 x3i − yi + 10 0 0 0).

T.A. Giresse, K.T. Crépin / Chaos, Solitons and Fractals 98 (2017) 88–100

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Fig. 10. Time evolution of the error of chaotic Mathieu-Van der Pol system with different order of derivative before introducing the control Fig. 10a and when the control is introduced Fig. 10b for (e1 ).

Fig. 11. Time evolution of the error of chaotic Mathieu-Van der Pol system with different order of derivative before introducing the control Fig. 11a and when the control is introduced Fig. 11b for (e2 ).

Fig. 12. Time evolution of the error of chaotic Mathieu-Van der Pol system with different order of derivative before introducing the control Fig. 12a and when the control is introduced Fig. 12b for (e3 ).

1, 2, · · · , n as follows:

Then, our control parameter should be chosen such that the non-linear part cancels in the error dynamics. This implies that we can choose it as follows:

⎧ α1 D e = z2 − y2 − u1 ⎪ ⎨ ∗α2 1 D∗ e2 = −(z1 − y1 ) − (z13 − y31 ) − h(z2 − y2 ) + i(z3 − y3 ) − u2 Dα3 e = z4 − y4 − u3 ⎪      ⎩ ∗α4 3 D∗ e4 = − j (z3 − y3 ) + k 1 − z32 z4 − 1 − y23 y4 + l (z1 − y1 ) − u4 , (5.11) By the GYC partial region stability theory [14], the Lyapunov exponent is chosen as a simple linear positive semi-definite function (see Section 5.1).

⎧ u = z2 − y2 + e1 ⎪ ⎨ 1 u2 = −(z1 − y1 ) − (z13 − y31 ) − h(z2 − y2 ) + i(z3 − y3 ) + e2 u = z4 − y4 + e3 ⎪      ⎩ 3 u4 = − j (z3 − y3 ) + k 1 − z32 z4 − 1 − y23 y4 + l (z1 − y1 ) + e4 ,

(5.12)

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Fig. 13. Time evolution of the error of chaotic Mathieu-Van der Pol system with different order of derivative before introducing the control Fig. 13a and when the control is introduced Fig. 13b for (e4 ).

The synchronization of the system leads to the same conclusion as in the case of Mathieu-Van der Pol system, the same aspects are observed. The representation of the phase portrait of the error dynamics leads also to the conclusion that the system is still chaotic. We would like now to consider the case where the generalized synchronization error function is defined by:

So, one can choose the control parameters as follows:

⎧ u1 = x2 − y2 + z2 + e1 ⎪ ⎪ ⎪ u = −[(a + bx3 )x1 − (a + by3 )y1 ] − [(a + bx3 )x31 − (a + by3 )y31 ] ⎪ ⎪ ⎨ 2 − c (x − y ) + d (x − y ) + F (z ) + e 2

2

3

3

2

u3 = x4 − y4 + z4 + e3 ⎪    ⎪ ⎪ u4 = −e(x3 − y3 ) + f 1 − x23 x4 − f 1 − y23 y4 + g(x1 − y1 ) − jz3 ⎪ ⎪   ⎩ + k 1 − z32 z4 + lz1 + e4 ,

(5.14) ei = xi − yi + zi + 100

i = 1, 2, 3, 4,

(5.13)

where xi and yi are all Mathieu-Van der Pol systems (see Section 5.1). zi denotes the Duffing-Van der Pol system, with fractional order derivative. The goal is to have limt → ∞ ei → 0 which implies that xi − yi + 100 → −zi ∀ i = 1, · · · , n. By the GYC partial region stability theory, the Lyapunov function as in the previous case is chosen as a positive semi-definite function as follows: V = e1 + e2 + e3 + e4 , from where, we have:



Dα∗ V = (x2 − y2 + z2 − u1 ) + − [(a + bx3 )x1 − (a + by3 )y1 ]



− [(a + bx3 )x31 − (a + by3 )y31 ] − c (x2 − y2 ) + d (x3 − y3 ) − z1





− z13 −hz2 + iz3 −u2 + x4 − y4 + z4 − u3 − e(x3 −y3 ) + [ f 1−x23 x4









− f 1 − y23 y4 ] + g(x1 − y1 ) − jz3 + k 1 − z32 z4 + lz1 − u4 ,

where F (z ) = −z1 −

z13

− hz2 + iz3 .

