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Synchronization between different dimensional chaotic systems using two scaling matrices
Optik 127 (2016) 959–963
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Synchronization between different dimensional chaotic systems using two scaling matrices Adel Ouannas a , M. Mossa Al-sawalha b,∗ a b
Department of Mathematics and Computer Science, Tebessa University, Algeria Mathematics Department, Faculty of Science, University of Hail, Saudi Arabia
a r t i c l e
i n f o
Article history: Received 28 January 2015 Accepted 27 October 2015 Keywords: Synchronization Scaling matrix Chaos
a b s t r a c t The epitome of this paper centers on chaos synchronization problem of different dimensional chaotic systems in different dimensions using two scaling matrices, the Lyapunov stability theory, and the stability theory of linear system. The controller is designed to assure that the synchronization of two different dimensional chaotic systems is achieved. Numerical examples and computer simulations are used to validate, numerically, the proposed synchronization schemes. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction Our natural world consists of physical systems that are undoubtedly nonlinear. It has been repeatedly demonstrated by scientists in the last recent decades that nonlinear systems, which models our real world, can display a variety of behaviors including chaos and hyperchaos. The most important characteristic of chaotic dynamics is its critical sensitivity to initial conditions, which is responsible for initially neighboring trajectories separating from each other exponentially in the course of time. This behavior made chaos undesirable and unwanted in many cases of research as it reduces their predictability over long time scales. But this special attribute may be a valuable advantage in certain areas of research. Chaotic dynamics has the ability to amplify small perturbations which improves their utility for reaching specific desired states with very high flexibility and low energy cost. In other words, we could try to control chaos for the benefit of our needs. Synchronization of different chaotic or hyperchaotic systems is one of the few main control methods popularly discussed recently. This is generally due to its prospective applications especially in chemical reactions, power converters, biological systems, information processing, secure communications, etc. [1]. The current problems of synchronization of chaos are very interesting, non-traditional, and indeed very challenging [2,3]. A wide variety of approaches
have been proposed for chaos synchronization such as adaptive control [4–9], linear and nonlinear feedback control [10–16], active control [17–21] complete synchronization [22] and projective synchronization [24–28]. To the best of our knowledge, most of the existing papers discuss the synchronization between two chaotic or hyperchaotic with the same dimension. However, in many real physics systems, the synchronization is carried out through the oscillators with different dimensions, especially the systems in biological science and social science. For example, in the cardiorespiratory system, the synchronization between the heart and the lung has been found even though their models have different dimensions [23]. In this paper, we proposed a method to synchronize two chaotic systems in different dimensions even though they have different dimensions, the synchronization controller is designed based on Lyapunov stability theory and the stability theory of linear system. An analytic expression of the controller is shown. Finally, illustrative examples of chaotic and hyperchaotic systems are used to show the effectiveness of the proposed method. 2. Theory Consider the drive system in the form of ˙ x(t) = f (x(t)), Rn
∗ Corresponding author at: Mathematics Department, Faculty of Science, University of Hail, Saudi Arabia. Tel.: +966 543156861. E-mail address: sawalha [email protected] (M.M. Al-sawalha). http://dx.doi.org/10.1016/j.ijleo.2015.10.174 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
(1) is the state vector of the drive system (1), f : Rn
→ Rn where x(t) ∈ defines a vector field in n-dimensional space. On the other hand, the response system is assumed by ˙ y(t) = g(y(t)) + U,
(2)
960
A. Ouannas, M.M. Al-sawalha / Optik 127 (2016) 959–963
where y(t) ∈ Rm is the state vector of the response system (2), g : Rm → Rm defines a vector field in m-dimensional space, U ∈ Rm is control input vector. Definition 1. The drive system (1) and the response system (2) are said to be synchronized in dimension d, with respect to scaling matrices and , if there exists a controller U ∈ Rm and given matrices = ()d×m and = ()d×n such that the synchronization error e(t) = y(t) − x(t), satisfies that lim e(t) = 0.
then the time derivative of V along the solution of error dynamical system equation (10) gives that ˙ V˙ (e(t)) = e˙ T (t)e(t) + eT (t)e(t) = e˙ T (t)(A − L1 )T e(t) + eT (t)(A − L1 )e(t) = eT (t)[(A − L1 )T + (A − L1 )]e(t) = −eT (t)Pe(t) < 0.
2.1. Synchronization between 3D drive and 4D response chaotic systems in 3D
It is clear that V is positive definite and V˙ is negative definite in the neighborhood of the zero solution for system (10). Therefore, the systems (3) and (4) are globally synchronized asymptotically, i.e lim e(t) = 0. This completes the proof. 䊐
In order to observe the synchronization behavior between between 3D drive and 4D response chaotic system in 3D, the drive and the response systems are defined below respectively,
2.2. Synchronization between 3D drive and 4D response chaotic systems in 4D
t→∞
t→∞
˙ x(t) = Ax(t) + f (x(t)),
(3)
R3
R3×3 ,
f : R3
→ R3
where x(t) ∈ is state vector, A ∈ are the linear part and the nonlinear part of system (3), respectively. Eq. (3) is considered as a drive system. By introducing an additive control U = (u1 , u2 , u3 , u4 )T ∈ R4 , then the controlled response system is given by ˙ y(t) = g(y(t)) + U,
(4)
R4
g : R4
→ R4 .
