Modified sliding mode synchronization of typical three-dimensional fractional-order chaotic systems

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Modified sliding mode synchronization of typical three-dimensional fractional–order chaotic systems Like Gao, Zhihui Wang, Ke Zhou, Wenji Zhu, Zhiding Wu, Tiedong Ma

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S0925-2312(15)00478-6 http://dx.doi.org/10.1016/j.neucom.2015.04.031 NEUCOM15389

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Received date: 30 January 2015 Revised date: 20 March 2015 Accepted date: 10 April 2015 Cite this article as: Like Gao, Zhihui Wang, Ke Zhou, Wenji Zhu, Zhiding Wu, Tiedong Ma, Modified sliding mode synchronization of typical threedimensional fractional–order chaotic systems, Neurocomputing, http://dx.doi. org/10.1016/j.neucom.2015.04.031 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Modified sliding mode synchronization of typical three-dimensional fractional-order chaotic systems Like Gao a, Zhihui Wang b , Ke Zhou a, Wenji Zhu a, Zhiding Wu a , Tiedong Ma b,c,∗ , a Electric

Power Research Institute, Guangxi Power Grid Corporation, Nanning 530023, China

b College

of Automation, Chongqing University, Chongqing 400044, China

c Key

Laboratory of Dependable Service Computing in Cyber Physical Society (Chongqing University), Ministry of Education, Chongqing 400044, China

Abstract In this paper, a modified sliding mode control scheme is proposed to realize complete synchronization of a class of three-dimensional fractional-order chaotic systems. By constructing the suitable sliding mode surface with fractional-order derivative, a single-state sliding mode controller is designed to realize the asymptotical stability of synchronization error system. Compared with the existing results, the main results in this paper are more reasonable and rigorous. Simulation results show the effectiveness and feasibility of the proposed sliding mode control method. Key words: Chaos synchronization; fractional-order chaotic system; fractional-order derivative; sliding mode control (SMC).

1

Introduction

Even though fractional differential calculus dates from 17th century, its application to physics and engineering only develops in recent years. It has been found that many physical phenomena, including certain types of electrical ∗ Corresponding author. Email address: [email protected], [email protected] (Tiedong Ma). Submitted to Neurocomputing for possible publication

16 April 2015

noises, relaxation behaviours of polarized impedances in dielectrics and interfaces, transmission lines, cardiac rhythm, and spectral densities of music, are known to exhibit fractional-order (or called noninteger-order) dynamics [1, 2]. Since the pioneering work of Pecora and Carroll [3], chaos synchronization has become more and more interesting to researchers in different fields [4–21]. Recently, due to the wide application in secure communication and process control, many different methods have been applied theoretically and experimentally to synchronize the fractional-order chaotic systems, such as active control method [22–25], sliding mode control method [26–29], adaptive control method [30–33], observer-based control method [34–37], impulsive control method [38–41], etc. Among these methods, sliding mode control (SMC) is an efficient method to deal with the robust control scheme. The main feature of SMC is to switch the control law to force the states of the system from the initial states onto some predefined sliding mode surface. The system on the surface has desired performance such as stability, disturbance rejection capability and tracking ability. In [42, 43], the authors investigate adaptive synchronization of a class of three-dimensional fractional-order chaotic systems via single driving variable. To further deal with the stability analysis of an integer-order system mixed with a fractional-order system, the fractional-order Lyapunov stability theorem is used in [44]. In the above papers, one assumption for the threedimensional fractional-order chaotic systems is given (i.e., if one state variable z(t) equal to zero, the rest two-dimensional subsystem is asymptotically stable), which is applicable to many typical fractional-order chaotic systems. However, the presented conditions in [42–44] just satisfy the asymptotically stability (i.e., z(t) → 0 as t → ∞), which is contradictory to the condition z(t) = 0 in the assumption. Motivated by the aforementioned discussion, in this paper we address the modified sliding mode synchronization scheme for a class of fractional-order chaotic systems. The proposed sliding surface and single state controller can achieve the finite-time convergence to zero, which can use the above assumption to ensure the asymptotical stability of the error dynamics. Compared with the existing results [42–44], the criteria derived in this paper are more reasonable and rigorous. Finally, three simulation examples are given to demonstrate the effectiveness of the proposed sliding mode control method. The rest of this paper is organized as follows. In Section 2, the problem formulation and main results for sliding mode synchronization are derived. In Section 3, two illustrative examples for fractional-order Arneodo’s and unified systems are presented to demonstrate the effectiveness and feasibility of the proposed method. Finally, concluding remarks are given in Section 4. 2

