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Modified generalized projective synchronization of a new fractional-order hyperchaotic system and its application to secure communication
Nonlinear Analysis: Real World Applications 13 (2012) 1441–1450
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Modified generalized projective synchronization of a new fractional-order hyperchaotic system and its application to secure communication Xiangjun Wu a,b,∗ , Hui Wang a , Hongtao Lu b a
Department of Computing Center, Institute of Complex Intelligent Network System, Henan University, Kaifeng 475004, China
b
Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
article
info
Article history: Received 15 September 2010 Accepted 15 November 2011 Keywords: Fractional order Hyperchaotic system Projective synchronization (PS) Modified generalized projective synchronization (MGPS) Secure communication
abstract This paper presents a new fractional-order hyperchaotic system. The chaotic behaviors of this system in phase portraits are analyzed by the fractional calculus theory and computer simulations. Numerical results have revealed that hyperchaos does exist in the new fractional-order four-dimensional system with order less than 4 and the lowest order to have hyperchaos in this system is 3.664. The existence of two positive Lyapunov exponents further verifies our results. Furthermore, a novel modified generalized projective synchronization (MGPS) for the fractional-order chaotic systems is proposed based on the stability theory of the fractional-order system, where the states of the drive and response systems are asymptotically synchronized up to a desired scaling matrix. The unpredictability of the scaling factors in projective synchronization can additionally enhance the security of communication. Thus MGPS of the new fractionalorder hyperchaotic system is applied to secure communication. Computer simulations are done to verify the proposed methods and the numerical results show that the obtained theoretic results are feasible and efficient. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction In spite of the 300-year history of fractional calculus [1,2], the applications of fractional calculus to physics, engineering and control processing are just a recent focus of interest [3,4]. It was found that many systems in interdisciplinary fields can be elegantly described with the help of fractional derivatives, for instance, viscoelastic systems [5], electromagnetic waves [6], dielectric polarization [7], quantitative finance [8] and quantum evolution of complex systems [9], and so forth. More examples for fractional-order dynamics can be found in [3] and references therein. These examples and many other similar samples perfectly demonstrate the importance of consideration and analysis of dynamical systems with fractionalorder models. There are many material differences between the ordinary differential equation systems (integer-order) and the corresponding fractional-order differential equation systems. Most of the properties or conclusions of the integer-order system cannot be simply extended to the case of the fractional-order one. To date, many fractional-order differential systems such as the fractional-order Rössler system [10], the fractional-order Chen system [11], the fractional-order Lü system [12], the fractional-order unified system [13], etc., display chaotic behavior. A hyperchaotic system is characterized as a chaotic attractor with more than one positive Lyapunov exponents which can enhance the randomness and higher unpredictability of the corresponding system. So the hyperchaos may be more
∗ Corresponding author at: Department of Computing Center, Institute of Complex Intelligent Network System, Henan University, Kaifeng 475004, China. Tel.: +86 378 13693784994; fax: +86 378 3883009. E-mail address: [email protected] (X. Wu). 1468-1218/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2011.11.008
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useful in some fields such as communication, encryption etc. Motivated by this, in our work, we numerically investigate the hyperchaotic behaviors of a new fractional-order four-dimensional system. It is found that hyperchaos does exist in the new fractional-order system with order as low as 3.664. The hyperchaotic dynamical behaviors of the system were demonstrated by computer simulations. Numerical evidence shows that the new fractional-order system has two positive Lyapunov exponents. On the other hand, chaos synchronization has attracted much attention since the seminal work of Pecora and Carroll [14]. Recently, synchronization of fractional-order chaotic systems starts to attract increasing attention due to its potential applications in secure communication and control processing. Various types of synchronization for the fractional-order chaotic systems have been investigated, such as complete synchronization (CS) [15], generalized synchronization (GS) [16], phase synchronization (PhS) [17], anti-synchronization (AS) [18], projective synchronization (PS) [19–21], etc. Amongst all kinds of chaos synchronization, projective synchronization, which was first reported by Mainieri and Rehacek [19], is one of the most noticeable one because it can obtain faster communication with its proportional feature [22,23]. In PS, the responses of the master (drive) and slave (response) systems synchronize up to a constant scaling factor. Recently, Wang and He [24] introduced projective synchronization of the fractional-order chaotic systems via linear separation. Then generalized projective synchronization (GPS) of the fractional-order chaotic systems was studied in [21,25]. However, in the above studies, all the states of the drive and response systems synchronize up to an identical constant scaling factor. In [26], Chen et al. proposed a new hyperchaotic system through adding a nonlinear controller to the third equation of the threedimensional autonomous Chen–Lee chaotic system. Furthermore, they considered the hybrid projective synchronization (HPS) of the new hyperchaotic systems by using a nonlinear feedback control. But they mainly focus on generalized projective synchronization of the integer-order hyperchaotic system. More recently, Zhou and Zhu [27] investigated the function projective synchronization between fractional-order chaotic systems based on the stability theory of fractionalorder systems and tracking control technique. The proposed method can be applied to achieve not only the synchronization between the drive system and the response system with different fractional orders, but also the synchronization between two nonidentical fractional-order chaotic systems. Motivated by the above discussions, in this paper, we propose a new synchronization phenomenon, modified generalized projective synchronization (MGPS), for a class of fractional-order chaotic systems, where the drive and response systems could be synchronized to a constant scaling matrix. By choosing the scaling factors in the scaling matrix, one can flex the scales of different states independently. The unpredictability of the scaling matrix in MGPS can additionally strengthen the security of communications, which could be employed to get more secure communications. Based on the stability theory of the fractional-order system, the controllers are designed to make the drive and response systems synchronized up to the desired scaling matrix. Moreover, by MGPS, a secure communication scheme is presented. The corresponding numerical simulations have verified the effectiveness of the theoretical results. This paper is organized as follows. In Section 2, a brief review of the fractional derivative and numerical algorithm for the fractional-order system is given. Dynamics of a new fractional-order hyperchaotic system is numerically studied and demonstrated by computer simulations. In Section 3, a general method of MGPS for coupled fractional-order chaotic systems is presented based on the stability theory of the fractional-order system. In Section 4, MGPS of the new fractional-order hyperchaotic system is derived and numerical simulations show the validity of the proposed synchronization scheme. A chaotic secure communication scheme using MGPS is given in Section 5. Finally, the conclusions of this paper are drawn in Section 6. 2. A new fractional-order four-dimensional system 2.1. Fractional derivative and its approximation method There are many definitions for the fractional differential operators [1]. The commonly used definition is the Riemann–Liouville definition, defined by Dα x(t ) =
dm dt m
J m−α x(t ),
α>0
(1)
where m = ⌈α⌉, i.e., m is the first integer which is not less than α, J β is the β -order Riemann–Liouville integral operator as described by J β z (t ) =
1
Γ (β)
z (τ )
t
∫ 0
(t − τ )1−β
dτ ,
0<β ≤1
(2)
where Γ (·) denotes the gamma function. Here and throughout, the following definition is applied: Dα∗ x(t ) = J m−α x(m) (t ),
α>0
(3)
where m = ⌈α⌉. It is common practice to call operator Dα∗ the Caputo differential operator of order α [28]. The Riemann–Liouville fractional derivative appears unsuitable to be treated by the Laplace transform technique in that,
X. Wu et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1441–1450
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it requires the knowledge of the non-integer order derivatives of the function at t = 0 [4]. This problem does not exist in the Caputo definition that is sometimes referred as smooth fractional derivative in literature [28]. In addition, Riemann–Liouville initial problems require homogeneous initial conditions, whereas Caputo initial problems allow us to specify inhomogeneous initial conditions too if this is desired [29]. It is known that under those homogeneous conditions, the problems with Riemann–Liouville operators are equivalent to those with Caputo operators [2]. So we would prefer the Caputo derivative to the Riemann–Liouville one. The approximate numerical techniques for fractional differential equations have been developed in the literature which are numerically stable and can be used to both linear and nonlinear fractional differential equations [30]. Here we choose the Caputo version and use an improved predictor–corrector algorithm for fractional differential equations [31], where the numerical approximation is more accurate and the computational cost is largely reduced. The fractional predictor–corrector algorithm is based on the analytical property that the following differential equation:
Dα∗ x = f (t , x), x
(k)
0≤t ≤T
(k)
(0) = x0 ,
k = 0, 1, 2, . . . , m − 1(m = ⌈α⌉),
is equivalent to the Volterra integral equation [32]: m−1
x(t ) =
−
(k) t
x0
k=0
k
k!