Having Eq. (5.14), we obtain Dα ∗ V = −e1 − e2 − e3 − e4 < 0 since ei ∀ i = 1, 2, · · · , n is positive definite-function in the first quadrant. Figs. 14 and 15 represent the evolution in time of xi − yi + 100 and −zi . we observe from these pictures a perfect synchronization of the systems after applying the control. This observation leads to the conclusion that the Duffing-Vander Pol can be used to control the Mathieu-Van der Pol system. In this Section deserved for results and simulations, we observed the following aspects:  The simulation of the synchronization of both the MathieuVan der Pol and the Duffing-Van der Pol systems using fractional order-derivative shows that the synchronization starts earlier compared to the synchronization with integer orderderivative,  Secondly, looking carefully at the phase portraits representation of the two systems and their largest Lyapunov expo-

Fig. 14. synchronization of coupled Mathieu-Van der Pol system with coupled Duffing-Van der Pol system using fractional order-derivative, q = α = 0.96.

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Fig. 15. synchronization of coupled Mathieu-Van der Pol system with coupled Duffing-Van der Pol system using fractional order-derivative, q = α = 0.96.

nent, we can conclude that the Mathieu-Van der Pol and Duffing-Van der Pol systems using fractional order-derivative have a least degree of chaos, due to the fact that they are more stable compared to integer order-derivative,  The time evolution representation of the error dynamics prove that, using an appropriate controller, these systems initially uncontrollable and unstable can be stabilized and controllable too. This effect is shown in this Section considering as master the Mathieu-Van der Pol and the Duffing-Van der Pol as the slave system.

and capability to keep an object in the same region as the time passes. The above results find their application in the communication processing tasks in the sense that the transmitted signal might be injected into the transmitter (chaotic system) and, simultaneously, transmitted to the receiver. By the proposed synchronization technique, a chaotic receiver is then derived to recover the information signal at the receiving end of the communication. In addition, we have discovered that with fractional order derivative the synchronization is quietly fast, that means the information will be recovered faster than using integer order derivative and with less error. This difference of time is of great importance in the technology of communication.

6. Conclusion In this paper, proposing a solution to the problem of synchronization and controllability of Mathieu-Van der Pol and DuffingVan der Pol systems with fractional order-derivative was our main objective. For this purpose, we considered the more general case where we have different order-derivative of coupled Mathieu-Van der Pol and coupled Duffing-Van der Pol systems that means α 1 = α 2 = α 3 = α 4 and we discovered after simulations that, approaching the synchronization by GYC partial region stability theory is more efficient, from the fact that, the error is minimized and approaches zero quickly once introducing our control appropriately chosen. Moreover, the simulations of the error dynamics with respect to different order of the derivative show the dependency of stability of the systems with respect to this parameter. In addition, it follows also from simulations that for both systems, the synchronization is very fast compared to the case where we consider the integer order-derivative. This is a great advantage when using fractional derivative in the synchronization process. The study of the stability of both systems shows that, the system with fractional order-derivative tends to be more stable compared to the case with integer order-derivative. We remarked in both cases that, these systems present some equilibrium points all unstable but which can be stabilized if some given conditions are fulfilled. This conclusion is in agreement with the results found by Kumar [21]. Furthermore, the phase portrait representation of both the Mathieu-Van der Pol and the Duffing-Van der Pol systems with fractional order-derivative guarantees that chaos occurs in the system after some time. But this chaos can occur in the system only under certain initial conditions and for parameters of the systems clearly defined. We realized that, for the values of α between (0.9 and 1), we get chaos in the systems. These graphs represent what we call in the nature strange-attractor from their strange behavior

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