2

Main results

Consider the following three-dimensional fractional-order chaotic error system [42]: D α x(t) = f1 (x, z), D α z(t) = f2 (x, z),

(1)

where D α = dα /dtα is generally called “α-order Caputo differential operator”, α ∈ (0, 1), x ∈ R2 , z ∈ R are the state error vectors, f1 and f2 are continuous differential nonlinear functions with f1 (0, 0) = f2 (0, 0) = 0. Assumption 1 [42] The function f1 (x, z) in (1) is smooth in a neighborhood of z = 0, and the subsystem D α x(t) = f1 (x, 0) is asymptotically stable about the origin x = 0 for all x. To study the synchronization of fractional-order chaotic systems, i.e., the asymptotical stability of system (1), we add the following feedback nonlinear controller u(x, z) ∈ R to the error system (1), D α x(t) = f1 (x, z), D α z(t) = f2 (x, z) + u(x, z),

(2)

The main idea of sliding mode control is designing a discontinuous control to force the system state trajectories to some predefined sliding mode surface, which is composed of two steps: first, constructing a sliding mode surface that has desired properties. Second, developing a switch control law to make the trajectories to reach the surface and remain on it evermore. We choose the following nonsingular fractional-order sliding mode surface as

s(t) = D α−1 z(t) + kD α−2 (z + sign(z)|z|γ ),

(3)

where sign(·) is the sign function, k > 0 and γ ∈ (0, 1) are constants. Once the system trajectory reaches the sliding surface (3), it yields the following equations: s(t) = 0 and s(t) ˙ = 0. (4) From (3) and (4), the dynamics of sliding mode can be achieved as follows:

it further yields

s(t) ˙ = D α z + kD α−1 (z + sign(z)|z|γ ) = 0,

(5)

D α z = −kD α−1 (z + sign(z)|z|γ ).

(6)

3

Before we give the main results, the fractional-order Lyapunov stability theorem can be stated as follows. Lemma 1 [45,46] Let x = 0 be an equilibrium point for the nonautonomous fractional-order system (7) D α x = f (x, t), where f (x, t) satisfies the Lipschitz condition with Lipschitz constant l > 0 and α ∈ (0, 1). Assume that there exists a Lyapunov function V (t, x(t)) satisfying α1 ||x||a ≤ V (t, x) ≤ α2 ||x||,

(8)

V˙ (t, x) ≤ −α3 ||x||,

(9)

where α1 , α2 , α3 and a are positive constants. Then the equilibrium point of the system (7) is Miattag-Leffler (asymptotic) stable. In the following theorem, the finite-time stability of the sliding surface (3) is proved. Theorem 1 The sliding mode dynamics (6) is asymptotic stable and its trajectories converge to the equilibrium z = 0 in a finite time. Proof Considering the following Lyapunov function: V (t) = |z(t)|,

(10)

and its derivative along the trajectory of system (6) is V˙ (t) = sign(z)z(t) ˙ = sign(z)D 1−α (D α z(t)) = sign(z)D 1−α (−kD α−1 (z + sign(z)|z|γ )) = −k(sign(z)z + sign(z)sign(z)|z|γ ) = −k(|z| + |z|γ ).

(11)

Since |z| and |z|γ are positive, it yields easily that V˙ (t) ≤ −k|z|.