+
t
∫
1
Γ (α)
0
f (τ , x) dτ . (t − τ )1−α
(4)
Now, set h = T /N , tn = nh(n = 0, 1, 2, . . . , N ∈ Z + ). Eq. (4) can be discretized as follows: m−1
xh (tn+1 ) =
−
hα
k
(k) tn+1
x0
k=0
k!
+
Γ (α + 2)
f (tn+1 , xθh (tn+1 )) +
hα
Γ (α + 2)
−
aj,n+1 f (tj , xh (tj )),
where the predicted value xθh (tn+1 ) is determined by xθh (tn+1 ) =
m−1
−
k
(k) tn+1
x0
k=0
k!
+
1
n −
Γ (α)
j =0
bj,n+1 f (tj , xh (tj )),
and
nα+1 − (n − α)(n + 1)α+1 ,
j=0
hα
((n − j + 1)α − (n − j)α ). α (n − j + 2)α+1 + (n − j)α+1 − 2(n − j + 1)α+1 , 1 ≤ j ≤ n, The estimation error is max x(tj ) − xh (tj ) = O(hθ ) (j = 0, 1, . . . , N ), where θ = min (2, 1 + α). aj , n + 1 =
bj,n+1 =
2.2. Dynamics analysis of a new fractional-order system Recently, Gao et al. proposed a new hyperchaotic system [33] by adding a nonlinear quadratic controller to the second equation of the three-dimensional autonomous modified Lorenz chaotic system [34], which is described by
x˙ = a(y − x) y˙ = bx + y − xz − w z˙ = xy − cz w ˙ = dyz ,
(5)
where x, y, z , w are state variables, and a, b, c , d are real constant parameters. When a = 10, b = 28, c = 8/3 and d = 0.1, the new four-dimensional system (5) is hyperchaotic as depicted in Fig. 1. Based on the above descriptions, we modify the derivative operator in Eq. (5) to be with respect to the fractional-order α(0 < α ≤ 1). Thus the fractional version of the new hyperchaotic system is given as follows:
α D∗ x = a(y − x) Dα y = bx + y − xz − w ∗
Dα z = xy − cz ∗α D∗ w = dyz .
(6)
According to the numerical algorithm for the fractional differential systems in Section 2.1, we find that hyperchaos does exist in the new four-dimensional system with fractional order. In the following simulations, the system parameters are always chosen as a = 10, b = 28, c = 8/3 and d = 0.1. The simulation results demonstrate that hyperchaos indeed exists in the fractional-order system (6) with order less than 4. Numeric evidence shows that when 0.916 ≤ α ≤ 1, the fractionalorder system (6) always exhibits hyperchaotic behaviors. For instance, when α = 0.94 and α = 0.916, hyperchaotic attractors are found and the phase portraits are displayed in Fig. 2(a) and (b), respectively. The two largest Lyapunov
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Fig. 1. Hyperchaotic attractors of the new system (5).
(a) α = 0.94.
(b) α = 0.916.
(c) α = 0.915.
Fig. 2. Phase portraits of the fractional-order system (6) with different fractional order α .
exponents λ1 and λ2 of the simulation time series are obtained as follows: when α = 0.94, λ1 = 0.862 and λ2 = 0.053; when α = 0.916, λ1 = 0.103 and λ2 = 0.012. Obviously, the fractional-order system (6) is hyperchaotic. However, if α < 0.916, the fractional-order system (6) has no hyperchaos. The phase portrait of system (6) for α = 0.915 is shown in Fig. 2(c). System (6) does not display hyperchaotic, but limit cycles appear which implies that there is no hyperchaos in the fractional-order system (6). Therefore, the lowest limit of the fractional order for this system to be hyperchaotic is α = 0.916 − 0.915. Thus, the lowest order we found for this system to yield hyperchaos is 3.664. 3. A general method for MGPS of the fractional-order chaotic systems Consider a general form of fractional-order chaotic system described by Dα∗ X = F (X ),
(7)
where the state vector X = (x1 , x2 , . . . , xn , w) . Divide the state vector X into two parts: a vector u = (x1 , x2 , . . . , xn ) and a variable w . Then system (7) can be rewritten by T
Dα∗ u = F1 (u, w)
Dα∗ w = F2 (u, w)
T
(8)
where F1 : Rn+1 → Rn is the continuous nonlinear vector function, and F2 : Rn+1 → R is a smooth function. It is well-known that many fractional-order systems, such as the Lorenz system, Rössler system, Chen system, Chua’s circuit, hyperchaotic Rössler system, hyperchaotic Chen and Lü system, can be written in the following form: Dα∗ Y = C Y + g (Y ) where Y ∈ Rn is the state vector, C ∈ Rn×n is the Jacobian matrix of the system at the origin, and g (Y ) is the nonlinear part. Therefore, without loss of generality, we can describe Eq. (8) as
Dα∗ u = Au + f (u, w) Dα∗ w = F2 (u, w).
(9)
Here A = (aij )n×n ∈ Rn×n is the Jacobian matrix of the system Dα∗ u at the origin, and f (u, w) is the nonlinear part. The coupled systems through the variable w with a controller U can be expressed in the form as
α D∗ um = F1 (um , w) Dα w = F2 (um , w) α∗ D∗ us = F1 (us , w) + U ,
(10)
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where the subscripts of m and s represent the drive system and the response system, respectively. Definition 1. For the drive and response systems in system (10), it is said to achieve MGPS if there exists a controller U such that limt →∞ ‖us − Λum ‖ = 0, where Λ ∈ Rn×n is a diagonal matrix, i.e., Λ = diag(β1 , β2 , . . . , βn ), and βi ̸= 0 for i = 1, 2, . . . , n. Remark 1. The matrix Λ is called a scaling matrix, and βi is called the scaling factor. In particular, if β1 = β2 = · · · = βn , MGPS is simplified to modified projective synchronization (MPS). The synchronization problem is to design a suitable controller U, which synchronizes the states of both the drive and response systems in the sense of MGPS. The main result is formed as the following Theorem 1. Theorem 1. For given synchronization scaling matrix Λ and any initial conditions um (0), w(0), us (0), MGPS between two coupled chaotic systems in the form as Eq. (10) can be obtained under the control law as follows:
¯ − K e − F1 (us , w) + ΛF1 (um , w), U = Ae
(11)
where A¯ ∈ R is a diagonal matrix where the diagonal elements are the same as those in the matrix A, i.e., A¯ = diag(a11 , a22 , . . . , ann ), where A = (aij )n×n ∈ Rn×n , and the control gain matrix K = diag(k1 , k2 , . . . , kn ) is a constant diagonal matrix and satisfies |arg(λi (A¯ − K ))| > 0.5απ (i = 1, 2, . . . , n). Here |arg(λi (M ))| means the argument of the eigenvalue λi of the matrix M. n×n
Proof. The synchronization error between the drive and response systems is defined as e = us − Λum .
(12)
The fractional derivative of Eq. (12) is Dα∗ e = Dα∗ us − ΛDα∗ um .
(13)
From Eq. (10), we can obtain the fractional-order error dynamical system as follows: Dα∗ e = Dα∗ (us − Λum ) = Dα∗ us − ΛDα∗ um
= F1 (us , w) − ΛF1 (um , w) + U .