where y(t) ∈ is state vector, The error system between the drive system (3) and the response system (4), can be derived as ˙ e(t) = (A − L1 )e(t) + R + U,
(5)
where = (ij ) ∈ R3×4 , = (ij ) ∈ R3×3 are scaling matrices and R = (L1 − A)y(t) − ((L1 − A) + A)x(t) + g(y(t)) − f (x(t)), (6) R3×3
and L1 ∈ is an unknown control matrix to be determined. Assume that U = (u1 , u2 , u3 , 0). Then, the error system (5), can be written as ˆ U, ˆ ˙ e(t) = (A − L1 )e(t) + R + where
⎛
11
ˆ =⎜ ⎝ 21 31
(7)
⎞
12
13
22
21 ⎠ ,
32
33
⎟
(8)
In order to observe the synchronization behavior between between 3D drive and 4D response chaotic system in 4D, the drive and the response systems are defined below, respectively, ˙ = f (x(t)), x(t)
(13)
where x(t) ∈ R3 is state vector, f : R3 → R3 . Eq. (13) is considered as a drive system. By introducing an additive control U = (u1 , u2 , u3 , u4 )T ∈ R4 , then the controlled response system is given by ˙ y(t) = By(t) + g(y(t)) + U,
(14)
where B is an 4 × 4 constant matrix, g : R4 → R4 is a nonlinear function and U ∈ R4 is a controller. The error system between the drive system (13) and the response system (14), can be derived as ˙ = (B − L2 )e(t) + R + U, e(t)
(15)
where = (ij ) ∈ R4×4 , = (ij ) ∈ R4×3 are scaling matrices, and R = ((B − L2 ) + B)y(t) − (B − L2 )x(t) + g(y(t)) − f (x(t)), (16) where L2 ∈ R4×4 is an unknown control matrix to be determined. To achieve synchronization between systems (13) and (14), we choose the controller U as U = −−1 R,
(17)
where −1 is the inverse matrix of .
ˆ = (u1 , u2 , u3 )T , is the new control law. To achieve synchroand U ˆ is chosen nization between systems (3) and (4), the controller U as ˆ −1 R, ˆ = − U
(9)
Theorem 2. If L2 is chosen such that all eigenvalues of B − L2 are strictly negative, then the drive system (13) and the response system (14) are globally synchronized with respect to and , under the control law (17).
ˆ −1 is the inverse of . ˆ By substituting Eq. (9) in Eq. (5), the where error system can be described as
Proof. By substituting Eq. (17) in Eq. (15), the error system can be written as
˙ = (A − L1 )e(t). e(t)
˙ e(t) = (B − L2 )e(t).
Theorem 1.
(10)
If there exists a positive definite matrix P, such that
T
(A − L1 ) + (A − L1 ) = −P.
(11)
Then, the drive system (3) and the response system (4) are globally synchronized, with respect to scaling matrices and , under the controller (9). Proof.
(18)
According to the stability criterion of linear system, if all eigenvalues of B − L2 are strictly negative, it is immediate that all solution of error system (18) go to zero as t→ ∞. Therefore, systems (13) and (14) are globally synchronized. This completes the proof. 䊐 3. Applications
Construct the candidate Lyapunov function in the form
V (e(t)) = eT (t)e(t),
(12)
In this section, we give two examples to show the effectiveness of our proposed synchronization schemes. We choose the Rössler
A. Ouannas, M.M. Al-sawalha / Optik 127 (2016) 959–963
Fig. 1. Typical dynamical behaviors of the chaotic Rössler system: (a) projection in (x, y, z) space; (b) projection in (x, y) space; (c) projection in (x, z) space; (d) projection in (y, z) space.
system as the drive system and the controlled hyperchaotic Liu system as the response system. The Rössler system [29] given by: x˙ = −x2 − x3 , x˙ 2 = x1 + ˛x2 ,
(19)
961
Fig. 2. Typical dynamical behaviors of the hyperchaotic Liu system: (a) projection in (x, y, z) space; (b) projection in (x, y, w) space; (c) projection in (y, z, w) space; (d) projection in (x, z, w) space.
In this case, the error system between the drive system (19) and the response system (20) can be derived as
⎛
⎞
⎛
3
y˙ 1 = a(y2 − y1 ) + u1 ,
=
e1 =
⎛
L1 =
3.1. Synchronization between the 3D drive Rössler system and the 4D response hyperchaotic Liu system in 3D
j=1
1
0
0
0
0
5 1
1
−1
1
0
0
⎛
0
A = ⎝1 0
−1
−1
˛
0
0
−ˇ
⎞
⎛
⎠ , f (x) = ⎝
0 0 x1 x3 +
⎞
⎠.