(12)

From Lemma 1, the Lyapunov function V (t) satisfies condition (8) (if let α1 = 0.5, α2 = 2, a = 1) and condition (9) (with α3 = k > 0), thus the error variable z(t) will converge to zero asymptotically. In the follows, the finite-time convergence of z(t) will be further investigated. From Eq.(11), one has d|z(t)| = −k(|z| + |z|γ ), V˙ (t) = dt 4

(13)

it is obvious that d|z(t)| k(|z| + |z|γ ) |z(t)|−γ d|z(t)| =− k(|z|1−γ + 1) d|z(t)|1−γ . =− k(1 − γ)(|z|1−γ + 1)

dt = −

(14)

Taking integral of both side of (14) from t1 (the reaching time for state z(t) to the sliding surface s(t)) to t2 (the convergence time for z(t) to zero), and considering z(t2 ) = 0, thus we have  z(t2 ) d|z(t)|1−γ 1 t2 − t1 = − k(1 − γ) z(t1 ) |z(t)|1−γ + 1 z(t2 ) 1  ln(|z(t)|1−γ + 1) =− z(t1 ) k(1 − γ) 1 (ln(|z(t2 )|1−γ + 1) − ln(|z(t1 )|1−γ + 1)) =− k(1 − γ) 1 ln(|z(t1 )|1−γ + 1). = k(1 − γ)

(15)

Therefore, the state trajectory z(t) can converge to zero in finite time t2 = t1 +

1 ln(|z(t1 )|1−γ + 1). k(1 − γ)

(16)

This completes the proof. After constructing a fractional-order sliding mode surface, the second step is to design nonlinear feedback controller u(x, z) to force the error state z(t) reach the sliding surface and remain on it evermore. In what follows, we propose the suitable controller to ensure the existence of the sliding motion. Theorem 2 If system (2) is controlled with the following controller: u(x, z) = −f2 (x, z) − D α−1 (kz + ksign(z)|z|γ ) − ξs − ξsign(s)|s|γ ,

(17)

where k > 0, ξ > 0 and γ ∈ (0, 1) are constants, then the error state z(t) will converge to the sliding surface s(t) = 0 in finite time. Proof Selecting a Lyapunov function in the form of V (t) = |s(t)| and the time derivative of V (t) is obtained as follows: V˙ (t) = sign(s)s(t) ˙ = sign(s)(D α z(t) + kD α−1 (z + sign(z)|z|γ )) 5

(18)

From error dynamics (2), Eqs.(17) and (18), one has V˙ (t) = sign(s)(f2 (x, z) + u(x, z) + kD α−1 (z + sign(z)|z|γ )) = sign(s)(−ξs − ξsign(s)|s|γ ) = −ξ(|s| + |s|γ ).

(19)

Obviously, the error state trajectories will converge to the sliding surface s(t) = 0 asymptotically. Similar to the proof of Theorem 1, we can also obtain the results that the sliding motion occurs in finite time. Referring to Eqs.(13) ∼ (15), the reaching time t1 can be derived as t1 = t0 +

1 ln(|s(t0 )|1−γ + 1), ξ(1 − γ)

(20)

where t0 is the initial time. The detailed proof is similar to that of Theorem 1, which is omitted here. This completes the proof. Remark 1 From Theorems 1 and 2, we can obtain that the error state trajectories of system (2) will first converge to s(t) = 0 in finite time t1 , and then the error state z(t) will converge to zero in finite time t2 . Based on Assumption 1, once z(t) is equal to zero, the subsystem D α (x) = f1 (x, 0) will be asymptotically stable about the origin x = 0. Remark 2 In [42–44], the same synchronization scheme is presented. For the first step of sliding mode synchronization (i.e., constructing a sliding mode surface that has desired properties), the given sliding surfaces in [42–44] just satisfy the asymptotically stability of error state z(t)(i.e. z(t) → 0 as t → ∞, or limt→∞ z = 0). However, as presented in Assumption 1 used in this paper and [42–44], the necessary condition for the stability of system Dα (x) = f1 (x, z) is z = 0, and not z(t) → 0 as t → ∞. Therefore, in [42–44], the result limt→∞ z = 0 can not implies the result limt→∞ x = 0 on the base of Assumption 1. By comparison, this paper constructs a suitable sliding surface (3), which can ensure the error state z(t) = 0 in finite time, and then by Assumption 1, the asymptotical stability of subsystem D α (x) = f1 (x, z) can be obtained. Therefore, the main results of this paper are more reasonable and rigorous than the existing results. Remark 3 For underlining the effectiveness of the proposed method, this paper studies the same synchronization scheme, i.e., sliding mode synchronization of typical three-dimensional fractional-order chaotic systems by single state controller. Although the single state controller is simple and easy to implement, the three-dimensional system need satisfy Assumption 1, which is a restrictive condition for application objective. To alleviate this problem, the sliding surface (3) can be replaced by the following form: s(t) = D α−1 e(t) + kD α−2 (e(t) + sign(e(t))|e(t)|γ ), 6