(14)
Substituting Eq. (11) into Eq. (14), the fractional-order error dynamical system can be rewritten by Dα∗ e = (A¯ − K )e.
(15)
Since |arg(λi (A¯ − K ))| > 0.5απ (i = 1, 2, . . . , n), according to the stability theory of the fractional-order systems [35], the error vector e asymptotically converges to zero as t → ∞. Therefore, MGPS between the drive and response systems is achieved by using the controller (11). This completes the proof. 4. MGPS of the new fractional-order hyperchaotic system In this section, we focus on investigating MGPS of the fractional-order chaotic systems. For the sake of convenience, we consider the new fractional-order hyperchaotic system (6) as an example to show the feasibility. The coupled fractionalorder hyperchaotic system through the variable w with the control is given by
α D∗ xm −a a Dα∗ ym = b 1 Dα∗ zm 0 0 Dα∗ w = dym zm α D∗ xs −a a α D y b 1 = s ∗α D∗ zs
0
0
0 0 −c
xm ym zm
+
0 −x m z m − w xm ym
(16) 0 0 −c
xs ys zs
+
0 −xs zs − w xs y s
u1 u2 u3
+
,
where the subscripts of m and s have the same meanings as those in Eq. (10), and ui (i = 1, 2, 3) are the controllers to be constructed so that the drive and response systems can be synchronized in the sense of MGPS. By Eq. (16), we can easily get
−a A=
b 0
a 1 0
0 0 , −c
f (x, y, z , w) =
0 −xz − w xy
.
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Fig. 3. The synchronized attractors of the drive and response systems in (x, y, z )-space with α = 0.94.
Define the synchronization errors as follows: e1 = xs − β1 xm e2 = ys − β2 ym e3 = zs − β3 zm ,
(17)
where Λ = diag(β1 , β2 , β3 ), and βi (i = 1, 2, 3) are the nonzero scaling factors. It is obvious that, if the errors (17) tend to zero, MGPS between the drive and response systems will occur with the desired scaling matrix accordingly. Taking the fractional derivative of Eq. (17) and by Eq. (16), the following error dynamical system is derived:
α D∗ e1 = −ae1 + ays − aβ1 ym + u1 Dα e2 = e2 + bxs − bβ2 xm − xs zs + (β2 − 1)w + β2 xm zm + u2 ∗α D∗ e3 = −ce3 + xs ys − β3 xm ym + u3 .
(18)
According to the controller (11) in Theorem 1, we design the controllers ui (i = 1, 2, 3) for the response system in system (16) as follows: u1 = −ays + aβ1 ym − k1 e1 u2 = −bxs + bβ2 xm + xs zs − (β2 − 1)w − β2 xm zm − k2 e2 u3 = −xs ys + β3 xm ym − k3 e3
(19)
where the control gains matrix K = diag(k1 , k2 , k3 ). Substituting Eq. (19) into Eq. (18), the fractional-order error system can be rewritten as Dα∗ e1
Dα∗ e2 =
−(a + k1 ) 0 0
Dα∗ e3
0
(1 − k 2 ) 0
0 0
−(c + k3 )
e1 e2 e3
= (A¯ − K )e,
(20)
where
−a A¯ =
0 0
0 1 0
0 0 . −c
If we choose the suitable control gains matrix K which meets the condition |arg(λi (A¯ − K ))| > 0.5απ (i = 1, 2, . . . , n), according to the Lyapunov stability theory of the fractional-order systems and Theorem 1, the zero point of the fractionalorder error system (20) is globally and asymptotically stable, i.e., MGPS between the drive and response systems is achieved. In the numerical simulations, we select the parameters as a = 10, b = 28, c = 8/3, d = 0.1, and the fractional order as α = 0.94. Thus the fractional-order system (6) behaves chaotically. The initial values of the drive and response systems are arbitrarily chosen as: (xm (0), ym (0), zm (0)) = (3, 4, 6), w(0) = 5 and (xs (0), ys (0), zs (0)) = (1, 2, 3), respectively. The desired scaling matrix is taken randomly as Λ = diag(1.5, −0.5, −3). The control gains are set arbitrarily as k1 = 0, k2 = 3 and k3 = 0. Corresponding eigenvalues of (A¯ − K ) are obtained as -10, -2 and -2.67 which satisfy the stability condition |arg λi (A¯ − K ) | > 0.5απ (i = 1, 2, 3). The corresponding numerical results are shown in Figs. 3 and 4, respectively. Fig. 3 depicts the synchronized attractors of the drive and response systems. The time evolutions of the states are plotted in Fig. 4(a)–(c). Fig. 4(d) displays the evolution of the MGPS error e = (e1 , e2 , e3 )T which tends to zero as t → ∞, which implies that the error dynamical system (20) between the drive and the response systems is globally and asymptotically stable. Thus, MGPS has been achieved with our designed controllers (19). We have also tested the proposed synchronization scheme for other fractional order α and control gains ki . Limited to the length of this paper, we omit these results here. What deserves to be mentioned is that the fractional order and control gains play an important role in the synchronization rate. The bigger the fractional order α and control gains ki are, the faster the synchronization rate becomes.
X. Wu et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1441–1450
(a) Time response of xm (t )(—) and xs (t )(- -).
(b) Time response of ym (t )(—) and ys (t )(- -).
(c) Time response of zm (t )(—) and zs (t )(- -).
(d) Time response of synchronization errors.
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Fig. 4. MGPS of the new fractional-order hyperchaotic system (6) with α = 0.94.
Fig. 5. The secure communication scheme based on MGPS of the fractional-order hyperchaotic systems.
5. A secure communication scheme based on MGPS An important application of chaotic synchronization is that the technique can be applied to secure communication easily. The secure communication system involves the development of a signal that contains the information that is to remain undetectable by interceptors within a carrier signal. We can ensure the security of this information by inserting it into a chaotic signal that is transmitted to a prescribed receiver that would be able to detect and recover the information from the chaotic signal. In this section, the application of MGPS in secure communication is investigated. Fig. 5 depicts a sketch designed for our communication scheme using MGPS, in which the transmitter system and the receiver system are the new fractional-order hyperchaotic systems (6). U represents the controller. In the transmitter, the original information signal S (t ) is modulated into the chaotic signal by employing an invertible function Φ , i.e., S ′ (t ) = Φ (xm , ym , zm , w, S (t )). Then we add the signal S ′ (t ) to one of the three variables xm , ym and zm , for instance, we inject the signal S ′ (t ) into the variable xm and derive a combined signal χ (t ) = xm (t ) + S ′ (t ). In the channel, the variables xm , ym , zm , w and the combined signal χ (t ) are transmitted to the receiver. In the receiver, MGPS between the drive and response systems can be obtained by constructing the controller U according to Eq. (11) in Theorem 1. If MGPS occurs, the state xs will tend to β1 xm where β1 is the desired ′ scaling factor. Thus S ′ (t ) can be derived through a simple transformation S (t ) = χ (t ) − xs /β1 . Further, the information − 1 ′ signal can be recovered by S˜ (t ) = Φ xs , ys , zs , w, S (t ) .
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(a) The information signal S (t ).
(b) The transmitted signal χ(t ).
(c) The recovered signal S˜ (t ).
(d) The error signal S˜ (t ) − S (t ).
Fig. 6. Simulation results of secure communication using MGPS when the information signal is a sinusoidal signal.
(a) The information signal S (t ).
(b) The transmitted signal χ(t ).
(c) The recovered signal S˜ (t ).
(d) The error signal S˜ (t ) − S (t ).
Fig. 7. Simulation results of secure communication using MGPS when the information signal is an impulse signal.