⎞
3
1j yj −
4
j=1
y − j=1 3j j
⎛
u4 1j xj ,
3
e2 =
(21)
4 j=1
2j yj −
x , and we choose j=1 3j j
1
0
1
0
1 1
⎞
⎞ ⎠,
0
then, R1 = (−a + d + 1)y1 + (a + 1)y2 − x1 + x2 + ˇx3 + x3 x1 + . R2 = (2b + d)y1 + 2y2 − 2y4 − 2x1 − ˛x2 + x3 + +2y1 y3 . R3 = dy1 + y2 − 5cy3 + 6y4 − x1 − (˛ + ˇ)x2 + x3 x1 − 5y1 y2 + . According to Eq. (9) the vector controller U = (u1 , u2 , u3 , u4 )T can be constructed as follows
⎛
1
(u1 , u2 , u3 ) = − ⎝ 0 T
In order to observe synchronization between the drive system (19) and the response system (20) in 3D, the Rössler system in Eq. (19) can be rewritten as the form x˙ = Ax + f (x) where
R3
⎞
−1
⎝1 1+˛ 0
where U = (u1 , u2 , u3 , u4 is a vector controller, y1 , y2 , y3 , y4 are state variables of the system and a, b, c, d are bifurcation parameters. The hyperchaotic Liu system is hyperchaotic (i.e., the system (20) with u1 = u2 = u3 = u4 = 0) when the parameter values are taken as a = 10, b = 35, c = 1.4 and d = 5. The projections of the hyperchaotic Liu system system attractor are shown in Fig. 2.
u1
⎝0 2 0 1⎠, = ⎝1 1 0⎠, ⎛
(20)
)T
4
x , and e3 = j=1 2j j
y˙ 2 = by1 + y1 y3 − y4 + u2 ,
y˙ 4 = dy1 + y2 + u4 ,
e3
where
where x1 , x2 , x3 are the state variables and ˛ = 0.2, = 0.2, ˇ = 5.7. The projections of the chaotic Rössler system attractor are shown in Fig. 1. The controlled hyperchaotic Liu system [30] is described by:
y˙ 3 = −cy3 − y1 y2 + y4 + u3 ,
⎛
⎞
e1 R1 e˙ 1 ⎜ ⎟ ⎜ u2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎝ e˙ 2 ⎠ = (A − L1 ) ⎝ e2 ⎠ + ⎝ R2 ⎠ + ⎜ ⎜u ⎟, ⎝ 3⎠ e˙ 3
x˙ 3 = −ˇx3 + x1 x3 + ,
⎞ ⎛
0
0
0.5
0
0
0.2
0
⎞
⎠ × (R1 , R2 , R3 )T , and u4 = 0. (22)
So,
⎛
−1
A − L1 = ⎝ 0 0
0
0
−1
0
0
−ˇ
⎞ ⎠.
962
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Fig. 3. Time evolution of synchronization errors e1 , e2 and e3 between Rössler system (19) and hyperchaotic Liu system (20) in 3D.
Using simple calculations, we can show that (A − L1 )T + (A − L1 ) is a negative definite matrix. Then the conditions of Theorem 1 are satisfied. Hence the synchronization between systems (19) and (20) is achieved. For the purpose of numerical simulation, fourth order Runge–Kutta integration method has been used to solve the systems of differential equations (19) and (20). In addition, time step size 0.001 has been employed. We select the parameters of Rössler system (19) as ˛ = 0.2, = 0.2, ˇ = 5.7, so that system (19) exhibits a chaotic behavior, and the parameters of Liu hyperchaotic system (20) as a = 10, b = 35, c = 1.4 and d = 5, so that system (20) exhibits a hyperchaotic behavior. The initial values of the driving and response systems are [x1 (0), x2 (0), x3 (0)] = [0.1, 0.6, 0.3] and [y1 (0), y2 (0), y3 (0), y4 (0)] = [0.1, 0.1, 0.14, 0.4], respectively, and the initial states of the error system are [e1 (0), e2 (0), e3 (0)] = [0.100, − 0.100, 0.200]. Fig. 3 displays the synchronization errors between systems (19) and (20) in 3D. 3.2. Synchronization between the 3D drive Rössler system and the 4D response hyperchaotic Liu system in 4D
⎛
−a
⎜ b B=⎜ ⎝ 0 d
a
0
⎞
0
0
⎞
0
0
0
In this case, the error system between the drive system (19) and the response system (20), can be described as
⎛
e˙ 1
⎞
⎛
e1
⎞ ⎛
R1
⎞
⎛
u1
⎞
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ e˙ 2 ⎟ ⎜e ⎟ ⎜R ⎟ ⎜u ⎟ ⎜ ⎟ = (B − L2 ) ⎜ 2 ⎟ + ⎜ 2 ⎟ + ⎜ 2 ⎟ , ⎜ e˙ ⎟ ⎜e ⎟ ⎜R ⎟ ⎜u ⎟ ⎝ 3⎠ ⎝ 3⎠ ⎝ 3⎠ ⎝ 3⎠ e˙ 4
e4 e1 =
where
3 2j xj , e3 = j=1 3 j=1
R4
j=1
3j yj −
(23)
u4
4 3 y − 1j xj , j=1 1j j 4 j=1 3 j=1
e2 =
3j xj and e4 =
4 2j yj − 4j=1 j=1
4j yj −
4j xj and we choose
⎛
1
⎜0 =⎝ 0 0
⎞
0
0
0
2
0
0⎟
0
3
0
0
0
4
⎠,
⎛
1
⎜0 =⎝ 1 0
⎞
0
1
1
0⎟
1
0
0
1
⎠,
⎛
0
⎜b L2 = ⎝ 0 d
R1 = ay2 + ax1 + x2 + (a + ˇ)x3 − − x3 x1 , R2 = 2by1 + 2y2 − 2y4 + (1 − ˛)x2 − x1 + 2y1 y3 , R3 = −3y4 + (c − 1)x1 + (c + 1 − ˛)x2 + x3 − y1 y2 , R4 = 4dy1 + 4y2 − 8y4 + (1 + ˇ)x3 − − x3 x1 . By using the same formula of Eq. (17), the vector controller U = (u1 , u2 , u3 , u4 )T is designed as
⎛
a
0
0
⎞
1 0 −1 ⎟ 0
1
1 0
0
1
1
⎠,
0 1 2
⎜ ⎜0 ⎜ ⎜ (u1 , u2 , u3 , u4 )T = − ⎜ ⎜0 0 ⎜ ⎝ So,
⎛
−a
⎜ 0 ⎝ 0
B − L2 = ⎜
0
⎜ y1 y3 ⎟ 0 0 −1 ⎟ ⎟ ⎟ , g(y) = ⎜ ⎜ ⎟. ⎠ 0 −c 1 ⎝ −y1 y2 ⎠ 1
then,
0
In order to observe synchronization between the drive system (19) and the response system (20) in 4D, the hyperchaotic Liu system (i.e., the uncontrolled system (20)) can be rewritten as the form y˙ = By + g(y) where
⎛
Fig. 4. Time evolution of synchronization errors e1 , e2 , e3 and e4 between Rössler system (19) and hyperchaotic Liu system (20) in 4D.