(21)

4

x3

2 0 −2 −4 10

20 10

0

x2

0 −10

−10 −20

x1

Fig. 1. The chaotic trajectories of the fractional-order Arneodo’s system

where e(t) = [xT (t), z(t)]T ∈ R3 . It’s easy to know that sliding surface (21) can ensure the asymptotical stability of error state e(t) without the restriction of Assumption 1. Similarly, the derived results are applicable to others fractionalorder chaotic systems with any dimensions. Remark 4 From Eqs. (16) and (20), it implies that larger control coefficients k and ξ help to faster convergence of synchronization error state, which can give us helpful guidelines in practical application.

3

Numerical simulations

In this section, our goal is to achieve sliding synchronization of two typical three-dimensional fractional-order chaotic systems. 3.1 Example 1 Considering the following fractional-order Arneodo’s system [47]: D α x1 = x2 , D α x2 = x3 , D α x3 = ax1 − bx2 − cx3 − dx31 .

(22)

When a = 5.5, b = 3.5, c = 0.4, d = 1 and α = 0.9, the chaotic trajectories of the fractional-order Arneodo’s system without the controller are shown in Fig.1. Let system (22) as the drive system, and the slave system is given as 7

e1

10 0 −10

0

20

40

60

80

100

0

20

40

60

80

100

0

20

40

60

80

100

e2

10 0 −10

e3

2 0 −2

time(sec)

Fig. 2. Time response of error system (26)

D α y1 = y2 , D α y2 = y3 , D α y3 = ay1 − by2 − cy3 − dy13.

(23)

From (22) and (23), the error dynamics can be obtained: D α e1 = e2 , D α e2 = e3 , D α e3 = ae1 − be2 − ce3 − d(x21 + x1 y1 + y12 )e1 .

(24)

If e1 = 0, the following two-dimensional subsystem of system (24) is asymptotically stable, D α e2 = e3 , (25) D α e3 = −be2 − ce3 . Then, from Assumption 1, let x = (e2 , e3 )T and z = e1 , the controlled error system is D α e1 = e2 + u(x, z), D α e2 = e3 , (26) α 2 2 D e3 = ae1 − be2 − ce3 − d(x1 + x1 y1 + y1 )e1 , where u(x, z) = u(e1 , e2 , e3 ) has the form as (17), and the sliding surface has the form as (3). The initial conditions of drive and response systems are taken as (1, −1, 1)T and (−1, 1, −1)T , respectively. Let k = 0.1, ξ = 0.5, γ = 0.95. By Theorems 1 and 2, the time response of synchronization error system (26) and the sliding surface are shown in Fig.2 and Fig.3, respectively. 8

6 4

s(t)

2 0 −2 −4 −6 −8

0

20

40

60

80

100

time(sec)

Fig. 3. Time response of sliding mode surface (3) for synchronization of fractional-order Arneodo’s systems

x2

30

0

−30 20 40

0

30 20

x1

−20

10

x3

Fig. 4. The chaotic trajectories of the fractional-order unified system

3.2 Example 2 Considering the following fractional-order unified system [48]: D α x1 = (25a + 10)(x2 − x1 ), D α x2 = (28 − 35a)x1 + (29a − 1)x2 − x1 x3 , D α x3 = x1 x2 − (a + 8)x3 /3.