X. Wu et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1441–1450
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In the following numerical simulations, for the sake of convenience, the system parameters, the fractional order α , the initial conditions of the drive and response systems, the scaling matrix and the control gains are chosen as the same values as those in Section 4. So MGPS between the drive and response systems can be realized under the controllers (19). In the following, we will apply the MGPS of the new fractional-order hyperchaotic system in the secure communication. First, the information signal is arbitrarily selected as S (t ) = 1 + 3 sin(π t /80) and the function Φ is given by S ′ (t ) = ym + S (t ). We assume that the signal S ′ (t ) is added to the variable xm . Simulation results for the application of MGPS in secure communication are shown in Fig. 6. The information signal S (t ) and the transmitted signal χ (t ) are shown in Fig. 6(a) and (b), respectively. Fig. 6(c) displays the recovered signal S˜ (t ). The error between the original information signal and the recovered one is shown in Fig. 6(d). From Fig. 6(d), it is easy to find that the information signal S (t ) is recovered accurately after a short transient. In addition, we also choose an impulse signal, which is depicted in Fig. 7(a), as the original information signal. We select the function Φ as S ′ (t ) = xm + S (t ) and input the signal S ′ (t ) into the variable ym . The results of this simulation are displayed in Fig. 7. Fig. 7(b) shows the transmitted signal χ (t ). Apparently, no effect of the embedded modulating information signal can be depicted. Fig. 7(c) depicts the recovered signal S˜ (t ). One can observe that the reconstructed signal S˜ (t ) coincides with the information signal S (t ) with good accuracy. This can be particularly seen in Fig. 7(d) which presents the difference signal S˜ (t ) − S (t ). As expected, the difference signal converges exactly to zero and the communication objective is attained. 6. Conclusions In this paper, modified generalized projective synchronization of a new fractional-order hyperchaotic system and its application in secure communication is investigated. First, we numerically study the hyperchaotic behaviors of a new fractional-order four-dimensional system according to the fractional calculus theory. We have found that hyperchaos does exist in the new fractional-order system with order as low as 3.664. The hyperchaotic dynamical behaviors of the system are illustrated by computer simulations. Our results have been validated by the existence of two positive Lyapunov exponents. Furthermore, based on the stability theory of the fractional-order systems, a novel modified generalized projective synchronization method for a class of fractional-order chaotic systems is presented. In MGPS, the drive and response systems could be asymptotically synchronized up to a desired scaling matrix, but not a constant. This feature can be applied to get more secure communications. Thus, a secure communication method by using MGPS of the new fractionalorder hyperchaotic system is introduced. For verifying the effectiveness and feasibility of the presented synchronization scheme and secure communication method, some numerical simulations are performed in our work. Acknowledgments This research is supported by the National Natural Science Foundation of China (Grant Nos. 61004006 and 60873133), the Natural Science Foundation of Henan Province, China (Grant No. 112300410009), the Foundation for University Young Key Teacher Program of Henan Province, China (Grant No. 2011GGJS-025) and the Natural Science Foundation of Educational Committee of He’nan Province, China (Grant No. 2011A520004). The authors would like to thank the referees for their helpful comments and suggestions which greatly improved the presentation of the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. P.L. Butzer, U. Westphal, An Introduction to Fractional Calculus, World Scientific, Singapore, 2000. R. Hifer, Applications of Fractional Calculus in Physics, World Scientific, New Jersey, 2001. S.G. Samko, A.A. Klibas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordan and Breach, Amsterdam, 1993. R.L. Bagley, R.A. Calico, Fractional order state equations for the control of viscoelastically damped structures, J. Guid. Control Dyn. 14 (1991) 304–311. O. Heaviside, Electromagnetic Theory, Chelsea, New York, 1971. H.H. Sun, A.A. Abdelwahad, B. Onaral, Linear approximation of transfer function with a pole of fractional order, IEEE Trans. Automat. Control 29 (1984) 441–444. N. Laskin, Fractional market dynamics, Physica A 287 (2000) 482–492. D. Kunsezov, A. Bulagc, G.D. Dang, Quantum Levy processes and fractional kinetics, Phys. Rev. Lett. 82 (1999) 1136–1139. C.G. Li, G. Chen, Chaos and hyperchaos in fractional order Rössler equations, Physica A 341 (2004) 55–61. C.G. Li, G. Chen, Chaos in the fractional order Chen system and its control, Chaos Solitons Fractals 22 (2004) 549–554. J.G. Lu, Chaotic dynamics of the fractional-order Lü system and its synchronization, Phys. Lett. A 354 (2006) 305–311. X. Wu, J. Li, G. Chen, Chaos in the fractional order unified system and its synchronization, J. Franklin Inst. 345 (2008) 392–401. L.M. Pecora, T.L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64 (1990) 821–825. H. Zhu, S. Zhou, J. Zhang, Chaos and synchronization of the fractional-order Chua’s system, Chaos Solitons Fractals 39 (2009) 1595–1603. X. Wu, H. Lu, S. Shen, Synchronization of a new fractional-order hyperchaotic system, Phys. Lett. A 373 (2009) 2329–2337. F.A. Breve, L. Zhao, M.G. Quiles, E.E.N. Macau, Chaotic phase synchronization and desynchronization in an oscillator network for object selection, Neural Netw. 22 (2009) 728–737. M.M. El-Dessoky, Anti-synchronization of four scroll attractor with fully unknown parameters, Nonlinear Anal. RWA 11 (2010) 778–783. R. Mainieri, J. Rehacek, Projective synchronization in three-dimensional chaotic systems, Phys. Rev. Lett. 82 (1999) 3042–3045. G. Peng, Y. Jiang, F. Chen, Projective synchronization of Chua’s chaotic systems with dead-zone in the control input, Math. Comput. Simulation 77 (2008) 374–382. X. Wu, Y. Lu, Generalized projective synchronization of the fractional-order Chen hyperchaotic system, Nonlinear Dynam. 57 (2009) 25–53. Z. Li, D. Xu, A secure communication scheme using projective chaos synchronization, Chaos Solitons Fractals 22 (2004) 477–481.
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[23] C.Y. Chee, D. Xu, Secure digital communication using controlled projective synchronization of chaos, Chaos Solitons Fractals 23 (2005) 1063–1070. [24] X. Wang, Y. He, Projective synchronization of fractional order chaotic system based on linear separation, Phys. Lett. A 372 (2008) 435–441. [25] D. Ghosh, S. Banerjee, A.R. Chowdhury, Generalized and projective synchronization in modulated time-delayed systems, Phys. Lett. A 374 (2010) 2143–2149. [26] C.-H. Chen, L.-J. Sheu, H.-K. Chen, J.-H. Chen, H.-C. Wang, Y.-C. Chao, Y.-K. Lin, A new hyper-chaotic system and its synchronization, Nonlinear Anal. RWA 10 (2009) 2088–2096. [27] P. Zhou, W. Zhu, Function projective synchronization for fractional-order chaotic systems, Nonlinear Anal. RWA 12 (2011) 811–816. [28] M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophys. J. R. Astron. Soc. 13 (1967) 529–539. [29] F. Keil, W. Mackens, J. Werther, Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties, Springer-Verlag, Heidelberg, 1999. [30] K. Diethelm, N.J. Ford, A.D. Freed, A predictor–corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam. 29 (2002) 3–22. [31] W. Deng, Numerical algorithm for the time fractional Fokker–Planck equation, J. Comput. Phys. 227 (2007) 1510–1522. [32] K. Diethelm, N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002) 229–248. [33] T. Gao, G. Chen, Z. Chen, S. Cang, The generation and circuit implementation of a new hyper-chaos based upon Lorenz system, Phys. Lett. A 361 (2007) 78–86. [34] S. Člikovský, G. Chen, On the generalized Lorenz canonical form, Chaos Solitons Fractals 26 (2005) 1271–1276. [35] C.T. Chen, Linear System Theory and Design, Rinehart and Winston, Holt, 1984.