0
0
0
0
−1
0
0
0
−c
0
0
0
−1
0
0
0
0
1 3 0
⎞
⎟ ⎟ ⎟ ⎟ ⎟ × (R1 , R2 , R3 , R4 )T . 0⎟ ⎟ ⎠
(24)
1 4
⎞ ⎟ ⎟. ⎠
Its easy to show that all eigenvalues of B − L2 are strictly negative. Then the conditions of Theorem 2 are satisfied. Therefore, the systems (19) and (20) are globally synchronized, in 4D. For the purpose of numerical simulation, fourth order Runge–Kutta integration method has been used to solve the systems of differential equations (19) and (20). In addition, time step size 0.001 has been employed. We select the parameters of Rössler system (19) as ˛ = 0.2, = 0.2, ˇ = 5.7, so that system (19) exhibits a chaotic behavior, and the parameters of Liu hyperchaotic system (20) as a = 10, b = 35, c = 1.4 and d = 5, so that system (20) exhibits a hyperchaotic behavior. The initial values of the driving and response systems are [x1 (0), x2 (0), x3 (0)] = [0.6, 0.1, 0.7] and [y1 (0), y2 (0), y3 (0), y4 (0)] = [1.2, 0.1, 0.4, 0.2], respectively, and the initial states of the error system are [e1 (0), e2 (0), e3 (0), e4 (0)] = [0.100, − 0.100, 0.100, − 0.100]. Fig. 4 displays the synchronization errors between systems (19) and (20) in 4D. 4. Conclusion In this paper, we have investigated the synchronization problem of different dimensional chaotic systems in different dimensions using two scaling matrices, the Lyapunov stability theory, and the stability theory of linear system. The nonlinear controller was
A. Ouannas, M.M. Al-sawalha / Optik 127 (2016) 959–963
designed to guaranteed the synchronization between 3D drive system and 4D response system in 3D by controlling the linear part of the drive system. However the synchronization between 3D drive system and 4D response system in 4D guaranteed by controlling the linear part of the response system. The new synchronization control approach was successfully applied to several examples: the Rössler system and hyperchaotic Liu system. Numerical simulations were presented to illustrate the proposed synchronization schemes. References [1] G. Chen, X. Dong, From Chaos to Order, World Scientific, Singapore, 1989. [2] A. Luo, A theory for synchronization of dynamical systems, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 1901–1951. [3] M. Rafikov, J. Balthazar, On control and synchronization in chaotic and hyperchaotic systems via linear feedback control, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 1246–1255. [4] Y. Sun, J. Cao, Adaptive synchronization between two different noise-perturbed chaotic systems with fully unknown parameters, Phys. A 376 (2007) 253–265. [5] J. Park, Adaptive synchronization of hyperchaotic Chen system with uncertain parameters, Chaos Solitons Fractals 26 (2005) 959–964. [6] X. Wu, H. Zhang, Synchronization of two hyperchaotic systems via adaptive control, Chaos Solitons Fractals 39 (2009) 2268–2273. [7] C. Shihua, J. Lü, Parameters identification and synchronization of chaotic systems based upon adaptive control, Phys. Lett. A 299 (2002) 353–358. [8] J. Lu, X. Wu, X. Han, J. Lü, Adaptive feedback synchronization of a unified chaotic system, Phys. Lett. A 329 (2004) 327–333. [9] J. Huang, Chaos synchronization between two novel different hyperchaotic systems with unknown parameters, Nonlinear Anal. Theory Methods Appl. 69 (2008) 4174–4181. [10] J. Lü, J. Lu, Controlling uncertain Lü system using linear feedback, Chaos Solitons Fractals 17 (2003) 127–133. [11] H. Chen, Global chaos synchronization of new chaotic systems via nonlinear control, Chaos Solitons Fractals 23 (2005) 1245–1251. [12] L. Huang, R. Feng, M. Wang, Synchronization of chaotic systems via nonlinear control, Phys. Lett. A 320 (2004) 271–275. [13] Q. Zhang, J. Lu, Chaos synchronization of a new chaotic system via nonlinear control, Chaos Solitons Fractals 37 (2008) 175–179.