(27)

When a ∈ [0, 1], system (27) becomes fractional-order Lorenz, L¨ u and Chen system. The trajectories of the fractional order unified system with a = 1 are shown in Fig.4. Let system (27) as the drive system, and the slave system as 9

e1

2 0 −2

0

20

40

60

80

100

0

20

40

60

80

100

0

20

40

60

80

100

e2

2 0 −2

e3

5 0 −5

time(sec)

Fig. 5. Time response of error system (30)

the same form, then the error dynamics can be obtained: D α e1 = (25a + 10)(e2 − e1 ), D α e2 = (28 − 35a)e1 + (29a − 1)e2 − x1 e3 − y3 e1 , D α e3 = x1 e2 + y2 e1 − (a + 8)e3 /3.

(28)

If e2 = 0, the following two-dimensional subsystem of system (27) is asymptotically stable, D α e1 = −(25a + 10)e1 , D α e3 = y2 e1 − (a + 8)e3 /3.

(29)

Then, from Assumption 1, let x = (e1 , e3 )T and z = e2 , the controlled error system is D α e1 = (25a + 10)(e2 − e1 ), D α e2 = (28 − 35a)e1 + (29a − 1)e2 − x1 e3 − y3 e1 + u(x, z), D α e3 = x1 e2 + y2 e1 − (a + 8)e3 /3.

(30)

where u(x, z) = u(e1 , e2 , e3 ) has the form as (17), and the sliding surface has the form as (3). The initial conditions of drive and response systems are taken as (1, −1, 10)T and (−1, 1, 15)T , respectively. Let k = 0.1, ξ = 0.5, γ = 0.95. By Theorems 1 and 2, the time response of synchronization error system (30) and the sliding surface are shown in Fig.5 and Fig.6, respectively. 10

1.5

s(t)

1

0.5

0

−0.5

0

20

40

60

80

100

time(sec)

Fig. 6. Time response of sliding mode surface (3) for synchronization of fractional-order unified systems

4

Conclusions

In this paper, we present a modified sliding mode control method to synchronize typical three-dimensional fractional-order chaotic systems. By constructing an efficient sliding mode surface and control law, some effective and practical sufficient conditions are presented to guarantee the synchronization error dynamics can converge to the origin. Compared with the existing results, the main results in this paper are more reasonable and rigorous. Finally, two numerical examples for fractional-order Arneodo’s and unified systems are given to demonstrate the effectiveness of the method.

5

Acknowledgements

This work was supported by the Major State Basic Research Development Program 973 (Grant No. 2012CB215202), the National Natural Science Foundation of China (Grant Nos. 61104080, 61134001), and the Fundamental Research Funds for the Central Universities (Grant No. CDJZR13 17 55 01).

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References [1] I. Podlubny, Fractional Differential Equations. New York: Academic, 1999. [2] R. Hilfer, Applications of Fractional Calculus in Physics. , Singapore: World Scientific, 2001. [3] L.M. Pecora, T.L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64(8) (1990) 821–824. [4] H.G. Zhang, D.R. Liu, Z.L. Wang, Controlling Chaos: Suppression, Synchronization and Chaotification. London: Springer-Verlag, 2009. [5] H.G. Zhang, W. Huang, Z.L. Wang, T.Y. Chai, Adaptive synchronization between two different chaotic systems with unknown parameters, Phys. Lett. A 350(5-6) (2006) 363-366. [6] H.G. Zhang, D.W. Gong, B. Chen, Z.W. Liu, Synchronization for coupled neural networks with interval delay: a novel augmented Lyapunov-Krasovskii functional method, IEEE Trans. Neur. Net. Lear. 24(1) (2013) 58–70. [7] H.G. Zhang, Z.W. Liu, Stability analysis for linear delayed systems via an optimally dividing delay interval approach, Automatica 47(9) (2011) 2126–2129. [8] Y.C. Wang, H.G. Zhang, X.Y. Wang, D.S. Yang, Networked Synchronization Control of Coupled Dynamic Networks With Time-Varying Delay, IEEE Trans. Syst. Man Cybern. B 40(6) (2010) 1468–1479. [9] H.G. Zhang, T.D. Ma, G.B. Huang, Z.L. Wang, Robust Global Exponential Synchronization of Uncertain Chaotic Delayed Neural Networks via Dual-Stage Impulsive Control, IEEE Trans. Syst. Man Cybern. B 40(3) (2010) 831–844. [10] T.D. Ma, J. Fu, Y. Sun, An improved impulsive control approach to robust lag synchronization between two different chaotic systems, Chin. Phys. B 19(9) (2010) 090502. [11] T.D. Ma, Synchronization of multi-agent stochastic impulsive perturbed chaotic delayed neural networks with switching topology, Neurocomputing 151 (2015) 1392–1406. [12] T.D. Ma, J. Zhang, Hybrid synchronization of coupled fractional-order complex networks, Neurocomputing 157 (2015) 166–172. [13] T.D. Ma, J. Zhang, Y.C. Zhou, H.Y. Wang, Adaptive Hybrid projective synchronization of two coupled fractional-order complex networks with different sizes, Neurocomputing DOI:10.1016/j.neucom.2015.02.071. [14] T.D. Ma, J. Fu, On the exponential synchronization of stochastic impulsive chaotic delayed neural networks, Neurocomputing 74(5) (2011) 857–862. [15] J. Fu, M. Yu, T.D. Ma, Modified impulsive synchronization of fractional order hyperchaotic systems, Chin. Phys. B. 20(12) (2011) 120508.