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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Modified generalized projective synchronization of a new fractional-order hyperchaotic system and its application to secure communication Xiangjun Wu a,b,∗ , Hui Wang a , Hongtao Lu b a
Department of Computing Center, Institute of Complex Intelligent Network System, Henan University, Kaifeng 475004, China
b
Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
article
info
Article history: Received 15 September 2010 Accepted 15 November 2011 Keywords: Fractional order Hyperchaotic system Projective synchronization (PS) Modified generalized projective synchronization (MGPS) Secure communication
abstract This paper presents a new fractional-order hyperchaotic system. The chaotic behaviors of this system in phase portraits are analyzed by the fractional calculus theory and computer simulations. Numerical results have revealed that hyperchaos does exist in the new fractional-order four-dimensional system with order less than 4 and the lowest order to have hyperchaos in this system is 3.664. The existence of two positive Lyapunov exponents further verifies our results. Furthermore, a novel modified generalized projective synchronization (MGPS) for the fractional-order chaotic systems is proposed based on the stability theory of the fractional-order system, where the states of the drive and response systems are asymptotically synchronized up to a desired scaling matrix. The unpredictability of the scaling factors in projective synchronization can additionally enhance the security of communication. Thus MGPS of the new fractionalorder hyperchaotic system is applied to secure communication. Computer simulations are done to verify the proposed methods and the numerical results show that the obtained theoretic results are feasible and efficient. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction In spite of the 300-year history of fractional calculus [1,2], the applications of fractional calculus to physics, engineering and control processing are just a recent focus of interest [3,4]. It was found that many systems in interdisciplinary fields can be elegantly described with the help of fractional derivatives, for instance, viscoelastic systems [5], electromagnetic waves [6], dielectric polarization [7], quantitative finance [8] and quantum evolution of complex systems [9], and so forth. More examples for fractional-order dynamics can be found in [3] and references therein. These examples and many other similar samples perfectly demonstrate the importance of consideration and analysis of dynamical systems with fractionalorder models. There are many material differences between the ordinary differential equation systems (integer-order) and the corresponding fractional-order differential equation systems. Most of the properties or conclusions of the integer-order system cannot be simply extended to the case of the fractional-order one. To date, many fractional-order differential systems such as the fractional-order Rössler system [10], the fractional-order Chen system [11], the fractional-order Lü system [12], the fractional-order unified system [13], etc., display chaotic behavior. A hyperchaotic system is characterized as a chaotic attractor with more than one positive Lyapunov exponents which can enhance the randomness and higher unpredictability of the corresponding system. So the hyperchaos may be more
∗ Corresponding author at: Department of Computing Center, Institute of Complex Intelligent Network System, Henan University, Kaifeng 475004, China. Tel.: +86 378 13693784994; fax: +86 378 3883009. E-mail address: [email protected] (X. Wu). 1468-1218/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2011.11.008
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useful in some fields such as communication, encryption etc. Motivated by this, in our work, we numerically investigate the hyperchaotic behaviors of a new fractional-order four-dimensional system. It is found that hyperchaos does exist in the new fractional-order system with order as low as 3.664. The hyperchaotic dynamical behaviors of the system were demonstrated by computer simulations. Numerical evidence shows that the new fractional-order system has two positive Lyapunov exponents. On the other hand, chaos synchronization has attracted much attention since the seminal work of Pecora and Carroll [14]. Recently, synchronization of fractional-order chaotic systems starts to attract increasing attention due to its potential applications in secure communication and control processing. Various types of synchronization for the fractional-order chaotic systems have been investigated, such as complete synchronization (CS) [15], generalized synchronization (GS) [16], phase synchronization (PhS) [17], anti-synchronization (AS) [18], projective synchronization (PS) [19–21], etc. Amongst all kinds of chaos synchronization, projective synchronization, which was first reported by Mainieri and Rehacek [19], is one of the most noticeable one because it can obtain faster communication with its proportional feature [22,23]. In PS, the responses of the master (drive) and slave (response) systems synchronize up to a constant scaling factor. Recently, Wang and He [24] introduced projective synchronization of the fractional-order chaotic systems via linear separation. Then generalized projective synchronization (GPS) of the fractional-order chaotic systems was studied in [21,25]. However, in the above studies, all the states of the drive and response systems synchronize up to an identical constant scaling factor. In [26], Chen et al. proposed a new hyperchaotic system through adding a nonlinear controller to the third equation of the threedimensional autonomous Chen–Lee chaotic system. Furthermore, they considered the hybrid projective synchronization (HPS) of the new hyperchaotic systems by using a nonlinear feedback control. But they mainly focus on generalized projective synchronization of the integer-order hyperchaotic system. More recently, Zhou and Zhu [27] investigated the function projective synchronization between fractional-order chaotic systems based on the stability theory of fractionalorder systems and tracking control technique. The proposed method can be applied to achieve not only the synchronization between the drive system and the response system with different fractional orders, but also the synchronization between two nonidentical fractional-order chaotic systems. Motivated by the above discussions, in this paper, we propose a new synchronization phenomenon, modified generalized projective synchronization (MGPS), for a class of fractional-order chaotic systems, where the drive and response systems could be synchronized to a constant scaling matrix. By choosing the scaling factors in the scaling matrix, one can flex the scales of different states independently. The unpredictability of the scaling matrix in MGPS can additionally strengthen the security of communications, which could be employed to get more secure communications. Based on the stability theory of the fractional-order system, the controllers are designed to make the drive and response systems synchronized up to the desired scaling matrix. Moreover, by MGPS, a secure communication scheme is presented. The corresponding numerical simulations have verified the effectiveness of the theoretical results. This paper is organized as follows. In Section 2, a brief review of the fractional derivative and numerical algorithm for the fractional-order system is given. Dynamics of a new fractional-order hyperchaotic system is numerically studied and demonstrated by computer simulations. In Section 3, a general method of MGPS for coupled fractional-order chaotic systems is presented based on the stability theory of the fractional-order system. In Section 4, MGPS of the new fractional-order hyperchaotic system is derived and numerical simulations show the validity of the proposed synchronization scheme. A chaotic secure communication scheme using MGPS is given in Section 5. Finally, the conclusions of this paper are drawn in Section 6. 2. A new fractional-order four-dimensional system 2.1. Fractional derivative and its approximation method There are many definitions for the fractional differential operators [1]. The commonly used definition is the Riemann–Liouville definition, defined by Dα x(t ) =
dm dt m
J m−α x(t ),
α>0
(1)
where m = ⌈α⌉, i.e., m is the first integer which is not less than α, J β is the β -order Riemann–Liouville integral operator as described by J β z (t ) =
1
Γ (β)
z (τ )
t
∫ 0
(t − τ )1−β
dτ ,
0<β ≤1
(2)
where Γ (·) denotes the gamma function. Here and throughout, the following definition is applied: Dα∗ x(t ) = J m−α x(m) (t ),
α>0
(3)
where m = ⌈α⌉. It is common practice to call operator Dα∗ the Caputo differential operator of order α [28]. The Riemann–Liouville fractional derivative appears unsuitable to be treated by the Laplace transform technique in that,
X. Wu et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1441–1450
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it requires the knowledge of the non-integer order derivatives of the function at t = 0 [4]. This problem does not exist in the Caputo definition that is sometimes referred as smooth fractional derivative in literature [28]. In addition, Riemann–Liouville initial problems require homogeneous initial conditions, whereas Caputo initial problems allow us to specify inhomogeneous initial conditions too if this is desired [29]. It is known that under those homogeneous conditions, the problems with Riemann–Liouville operators are equivalent to those with Caputo operators [2]. So we would prefer the Caputo derivative to the Riemann–Liouville one. The approximate numerical techniques for fractional differential equations have been developed in the literature which are numerically stable and can be used to both linear and nonlinear fractional differential equations [30]. Here we choose the Caputo version and use an improved predictor–corrector algorithm for fractional differential equations [31], where the numerical approximation is more accurate and the computational cost is largely reduced. The fractional predictor–corrector algorithm is based on the analytical property that the following differential equation:
Dα∗ x = f (t , x), x
(k)
0≤t ≤T
(k)
(0) = x0 ,
k = 0, 1, 2, . . . , m − 1(m = ⌈α⌉),
is equivalent to the Volterra integral equation [32]: m−1
x(t ) =
−
(k) t
x0
k=0
k
k!