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Optik journal homepage: www.elsevier.de/ijleo
Synchronization between different dimensional chaotic systems using two scaling matrices Adel Ouannas a , M. Mossa Al-sawalha b,∗ a b
Department of Mathematics and Computer Science, Tebessa University, Algeria Mathematics Department, Faculty of Science, University of Hail, Saudi Arabia
a r t i c l e
i n f o
Article history: Received 28 January 2015 Accepted 27 October 2015 Keywords: Synchronization Scaling matrix Chaos
a b s t r a c t The epitome of this paper centers on chaos synchronization problem of different dimensional chaotic systems in different dimensions using two scaling matrices, the Lyapunov stability theory, and the stability theory of linear system. The controller is designed to assure that the synchronization of two different dimensional chaotic systems is achieved. Numerical examples and computer simulations are used to validate, numerically, the proposed synchronization schemes. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction Our natural world consists of physical systems that are undoubtedly nonlinear. It has been repeatedly demonstrated by scientists in the last recent decades that nonlinear systems, which models our real world, can display a variety of behaviors including chaos and hyperchaos. The most important characteristic of chaotic dynamics is its critical sensitivity to initial conditions, which is responsible for initially neighboring trajectories separating from each other exponentially in the course of time. This behavior made chaos undesirable and unwanted in many cases of research as it reduces their predictability over long time scales. But this special attribute may be a valuable advantage in certain areas of research. Chaotic dynamics has the ability to amplify small perturbations which improves their utility for reaching specific desired states with very high flexibility and low energy cost. In other words, we could try to control chaos for the benefit of our needs. Synchronization of different chaotic or hyperchaotic systems is one of the few main control methods popularly discussed recently. This is generally due to its prospective applications especially in chemical reactions, power converters, biological systems, information processing, secure communications, etc. [1]. The current problems of synchronization of chaos are very interesting, non-traditional, and indeed very challenging [2,3]. A wide variety of approaches
have been proposed for chaos synchronization such as adaptive control [4–9], linear and nonlinear feedback control [10–16], active control [17–21] complete synchronization [22] and projective synchronization [24–28]. To the best of our knowledge, most of the existing papers discuss the synchronization between two chaotic or hyperchaotic with the same dimension. However, in many real physics systems, the synchronization is carried out through the oscillators with different dimensions, especially the systems in biological science and social science. For example, in the cardiorespiratory system, the synchronization between the heart and the lung has been found even though their models have different dimensions [23]. In this paper, we proposed a method to synchronize two chaotic systems in different dimensions even though they have different dimensions, the synchronization controller is designed based on Lyapunov stability theory and the stability theory of linear system. An analytic expression of the controller is shown. Finally, illustrative examples of chaotic and hyperchaotic systems are used to show the effectiveness of the proposed method. 2. Theory Consider the drive system in the form of ˙ x(t) = f (x(t)), Rn
∗ Corresponding author at: Mathematics Department, Faculty of Science, University of Hail, Saudi Arabia. Tel.: +966 543156861. E-mail address: sawalha [email protected] (M.M. Al-sawalha). http://dx.doi.org/10.1016/j.ijleo.2015.10.174 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
(1) is the state vector of the drive system (1), f : Rn
→ Rn where x(t) ∈ defines a vector field in n-dimensional space. On the other hand, the response system is assumed by ˙ y(t) = g(y(t)) + U,
(2)
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A. Ouannas, M.M. Al-sawalha / Optik 127 (2016) 959–963
where y(t) ∈ Rm is the state vector of the response system (2), g : Rm → Rm defines a vector field in m-dimensional space, U ∈ Rm is control input vector. Definition 1. The drive system (1) and the response system (2) are said to be synchronized in dimension d, with respect to scaling matrices and , if there exists a controller U ∈ Rm and given matrices = ()d×m and = ()d×n such that the synchronization error e(t) = y(t) − x(t), satisfies that lim e(t) = 0.
then the time derivative of V along the solution of error dynamical system equation (10) gives that ˙ V˙ (e(t)) = e˙ T (t)e(t) + eT (t)e(t) = e˙ T (t)(A − L1 )T e(t) + eT (t)(A − L1 )e(t) = eT (t)[(A − L1 )T + (A − L1 )]e(t) = −eT (t)Pe(t) < 0.
2.1. Synchronization between 3D drive and 4D response chaotic systems in 3D
It is clear that V is positive definite and V˙ is negative definite in the neighborhood of the zero solution for system (10). Therefore, the systems (3) and (4) are globally synchronized asymptotically, i.e lim e(t) = 0. This completes the proof. 䊐
In order to observe the synchronization behavior between between 3D drive and 4D response chaotic system in 3D, the drive and the response systems are defined below respectively,
2.2. Synchronization between 3D drive and 4D response chaotic systems in 4D
t→∞
t→∞
˙ x(t) = Ax(t) + f (x(t)),
(3)
R3
R3×3 ,
f : R3
→ R3
where x(t) ∈ is state vector, A ∈ are the linear part and the nonlinear part of system (3), respectively. Eq. (3) is considered as a drive system. By introducing an additive control U = (u1 , u2 , u3 , u4 )T ∈ R4 , then the controlled response system is given by ˙ y(t) = g(y(t)) + U,
(4)
R4
g : R4
→ R4 .