12

[16] T.D. Ma, J. Fu, Global exponential synchronization between L system and Chen system with unknown parameters and channel time-delay, Chin. Phys. B 20(5) (2011) 050511. [17] T.D. Ma, H.G. Zhang, J. Fu, Exponential synchronization of stochastic impulsive perturbed chaotic Lur’e systems with time-varying delay and parametric uncertainty, Chin. Phys. B 17(12) (2008) 4407–4417. [18] T.D. Ma, H.G. Zhang, Z.L. Wang, Impulsive synchronization for unified chaotic systems with channel time-delay and parameter uncertainty, Acta Phys. Sin. 56(7) (2007) 3796–3802. [19] H.G. Zhang, T.D. Ma, J. Fu, S.C. Tong, Global impulsive exponential synchronization of stochastic perturbed chaotic delayed neural networks, Chin. Phys. B 18(9) (2009) 3742–3750. [20] H.G. Zhang, T.D. Ma, W. Yu, J. Fu, A practical approach to robust impulsive lag synchronization between different chaotic systems, Chin. Phys. B 17(10) (2008) 3616–3622. [21] H.G. Zhang, T.D. Ma, J. Fu, S.C. Tong, Robust lag synchronization of two different chaotic systems via dual-stage impulsive control, Chin. Phys. B 18(9) (2009) 3751–3757. [22] M. Srivastava, S.P. Ansari, S.K. Agrawal, S. Das, A.Y.T. Leung, Antisynchronization between identical and non-identical fractional-order chaotic systems using active control method, Nonlinear Dynamics 76(2) (2014) 905– 914. [23] S.K. Agrawal, M. Srivastava, S. Das, Synchronization of fractional order chaotic systems using active control method, Chaos, Solitons & Fractals 45(6) (2012) 737–752. [24] S. Bhalekar, V. Daftardar-Gejji, Synchronization of different fractional order chaotic systems using active control, Commun. Nonlinear Sci. Numer. Simulat. 15(11) (2010) 3536–3546. [25] H. Taghvafard, G.H. Erjaee, Phase and anti-phase synchronization of fractional order chaotic systems via active control, Commun. Nonlinear Sci. Numer. Simulat. 16(10) (2011) 4079–4088. [26] L. Liu, W. Ding, C.X. Liu, H.G. Ji, C.Q. Cao, Hyperchaos synchronization of fractional-order arbitrary dimensional dynamical systems via modified sliding mode control, Nonlinear Dynamics 76(4) (2014) 2059–2071. [27] D.Y. Chen, R.F. Zhang, J.C. Sprott, X.Y. Ma, Synchronization between integerorder chaotic systems and a class of fractional-order chaotic system based on fuzzy sliding mode control, Nonlinear Dynamics 70(2) (2012) 1549–1561. [28] D.Y. Chen, R.F. Zhang, J.C. Sprott, H.T. Chen, X.Y. Ma, Synchronization between integer-order chaotic systems and a class of fractional-order chaotic systems via sliding mode control, Chaos 22(2) (2012) 023130.