+
t
∫
1
Γ (α)
0
f (τ , x) dτ . (t − τ )1−α
(4)
Now, set h = T /N , tn = nh(n = 0, 1, 2, . . . , N ∈ Z + ). Eq. (4) can be discretized as follows: m−1
xh (tn+1 ) =
−
hα
k
(k) tn+1
x0
k=0
k!
+
Γ (α + 2)
f (tn+1 , xθh (tn+1 )) +
hα
Γ (α + 2)
−
aj,n+1 f (tj , xh (tj )),
where the predicted value xθh (tn+1 ) is determined by xθh (tn+1 ) =
m−1
−
k
(k) tn+1
x0
k=0
k!
+
1
n −
Γ (α)
j =0
bj,n+1 f (tj , xh (tj )),
and
nα+1 − (n − α)(n + 1)α+1 ,
j=0
hα
((n − j + 1)α − (n − j)α ). α (n − j + 2)α+1 + (n − j)α+1 − 2(n − j + 1)α+1 , 1 ≤ j ≤ n, The estimation error is max x(tj ) − xh (tj ) = O(hθ ) (j = 0, 1, . . . , N ), where θ = min (2, 1 + α). aj , n + 1 =
bj,n+1 =
2.2. Dynamics analysis of a new fractional-order system Recently, Gao et al. proposed a new hyperchaotic system [33] by adding a nonlinear quadratic controller to the second equation of the three-dimensional autonomous modified Lorenz chaotic system [34], which is described by
x˙ = a(y − x) y˙ = bx + y − xz − w z˙ = xy − cz w ˙ = dyz ,
(5)
where x, y, z , w are state variables, and a, b, c , d are real constant parameters. When a = 10, b = 28, c = 8/3 and d = 0.1, the new four-dimensional system (5) is hyperchaotic as depicted in Fig. 1. Based on the above descriptions, we modify the derivative operator in Eq. (5) to be with respect to the fractional-order α(0 < α ≤ 1). Thus the fractional version of the new hyperchaotic system is given as follows:
α D∗ x = a(y − x) Dα y = bx + y − xz − w ∗
Dα z = xy − cz ∗α D∗ w = dyz .
(6)
According to the numerical algorithm for the fractional differential systems in Section 2.1, we find that hyperchaos does exist in the new four-dimensional system with fractional order. In the following simulations, the system parameters are always chosen as a = 10, b = 28, c = 8/3 and d = 0.1. The simulation results demonstrate that hyperchaos indeed exists in the fractional-order system (6) with order less than 4. Numeric evidence shows that when 0.916 ≤ α ≤ 1, the fractionalorder system (6) always exhibits hyperchaotic behaviors. For instance, when α = 0.94 and α = 0.916, hyperchaotic attractors are found and the phase portraits are displayed in Fig. 2(a) and (b), respectively. The two largest Lyapunov
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Fig. 1. Hyperchaotic attractors of the new system (5).
(a) α = 0.94.
(b) α = 0.916.
(c) α = 0.915.
Fig. 2. Phase portraits of the fractional-order system (6) with different fractional order α .
exponents λ1 and λ2 of the simulation time series are obtained as follows: when α = 0.94, λ1 = 0.862 and λ2 = 0.053; when α = 0.916, λ1 = 0.103 and λ2 = 0.012. Obviously, the fractional-order system (6) is hyperchaotic. However, if α < 0.916, the fractional-order system (6) has no hyperchaos. The phase portrait of system (6) for α = 0.915 is shown in Fig. 2(c). System (6) does not display hyperchaotic, but limit cycles appear which implies that there is no hyperchaos in the fractional-order system (6). Therefore, the lowest limit of the fractional order for this system to be hyperchaotic is α = 0.916 − 0.915. Thus, the lowest order we found for this system to yield hyperchaos is 3.664. 3. A general method for MGPS of the fractional-order chaotic systems Consider a general form of fractional-order chaotic system described by Dα∗ X = F (X ),
(7)
where the state vector X = (x1 , x2 , . . . , xn , w) . Divide the state vector X into two parts: a vector u = (x1 , x2 , . . . , xn ) and a variable w . Then system (7) can be rewritten by T
Dα∗ u = F1 (u, w)
Dα∗ w = F2 (u, w)
T
(8)
where F1 : Rn+1 → Rn is the continuous nonlinear vector function, and F2 : Rn+1 → R is a smooth function. It is well-known that many fractional-order systems, such as the Lorenz system, Rössler system, Chen system, Chua’s circuit, hyperchaotic Rössler system, hyperchaotic Chen and Lü system, can be written in the following form: Dα∗ Y = C Y + g (Y ) where Y ∈ Rn is the state vector, C ∈ Rn×n is the Jacobian matrix of the system at the origin, and g (Y ) is the nonlinear part. Therefore, without loss of generality, we can describe Eq. (8) as
Dα∗ u = Au + f (u, w) Dα∗ w = F2 (u, w).
(9)
Here A = (aij )n×n ∈ Rn×n is the Jacobian matrix of the system Dα∗ u at the origin, and f (u, w) is the nonlinear part. The coupled systems through the variable w with a controller U can be expressed in the form as
α D∗ um = F1 (um , w) Dα w = F2 (um , w) α∗ D∗ us = F1 (us , w) + U ,
(10)
X. Wu et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1441–1450
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where the subscripts of m and s represent the drive system and the response system, respectively. Definition 1. For the drive and response systems in system (10), it is said to achieve MGPS if there exists a controller U such that limt →∞ ‖us − Λum ‖ = 0, where Λ ∈ Rn×n is a diagonal matrix, i.e., Λ = diag(β1 , β2 , . . . , βn ), and βi ̸= 0 for i = 1, 2, . . . , n. Remark 1. The matrix Λ is called a scaling matrix, and βi is called the scaling factor. In particular, if β1 = β2 = · · · = βn , MGPS is simplified to modified projective synchronization (MPS). The synchronization problem is to design a suitable controller U, which synchronizes the states of both the drive and response systems in the sense of MGPS. The main result is formed as the following Theorem 1. Theorem 1. For given synchronization scaling matrix Λ and any initial conditions um (0), w(0), us (0), MGPS between two coupled chaotic systems in the form as Eq. (10) can be obtained under the control law as follows:
¯ − K e − F1 (us , w) + ΛF1 (um , w), U = Ae
(11)
where A¯ ∈ R is a diagonal matrix where the diagonal elements are the same as those in the matrix A, i.e., A¯ = diag(a11 , a22 , . . . , ann ), where A = (aij )n×n ∈ Rn×n , and the control gain matrix K = diag(k1 , k2 , . . . , kn ) is a constant diagonal matrix and satisfies |arg(λi (A¯ − K ))| > 0.5απ (i = 1, 2, . . . , n). Here |arg(λi (M ))| means the argument of the eigenvalue λi of the matrix M. n×n
Proof. The synchronization error between the drive and response systems is defined as e = us − Λum .
(12)
The fractional derivative of Eq. (12) is Dα∗ e = Dα∗ us − ΛDα∗ um .
(13)
From Eq. (10), we can obtain the fractional-order error dynamical system as follows: Dα∗ e = Dα∗ (us − Λum ) = Dα∗ us − ΛDα∗ um
= F1 (us , w) − ΛF1 (um , w) + U .