where y(t) ∈ is state vector, The error system between the drive system (3) and the response system (4), can be derived as ˙ e(t) = (A − L1 )e(t) + R + U,
(5)
where = (ij ) ∈ R3×4 , = (ij ) ∈ R3×3 are scaling matrices and R = (L1 − A)y(t) − ((L1 − A) + A)x(t) + g(y(t)) − f (x(t)), (6) R3×3
and L1 ∈ is an unknown control matrix to be determined. Assume that U = (u1 , u2 , u3 , 0). Then, the error system (5), can be written as ˆ U, ˆ ˙ e(t) = (A − L1 )e(t) + R + where
⎛
11
ˆ =⎜ ⎝ 21 31
(7)
⎞
12
13
22
21 ⎠ ,
32
33
⎟
(8)
In order to observe the synchronization behavior between between 3D drive and 4D response chaotic system in 4D, the drive and the response systems are defined below, respectively, ˙ = f (x(t)), x(t)
(13)
where x(t) ∈ R3 is state vector, f : R3 → R3 . Eq. (13) is considered as a drive system. By introducing an additive control U = (u1 , u2 , u3 , u4 )T ∈ R4 , then the controlled response system is given by ˙ y(t) = By(t) + g(y(t)) + U,
(14)
where B is an 4 × 4 constant matrix, g : R4 → R4 is a nonlinear function and U ∈ R4 is a controller. The error system between the drive system (13) and the response system (14), can be derived as ˙ = (B − L2 )e(t) + R + U, e(t)
(15)
where = (ij ) ∈ R4×4 , = (ij ) ∈ R4×3 are scaling matrices, and R = ((B − L2 ) + B)y(t) − (B − L2 )x(t) + g(y(t)) − f (x(t)), (16) where L2 ∈ R4×4 is an unknown control matrix to be determined. To achieve synchronization between systems (13) and (14), we choose the controller U as U = −−1 R,
(17)
where −1 is the inverse matrix of .
ˆ = (u1 , u2 , u3 )T , is the new control law. To achieve synchroand U ˆ is chosen nization between systems (3) and (4), the controller U as ˆ −1 R, ˆ = − U
(9)
Theorem 2. If L2 is chosen such that all eigenvalues of B − L2 are strictly negative, then the drive system (13) and the response system (14) are globally synchronized with respect to and , under the control law (17).
ˆ −1 is the inverse of . ˆ By substituting Eq. (9) in Eq. (5), the where error system can be described as
Proof. By substituting Eq. (17) in Eq. (15), the error system can be written as
˙ = (A − L1 )e(t). e(t)
˙ e(t) = (B − L2 )e(t).
Theorem 1.
(10)
If there exists a positive definite matrix P, such that
T
(A − L1 ) + (A − L1 ) = −P.
(11)
Then, the drive system (3) and the response system (4) are globally synchronized, with respect to scaling matrices and , under the controller (9). Proof.
(18)
According to the stability criterion of linear system, if all eigenvalues of B − L2 are strictly negative, it is immediate that all solution of error system (18) go to zero as t→ ∞. Therefore, systems (13) and (14) are globally synchronized. This completes the proof. 䊐 3. Applications
Construct the candidate Lyapunov function in the form
V (e(t)) = eT (t)e(t),
(12)
In this section, we give two examples to show the effectiveness of our proposed synchronization schemes. We choose the Rössler
A. Ouannas, M.M. Al-sawalha / Optik 127 (2016) 959–963
Fig. 1. Typical dynamical behaviors of the chaotic Rössler system: (a) projection in (x, y, z) space; (b) projection in (x, y) space; (c) projection in (x, z) space; (d) projection in (y, z) space.
system as the drive system and the controlled hyperchaotic Liu system as the response system. The Rössler system [29] given by: x˙ = −x2 − x3 , x˙ 2 = x1 + ˛x2 ,
(19)
961
Fig. 2. Typical dynamical behaviors of the hyperchaotic Liu system: (a) projection in (x, y, z) space; (b) projection in (x, y, w) space; (c) projection in (y, z, w) space; (d) projection in (x, z, w) space.
In this case, the error system between the drive system (19) and the response system (20) can be derived as
⎛
⎞
⎛
3
y˙ 1 = a(y2 − y1 ) + u1 ,
=
e1 =
⎛
L1 =
3.1. Synchronization between the 3D drive Rössler system and the 4D response hyperchaotic Liu system in 3D
j=1
1
0
0
0
0
5 1
1
−1
1
0
0
⎛
0
A = ⎝1 0
−1
−1
˛
0
0
−ˇ
⎞
⎛
⎠ , f (x) = ⎝
0 0 x1 x3 +
⎞
⎠.