13

[29] CS.H. Hosseinniaa, R. Ghaderia, N.A. Ranjbar, M. Mahmoudiana, S. Momani, Sliding mode synchronization of an uncertain fractional order chaotic system, Computers & Mathematics with Applications 59(5) (2010) 1637–1643. [30] W.Y. Ma, C.P. Li, Y.J. Wu, Y.Q. Wu, Adaptive Synchronization of Fractional Neural Networks with Unknown Parameters and Time Delays, Entropy 16(12) (2014) 6286–6299. [31] S.K. Agrawal, S. Das, A modified adaptive control method for synchronization of some fractional chaotic systems with unknown parameters, Nonlinear Dynamics 73(1-2) (2013) 907–919. [32] T.D. Ma, W.B. Jiang, J. Fu, Y. Chai, L.P. Chen, F.Z. Xue, Adaptive synchronization of a class of fractional-order chaotic systems, Acta Physica Sinica 61(16) 160506 (2012) [33] R.X. Zhang, S.P. Yang, Adaptive synchronization of fractional-order chaotic systems, Chinese Physics B 19(2) (2010) 020510. [34] J.W. Wang, Q.H. Ma, L. Zeng, Observer-based synchronization in fractionalorder leader-follower complex networks, Nonlinear Dynamics 73(1-2) (2013) 921–929. [35] L. Liu, D.L. Liang, C.X. Liu, Q. Zhang, A Nonlinear State Observer Design for Projective Synchronization of Fractional-Order Permanent Magnet Synchronous Motor, International Journal of Modern Physics B 26(30) (2012) 1250166. [36] X.Y. Wang, Z.W. Hu, Projective Synchronization of Fractional Order Chaotic Systems Based on State Observer, International Journal of Modern Physics B 26(30) (2012) 1250176. [37] D.M. Senejohnny, H. Delavari, Active sliding observer scheme based fractional chaos synchronization, Communication in Nonlinear Science and Numerical Simulation 17(11) (2012) 4373–4383. [38] I. Stamova, Global Mittag-Leffler stability and synchronization of impulsive fractional-order neural networks with time-varying delays, Nonlinear Dynamics 77(4) (2014) 1251–1260. [39] X.Y. Wang, Y.L. Zhang, D. Li, N. Zhang, Impulsive synchronization of a class of fractional-order hyperchaotic systems, Chinese Physics B 20(3) (2011) 030506. [40] T.D. Ma, W.B. Jiang, J. Fu, Impulsive synchronization of fractional order hyperchaotic systems based on comparison system, Acta Physica Sinica 61(9) (2012) 090503. [41] T.D. Ma, W.B. Jiang, J. Fu, F.Z. Xue, Synchronization of hyperchaotic systems via improved impulsive control method, Acta Physica Sinica 61(10) (2012) 100507. [42] R.X. Zhang, S.P. Yang, Adaptive synchronization of fractional-order chaotic systems via a single driving variable, Nonlinear Dynamics 66(4) (2011) 831– 837.

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[43] R.X. Zhang, S.P. Yang, Response to the comments on “Adaptive synchronization of fractional-order chaotic systems via a single driving variable”, Nonlinear Dynamics 66(4) (2011) 843–844. [44] M.P. Aghababa, Comments on “Adaptive synchronization of fractionalorder chaotic systems via a single driving variable” [Nonlinear Dyn. (2011), doi:10.1007/s11071-011-9944-2], Nonlinear Dynamics 66(4) (2011) 839–842. [45] Y. Li, Y.Q. Chen, I. Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica 45(8) (2009) 1965–1969. [46] Y. Li, Y.Q. Chen, I. Podlubny, Stability of fractional order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Comput. Math. Appl. 59(5) (2010) 1810–1821. [47] J.G. Lu, Chaotic dynamics and synchronization of fractional-order Arneodo’s systems, Chaos Solitons Fractals 26(4) (2005) 1125–1133. [48] X.J. Wang, J. Li, G.R. Chen, Chaos in the fractional order unified system and its synchronization, J. Franklin Inst. 345(4) (2008) 392–401.

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