(14)
Substituting Eq. (11) into Eq. (14), the fractional-order error dynamical system can be rewritten by Dα∗ e = (A¯ − K )e.
(15)
Since |arg(λi (A¯ − K ))| > 0.5απ (i = 1, 2, . . . , n), according to the stability theory of the fractional-order systems [35], the error vector e asymptotically converges to zero as t → ∞. Therefore, MGPS between the drive and response systems is achieved by using the controller (11). This completes the proof. 4. MGPS of the new fractional-order hyperchaotic system In this section, we focus on investigating MGPS of the fractional-order chaotic systems. For the sake of convenience, we consider the new fractional-order hyperchaotic system (6) as an example to show the feasibility. The coupled fractionalorder hyperchaotic system through the variable w with the control is given by
α D∗ xm −a a Dα∗ ym = b 1 Dα∗ zm 0 0 Dα∗ w = dym zm α D∗ xs −a a α D y b 1 = s ∗α D∗ zs
0
0
0 0 −c
xm ym zm
+
0 −x m z m − w xm ym
(16) 0 0 −c
xs ys zs
+
0 −xs zs − w xs y s
u1 u2 u3
+
,
where the subscripts of m and s have the same meanings as those in Eq. (10), and ui (i = 1, 2, 3) are the controllers to be constructed so that the drive and response systems can be synchronized in the sense of MGPS. By Eq. (16), we can easily get
−a A=
b 0
a 1 0
0 0 , −c
f (x, y, z , w) =
0 −xz − w xy
.
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Fig. 3. The synchronized attractors of the drive and response systems in (x, y, z )-space with α = 0.94.
Define the synchronization errors as follows: e1 = xs − β1 xm e2 = ys − β2 ym e3 = zs − β3 zm ,
(17)
where Λ = diag(β1 , β2 , β3 ), and βi (i = 1, 2, 3) are the nonzero scaling factors. It is obvious that, if the errors (17) tend to zero, MGPS between the drive and response systems will occur with the desired scaling matrix accordingly. Taking the fractional derivative of Eq. (17) and by Eq. (16), the following error dynamical system is derived:
α D∗ e1 = −ae1 + ays − aβ1 ym + u1 Dα e2 = e2 + bxs − bβ2 xm − xs zs + (β2 − 1)w + β2 xm zm + u2 ∗α D∗ e3 = −ce3 + xs ys − β3 xm ym + u3 .
(18)
According to the controller (11) in Theorem 1, we design the controllers ui (i = 1, 2, 3) for the response system in system (16) as follows: u1 = −ays + aβ1 ym − k1 e1 u2 = −bxs + bβ2 xm + xs zs − (β2 − 1)w − β2 xm zm − k2 e2 u3 = −xs ys + β3 xm ym − k3 e3
(19)
where the control gains matrix K = diag(k1 , k2 , k3 ). Substituting Eq. (19) into Eq. (18), the fractional-order error system can be rewritten as Dα∗ e1
Dα∗ e2 =
−(a + k1 ) 0 0
Dα∗ e3
0
(1 − k 2 ) 0
0 0
−(c + k3 )
e1 e2 e3
= (A¯ − K )e,
(20)
where
−a A¯ =
0 0
0 1 0
0 0 . −c
If we choose the suitable control gains matrix K which meets the condition |arg(λi (A¯ − K ))| > 0.5απ (i = 1, 2, . . . , n), according to the Lyapunov stability theory of the fractional-order systems and Theorem 1, the zero point of the fractionalorder error system (20) is globally and asymptotically stable, i.e., MGPS between the drive and response systems is achieved. In the numerical simulations, we select the parameters as a = 10, b = 28, c = 8/3, d = 0.1, and the fractional order as α = 0.94. Thus the fractional-order system (6) behaves chaotically. The initial values of the drive and response systems are arbitrarily chosen as: (xm (0), ym (0), zm (0)) = (3, 4, 6), w(0) = 5 and (xs (0), ys (0), zs (0)) = (1, 2, 3), respectively. The desired scaling matrix is taken randomly as Λ = diag(1.5, −0.5, −3). The control gains are set arbitrarily as k1 = 0, k2 = 3 and k3 = 0. Corresponding eigenvalues of (A¯ − K ) are obtained as -10, -2 and -2.67 which satisfy the stability condition |arg λi (A¯ − K ) | > 0.5απ (i = 1, 2, 3). The corresponding numerical results are shown in Figs. 3 and 4, respectively. Fig. 3 depicts the synchronized attractors of the drive and response systems. The time evolutions of the states are plotted in Fig. 4(a)–(c). Fig. 4(d) displays the evolution of the MGPS error e = (e1 , e2 , e3 )T which tends to zero as t → ∞, which implies that the error dynamical system (20) between the drive and the response systems is globally and asymptotically stable. Thus, MGPS has been achieved with our designed controllers (19). We have also tested the proposed synchronization scheme for other fractional order α and control gains ki . Limited to the length of this paper, we omit these results here. What deserves to be mentioned is that the fractional order and control gains play an important role in the synchronization rate. The bigger the fractional order α and control gains ki are, the faster the synchronization rate becomes.
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(a) Time response of xm (t )(—) and xs (t )(- -).
(b) Time response of ym (t )(—) and ys (t )(- -).
(c) Time response of zm (t )(—) and zs (t )(- -).
(d) Time response of synchronization errors.
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Fig. 4. MGPS of the new fractional-order hyperchaotic system (6) with α = 0.94.
Fig. 5. The secure communication scheme based on MGPS of the fractional-order hyperchaotic systems.
5. A secure communication scheme based on MGPS An important application of chaotic synchronization is that the technique can be applied to secure communication easily. The secure communication system involves the development of a signal that contains the information that is to remain undetectable by interceptors within a carrier signal. We can ensure the security of this information by inserting it into a chaotic signal that is transmitted to a prescribed receiver that would be able to detect and recover the information from the chaotic signal. In this section, the application of MGPS in secure communication is investigated. Fig. 5 depicts a sketch designed for our communication scheme using MGPS, in which the transmitter system and the receiver system are the new fractional-order hyperchaotic systems (6). U represents the controller. In the transmitter, the original information signal S (t ) is modulated into the chaotic signal by employing an invertible function Φ , i.e., S ′ (t ) = Φ (xm , ym , zm , w, S (t )). Then we add the signal S ′ (t ) to one of the three variables xm , ym and zm , for instance, we inject the signal S ′ (t ) into the variable xm and derive a combined signal χ (t ) = xm (t ) + S ′ (t ). In the channel, the variables xm , ym , zm , w and the combined signal χ (t ) are transmitted to the receiver. In the receiver, MGPS between the drive and response systems can be obtained by constructing the controller U according to Eq. (11) in Theorem 1. If MGPS occurs, the state xs will tend to β1 xm where β1 is the desired ′ scaling factor. Thus S ′ (t ) can be derived through a simple transformation S (t ) = χ (t ) − xs /β1 . Further, the information − 1 ′ signal can be recovered by S˜ (t ) = Φ xs , ys , zs , w, S (t ) .
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(a) The information signal S (t ).
(b) The transmitted signal χ(t ).
(c) The recovered signal S˜ (t ).
(d) The error signal S˜ (t ) − S (t ).
Fig. 6. Simulation results of secure communication using MGPS when the information signal is a sinusoidal signal.
(a) The information signal S (t ).
(b) The transmitted signal χ(t ).
(c) The recovered signal S˜ (t ).
(d) The error signal S˜ (t ) − S (t ).
Fig. 7. Simulation results of secure communication using MGPS when the information signal is an impulse signal.