⎞
3
1j yj −
4
j=1
y − j=1 3j j
⎛
u4 1j xj ,
3
e2 =
(21)
4 j=1
2j yj −
x , and we choose j=1 3j j
1
0
1
0
1 1
⎞
⎞ ⎠,
0
then, R1 = (−a + d + 1)y1 + (a + 1)y2 − x1 + x2 + ˇx3 + x3 x1 + . R2 = (2b + d)y1 + 2y2 − 2y4 − 2x1 − ˛x2 + x3 + +2y1 y3 . R3 = dy1 + y2 − 5cy3 + 6y4 − x1 − (˛ + ˇ)x2 + x3 x1 − 5y1 y2 + . According to Eq. (9) the vector controller U = (u1 , u2 , u3 , u4 )T can be constructed as follows
⎛
1
(u1 , u2 , u3 ) = − ⎝ 0 T
In order to observe synchronization between the drive system (19) and the response system (20) in 3D, the Rössler system in Eq. (19) can be rewritten as the form x˙ = Ax + f (x) where
R3
⎞
−1
⎝1 1+˛ 0
where U = (u1 , u2 , u3 , u4 is a vector controller, y1 , y2 , y3 , y4 are state variables of the system and a, b, c, d are bifurcation parameters. The hyperchaotic Liu system is hyperchaotic (i.e., the system (20) with u1 = u2 = u3 = u4 = 0) when the parameter values are taken as a = 10, b = 35, c = 1.4 and d = 5. The projections of the hyperchaotic Liu system system attractor are shown in Fig. 2.
u1
⎝0 2 0 1⎠, = ⎝1 1 0⎠, ⎛
(20)
)T
4
x , and e3 = j=1 2j j
y˙ 2 = by1 + y1 y3 − y4 + u2 ,
y˙ 4 = dy1 + y2 + u4 ,
e3
where
where x1 , x2 , x3 are the state variables and ˛ = 0.2, = 0.2, ˇ = 5.7. The projections of the chaotic Rössler system attractor are shown in Fig. 1. The controlled hyperchaotic Liu system [30] is described by:
y˙ 3 = −cy3 − y1 y2 + y4 + u3 ,
⎛
⎞
e1 R1 e˙ 1 ⎜ ⎟ ⎜ u2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎝ e˙ 2 ⎠ = (A − L1 ) ⎝ e2 ⎠ + ⎝ R2 ⎠ + ⎜ ⎜u ⎟, ⎝ 3⎠ e˙ 3
x˙ 3 = −ˇx3 + x1 x3 + ,
⎞ ⎛
0
0
0.5
0
0
0.2
0
⎞
⎠ × (R1 , R2 , R3 )T , and u4 = 0. (22)
So,
⎛
−1
A − L1 = ⎝ 0 0
0
0
−1
0
0
−ˇ
⎞ ⎠.
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Fig. 3. Time evolution of synchronization errors e1 , e2 and e3 between Rössler system (19) and hyperchaotic Liu system (20) in 3D.
Using simple calculations, we can show that (A − L1 )T + (A − L1 ) is a negative definite matrix. Then the conditions of Theorem 1 are satisfied. Hence the synchronization between systems (19) and (20) is achieved. For the purpose of numerical simulation, fourth order Runge–Kutta integration method has been used to solve the systems of differential equations (19) and (20). In addition, time step size 0.001 has been employed. We select the parameters of Rössler system (19) as ˛ = 0.2, = 0.2, ˇ = 5.7, so that system (19) exhibits a chaotic behavior, and the parameters of Liu hyperchaotic system (20) as a = 10, b = 35, c = 1.4 and d = 5, so that system (20) exhibits a hyperchaotic behavior. The initial values of the driving and response systems are [x1 (0), x2 (0), x3 (0)] = [0.1, 0.6, 0.3] and [y1 (0), y2 (0), y3 (0), y4 (0)] = [0.1, 0.1, 0.14, 0.4], respectively, and the initial states of the error system are [e1 (0), e2 (0), e3 (0)] = [0.100, − 0.100, 0.200]. Fig. 3 displays the synchronization errors between systems (19) and (20) in 3D. 3.2. Synchronization between the 3D drive Rössler system and the 4D response hyperchaotic Liu system in 4D
⎛
−a
⎜ b B=⎜ ⎝ 0 d
a
0
⎞
0
0
⎞
0
0
0
In this case, the error system between the drive system (19) and the response system (20), can be described as
⎛
e˙ 1
⎞
⎛
e1
⎞ ⎛
R1
⎞
⎛
u1
⎞
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ e˙ 2 ⎟ ⎜e ⎟ ⎜R ⎟ ⎜u ⎟ ⎜ ⎟ = (B − L2 ) ⎜ 2 ⎟ + ⎜ 2 ⎟ + ⎜ 2 ⎟ , ⎜ e˙ ⎟ ⎜e ⎟ ⎜R ⎟ ⎜u ⎟ ⎝ 3⎠ ⎝ 3⎠ ⎝ 3⎠ ⎝ 3⎠ e˙ 4
e4 e1 =
where
3 2j xj , e3 = j=1 3 j=1
R4
j=1
3j yj −
(23)
u4
4 3 y − 1j xj , j=1 1j j 4 j=1 3 j=1
e2 =
3j xj and e4 =
4 2j yj − 4j=1 j=1
4j yj −
4j xj and we choose
⎛
1
⎜0 =⎝ 0 0
⎞
0
0
0
2
0
0⎟
0
3
0
0
0
4
⎠,
⎛
1
⎜0 =⎝ 1 0
⎞
0
1
1
0⎟
1
0
0
1
⎠,
⎛
0
⎜b L2 = ⎝ 0 d
R1 = ay2 + ax1 + x2 + (a + ˇ)x3 − − x3 x1 , R2 = 2by1 + 2y2 − 2y4 + (1 − ˛)x2 − x1 + 2y1 y3 , R3 = −3y4 + (c − 1)x1 + (c + 1 − ˛)x2 + x3 − y1 y2 , R4 = 4dy1 + 4y2 − 8y4 + (1 + ˇ)x3 − − x3 x1 . By using the same formula of Eq. (17), the vector controller U = (u1 , u2 , u3 , u4 )T is designed as
⎛
a
0
0
⎞
1 0 −1 ⎟ 0
1
1 0
0
1
1
⎠,
0 1 2
⎜ ⎜0 ⎜ ⎜ (u1 , u2 , u3 , u4 )T = − ⎜ ⎜0 0 ⎜ ⎝ So,
⎛
−a
⎜ 0 ⎝ 0
B − L2 = ⎜
0
⎜ y1 y3 ⎟ 0 0 −1 ⎟ ⎟ ⎟ , g(y) = ⎜ ⎜ ⎟. ⎠ 0 −c 1 ⎝ −y1 y2 ⎠ 1
then,
0
In order to observe synchronization between the drive system (19) and the response system (20) in 4D, the hyperchaotic Liu system (i.e., the uncontrolled system (20)) can be rewritten as the form y˙ = By + g(y) where
⎛
Fig. 4. Time evolution of synchronization errors e1 , e2 , e3 and e4 between Rössler system (19) and hyperchaotic Liu system (20) in 4D.