X. Wu et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1441–1450
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In the following numerical simulations, for the sake of convenience, the system parameters, the fractional order α , the initial conditions of the drive and response systems, the scaling matrix and the control gains are chosen as the same values as those in Section 4. So MGPS between the drive and response systems can be realized under the controllers (19). In the following, we will apply the MGPS of the new fractional-order hyperchaotic system in the secure communication. First, the information signal is arbitrarily selected as S (t ) = 1 + 3 sin(π t /80) and the function Φ is given by S ′ (t ) = ym + S (t ). We assume that the signal S ′ (t ) is added to the variable xm . Simulation results for the application of MGPS in secure communication are shown in Fig. 6. The information signal S (t ) and the transmitted signal χ (t ) are shown in Fig. 6(a) and (b), respectively. Fig. 6(c) displays the recovered signal S˜ (t ). The error between the original information signal and the recovered one is shown in Fig. 6(d). From Fig. 6(d), it is easy to find that the information signal S (t ) is recovered accurately after a short transient. In addition, we also choose an impulse signal, which is depicted in Fig. 7(a), as the original information signal. We select the function Φ as S ′ (t ) = xm + S (t ) and input the signal S ′ (t ) into the variable ym . The results of this simulation are displayed in Fig. 7. Fig. 7(b) shows the transmitted signal χ (t ). Apparently, no effect of the embedded modulating information signal can be depicted. Fig. 7(c) depicts the recovered signal S˜ (t ). One can observe that the reconstructed signal S˜ (t ) coincides with the information signal S (t ) with good accuracy. This can be particularly seen in Fig. 7(d) which presents the difference signal S˜ (t ) − S (t ). As expected, the difference signal converges exactly to zero and the communication objective is attained. 6. Conclusions In this paper, modified generalized projective synchronization of a new fractional-order hyperchaotic system and its application in secure communication is investigated. First, we numerically study the hyperchaotic behaviors of a new fractional-order four-dimensional system according to the fractional calculus theory. We have found that hyperchaos does exist in the new fractional-order system with order as low as 3.664. The hyperchaotic dynamical behaviors of the system are illustrated by computer simulations. Our results have been validated by the existence of two positive Lyapunov exponents. Furthermore, based on the stability theory of the fractional-order systems, a novel modified generalized projective synchronization method for a class of fractional-order chaotic systems is presented. In MGPS, the drive and response systems could be asymptotically synchronized up to a desired scaling matrix, but not a constant. This feature can be applied to get more secure communications. Thus, a secure communication method by using MGPS of the new fractionalorder hyperchaotic system is introduced. For verifying the effectiveness and feasibility of the presented synchronization scheme and secure communication method, some numerical simulations are performed in our work. Acknowledgments This research is supported by the National Natural Science Foundation of China (Grant Nos. 61004006 and 60873133), the Natural Science Foundation of Henan Province, China (Grant No. 112300410009), the Foundation for University Young Key Teacher Program of Henan Province, China (Grant No. 2011GGJS-025) and the Natural Science Foundation of Educational Committee of He’nan Province, China (Grant No. 2011A520004). The authors would like to thank the referees for their helpful comments and suggestions which greatly improved the presentation of the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. P.L. Butzer, U. Westphal, An Introduction to Fractional Calculus, World Scientific, Singapore, 2000. R. Hifer, Applications of Fractional Calculus in Physics, World Scientific, New Jersey, 2001. S.G. Samko, A.A. Klibas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordan and Breach, Amsterdam, 1993. R.L. Bagley, R.A. Calico, Fractional order state equations for the control of viscoelastically damped structures, J. Guid. Control Dyn. 14 (1991) 304–311. O. Heaviside, Electromagnetic Theory, Chelsea, New York, 1971. H.H. Sun, A.A. Abdelwahad, B. Onaral, Linear approximation of transfer function with a pole of fractional order, IEEE Trans. Automat. Control 29 (1984) 441–444. N. Laskin, Fractional market dynamics, Physica A 287 (2000) 482–492. D. Kunsezov, A. Bulagc, G.D. Dang, Quantum Levy processes and fractional kinetics, Phys. Rev. Lett. 82 (1999) 1136–1139. C.G. Li, G. Chen, Chaos and hyperchaos in fractional order Rössler equations, Physica A 341 (2004) 55–61. C.G. Li, G. Chen, Chaos in the fractional order Chen system and its control, Chaos Solitons Fractals 22 (2004) 549–554. J.G. Lu, Chaotic dynamics of the fractional-order Lü system and its synchronization, Phys. Lett. A 354 (2006) 305–311. X. Wu, J. Li, G. Chen, Chaos in the fractional order unified system and its synchronization, J. Franklin Inst. 345 (2008) 392–401. L.M. Pecora, T.L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64 (1990) 821–825. H. Zhu, S. Zhou, J. Zhang, Chaos and synchronization of the fractional-order Chua’s system, Chaos Solitons Fractals 39 (2009) 1595–1603. X. Wu, H. Lu, S. Shen, Synchronization of a new fractional-order hyperchaotic system, Phys. Lett. A 373 (2009) 2329–2337. F.A. Breve, L. Zhao, M.G. Quiles, E.E.N. Macau, Chaotic phase synchronization and desynchronization in an oscillator network for object selection, Neural Netw. 22 (2009) 728–737. M.M. El-Dessoky, Anti-synchronization of four scroll attractor with fully unknown parameters, Nonlinear Anal. RWA 11 (2010) 778–783. R. Mainieri, J. Rehacek, Projective synchronization in three-dimensional chaotic systems, Phys. Rev. Lett. 82 (1999) 3042–3045. G. Peng, Y. Jiang, F. Chen, Projective synchronization of Chua’s chaotic systems with dead-zone in the control input, Math. Comput. Simulation 77 (2008) 374–382. X. Wu, Y. Lu, Generalized projective synchronization of the fractional-order Chen hyperchaotic system, Nonlinear Dynam. 57 (2009) 25–53. Z. Li, D. Xu, A secure communication scheme using projective chaos synchronization, Chaos Solitons Fractals 22 (2004) 477–481.
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X. Wu et al. / Nonlinear Analysis: Real World Applications 13 (2012) 1441–1450
[23] C.Y. Chee, D. Xu, Secure digital communication using controlled projective synchronization of chaos, Chaos Solitons Fractals 23 (2005) 1063–1070. [24] X. Wang, Y. He, Projective synchronization of fractional order chaotic system based on linear separation, Phys. Lett. A 372 (2008) 435–441. [25] D. Ghosh, S. Banerjee, A.R. Chowdhury, Generalized and projective synchronization in modulated time-delayed systems, Phys. Lett. A 374 (2010) 2143–2149. [26] C.-H. Chen, L.-J. Sheu, H.-K. Chen, J.-H. Chen, H.-C. Wang, Y.-C. Chao, Y.-K. Lin, A new hyper-chaotic system and its synchronization, Nonlinear Anal. RWA 10 (2009) 2088–2096. [27] P. Zhou, W. Zhu, Function projective synchronization for fractional-order chaotic systems, Nonlinear Anal. RWA 12 (2011) 811–816. [28] M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophys. J. R. Astron. Soc. 13 (1967) 529–539. [29] F. Keil, W. Mackens, J. Werther, Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties, Springer-Verlag, Heidelberg, 1999. [30] K. Diethelm, N.J. Ford, A.D. Freed, A predictor–corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam. 29 (2002) 3–22. [31] W. Deng, Numerical algorithm for the time fractional Fokker–Planck equation, J. Comput. Phys. 227 (2007) 1510–1522. [32] K. Diethelm, N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002) 229–248. [33] T. Gao, G. Chen, Z. Chen, S. Cang, The generation and circuit implementation of a new hyper-chaos based upon Lorenz system, Phys. Lett. A 361 (2007) 78–86. [34] S. Člikovský, G. Chen, On the generalized Lorenz canonical form, Chaos Solitons Fractals 26 (2005) 1271–1276. [35] C.T. Chen, Linear System Theory and Design, Rinehart and Winston, Holt, 1984.