0
0
0
0
−1
0
0
0
−c
0
0
0
−1
0
0
0
0
1 3 0
⎞
⎟ ⎟ ⎟ ⎟ ⎟ × (R1 , R2 , R3 , R4 )T . 0⎟ ⎟ ⎠
(24)
1 4
⎞ ⎟ ⎟. ⎠
Its easy to show that all eigenvalues of B − L2 are strictly negative. Then the conditions of Theorem 2 are satisfied. Therefore, the systems (19) and (20) are globally synchronized, in 4D. For the purpose of numerical simulation, fourth order Runge–Kutta integration method has been used to solve the systems of differential equations (19) and (20). In addition, time step size 0.001 has been employed. We select the parameters of Rössler system (19) as ˛ = 0.2, = 0.2, ˇ = 5.7, so that system (19) exhibits a chaotic behavior, and the parameters of Liu hyperchaotic system (20) as a = 10, b = 35, c = 1.4 and d = 5, so that system (20) exhibits a hyperchaotic behavior. The initial values of the driving and response systems are [x1 (0), x2 (0), x3 (0)] = [0.6, 0.1, 0.7] and [y1 (0), y2 (0), y3 (0), y4 (0)] = [1.2, 0.1, 0.4, 0.2], respectively, and the initial states of the error system are [e1 (0), e2 (0), e3 (0), e4 (0)] = [0.100, − 0.100, 0.100, − 0.100]. Fig. 4 displays the synchronization errors between systems (19) and (20) in 4D. 4. Conclusion In this paper, we have investigated the synchronization problem of different dimensional chaotic systems in different dimensions using two scaling matrices, the Lyapunov stability theory, and the stability theory of linear system. The nonlinear controller was
A. Ouannas, M.M. Al-sawalha / Optik 127 (2016) 959–963
designed to guaranteed the synchronization between 3D drive system and 4D response system in 3D by controlling the linear part of the drive system. However the synchronization between 3D drive system and 4D response system in 4D guaranteed by controlling the linear part of the response system. The new synchronization control approach was successfully applied to several examples: the Rössler system and hyperchaotic Liu system. Numerical simulations were presented to illustrate the proposed synchronization schemes. References [1] G. Chen, X. Dong, From Chaos to Order, World Scientific, Singapore, 1989. [2] A. Luo, A theory for synchronization of dynamical systems, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 1901–1951. [3] M. Rafikov, J. Balthazar, On control and synchronization in chaotic and hyperchaotic systems via linear feedback control, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 1246–1255. [4] Y. Sun, J. Cao, Adaptive synchronization between two different noise-perturbed chaotic systems with fully unknown parameters, Phys. A 376 (2007) 253–265. [5] J. Park, Adaptive synchronization of hyperchaotic Chen system with uncertain parameters, Chaos Solitons Fractals 26 (2005) 959–964. [6] X. Wu, H. Zhang, Synchronization of two hyperchaotic systems via adaptive control, Chaos Solitons Fractals 39 (2009) 2268–2273. [7] C. Shihua, J. Lü, Parameters identification and synchronization of chaotic systems based upon adaptive control, Phys. Lett. A 299 (2002) 353–358. [8] J. Lu, X. Wu, X. Han, J. Lü, Adaptive feedback synchronization of a unified chaotic system, Phys. Lett. A 329 (2004) 327–333. [9] J. Huang, Chaos synchronization between two novel different hyperchaotic systems with unknown parameters, Nonlinear Anal. Theory Methods Appl. 69 (2008) 4174–4181. [10] J. Lü, J. Lu, Controlling uncertain Lü system using linear feedback, Chaos Solitons Fractals 17 (2003) 127–133. [11] H. Chen, Global chaos synchronization of new chaotic systems via nonlinear control, Chaos Solitons Fractals 23 (2005) 1245–1251. [12] L. Huang, R. Feng, M. Wang, Synchronization of chaotic systems via nonlinear control, Phys. Lett. A 320 (2004) 271–275. [13] Q. Zhang, J. Lu, Chaos synchronization of a new chaotic system via nonlinear control, Chaos Solitons Fractals 37 (2008) 175–179.
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