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Synchronization of generalised linearly bidirectionally coupled unified chaotic system
Available online at www.sciencedirect.com
Chaos, Solitons and Fractals 40 (2009) 885–892 www.elsevier.com/locate/chaos
Synchronization of generalised linearly bidirectionally coupled unified chaotic system Anindita Tarai (Poria) a, Swarup Poria b, Prasanta Chatterjee a
c,*
Department of Mathematics, Aligunj R.R.B. High School Midnapore (West), West Bengal, India b Department of Mathematics, Midnapore College, Midnapore (West), West Bengal, India c Department of Mathematics, Visva Bharati University, Santiniketan, India Accepted 15 August 2007
Abstract Several important properties of chaos synchronization of bidirectionally coupled systems remain still unexplored. This paper investigates identical synchronization scheme for generalised linearly bidirectionally coupled unified chaotic system. The Lyapunov stability theory is used to substantiate the results. The study of linearly bidirectionally coupled unified chaotic systems are done first and conditions on coupling parameters for synchronization are derived. Finally, numerical simulation results are presented to show the feasibility and effectiveness of the approach. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Since the pioneering work by Pecora and Carroll [1] in 1990, chaos synchronization has received much attention because of its fundamental importance in nonlinear dynamics and potential applications to laser dynamics, electronic circuits, chemical and biological systems, and secure communications. Experimental realization of chaos synchronization have been achieved with a magnetic ribbon, a heart, a thermal convection loop, a diode oscillator, a Belousov– Zhabotinski reaction diffusion chemical system, and many other experiments. Many chaos synchronization and control methods have been developed, such as backstepping design method [2], impulsive control method [3], differential geometric method [4], invariant manifold method [5], adaptive control method [6], linear and nonlinear feedback control method [7], active control method [8] and synchronization in unidirectionally and bidirectionally coupled systems [9]. Chaotic synchronization means that, given two coupled dynamical systems (which may or may not be identical), there exist a smooth and invertible map g which carries trajectories on the attractor of the first system to trajectories on the attractor of the second system with the property that if an orbit of the first system approaches a trajectory r1(t) on the attractor of the first system, then the corresponding orbit of the second system approaches the trajectory r2(t) = g(r1(t)). Now consider a pair of coupled dynamical systems x_1 ¼ f ðx1 ; x2 Þ;
*
x_2 ¼ gðx1 ; x2 Þ
Corresponding author. E-mail address: prasantachatterjee1@rediffmail.com (P. Chatterjee).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.08.039
ð1Þ
886
A. Tarai (Poria) et al. / Chaos, Solitons and Fractals 40 (2009) 885–892
where f, g: R2n ! R2n are smooth functions. Synchronization between the systems x1 and x2 is defined in the following way: the systems x1 and x2 synchronize if there exists a compact, diagonal like, smooth n manifold M with boundary which is invarient under the flow, inflowing, and locally attracting. M will be referred to as the synchronization manifold. The synchronization manifold M can be viewed as the graph of the function g and so the existence of the manifold M implies the existence of the function g. Synchronization manifold may have both positive and negative Lyapunov exponents. In order to ensure synchronization under perturbation, we require that the rate at which trajectories are attracted towards the manifold is greater than the expansion or contraction rates within the manifold. The generalize Lyapunov number measures the rate of contraction or expansion under the flow in the direction transversal to the manifold. A sufficient condition for synchronization is that this generalised Lyapunov number be negative. Given a pair of coupled dynamical systems, the coupling is said to be unidirectional if one of the equations is independent of the other [i.e., Eq. (1) can be written as X_ ¼ f ðXÞ and Y_ ¼ gðX; YÞ]. The coupling can be unidirectional in only part of the phase space. Physically, this means that in part of the phase space, the behavior of one system has no influence on the behavior of the other. If coupling is not unidirectional then it must be bidirectional. There have been very few results about bidirectionally coupled synchronization of chaotic systems. But most of the natural systems are bidirectionally coupled. Therefore, the study of bidirectionally coupled system is necessary. We attempt to begin such a study in the present paper. Recently, Zhou et al. [10] investigated linearly and nonlinearly bidirectionally coupled chaotic systems. In this paper, we consider the most general type of linear bidirectional coupling and we have taken unified chaotic systems [11] for the study because unified chaotic system contains the Lorenz system [12], Chen system [13] and Lu system [14] for special choice of parameter values. We derive the restrictions on coupling parameters for both the cases. Finally, the numerical simulation results are presented to show power of the method.
2. The unified chaotic system Lorenz [12] had found the first classical chaotic system in 1963. Chen and Ueta [13] have found a chotic system which is similar to Lorenz system but not topologically equivalent to Lorenz system in 1999. Recently, a chaotic system is presented by Lu et al. [14], which bridged the gap between the Lorenz and Chen systems. A new unified chaotic system with continuous periodic switch between the Lorenz and Chen system is presented by Lu and Wu [11] in 2004. The unified chaotic system can be described by the following system of differential equations: dx ¼ ð25a þ 10Þðy xÞ; dt dy ¼ ð28 35aÞx xz þ ð29a 1Þy; dt dz 8þa ¼ xy z; dt 3
ð2Þ
where a 2 [0,1]. When a = 0, 0.8, 1 this system represents the Lorenz chaotic system, Lu chaotic system and Chen chaotic system, respectively. Practically, unified system is chaotic for any a 2 [0, 1]. Consider two unified hyperchaotic systems: X_ ¼ AX þ f ðX Þ þ DðY X Þ; Y_ ¼ AY þ f ðY Þ þ DðX Y Þ; where X = (x1, y1, 0 a1 B A ¼ @ a2 0
T
ð3Þ ð4Þ T
z1) , Y = (x2, y2, z2) , f(X) = (0, x1z1, x1y1) and 1 1 0 d 11 d 12 d 13 a1 0 C C B a3 0 A; D ¼ @ d 21 d 22 d 23 A: 0 a4 d 31 d 32 d 33
ð5Þ
Here a1 = (25a + 10), a2 = (28 35a), a3 = (29a 1) and a4 ¼ 8þa and dij’s are the coupling coefficients. Notice that D 3 is the generalised linear bidirectional coupling coefficient matrix. These coefficients are constants for the case of linear coupling and are functions of state variable in the case of nonlinear coupling. Systems (3) and (4) asymptotically synchronizes if limt!1kY(t) X(t)k = 0, where k Æ k denotes the Euclidean norm. So our tusk is to find suitable coupling coefficients such that systems (3) and (4) synchronizes.
A. Tarai (Poria) et al. / Chaos, Solitons and Fractals 40 (2009) 885–892
887
3. Synchronization of linearly coupled unified systems In this case, the coupling coefficients dij’s are constants. Here our aim is to find the validity range of these coefficients for synchronization. Obviously the validity ranges are dependent on the parameter a of the unified system. Let e = Y(t) X(t). Now using systems (3) and (4) we find the dynamical system of the synchronization error as e_ ¼ Ae þ f ðY Þ f ðX Þ 2De:
ð6Þ
Let us write system (6) as e_ ¼ ðA þ B 2DÞe; where
0
0
B B ¼ @ z1 y1
0 0 x2
0
ð7Þ 1
C x2 A: 0
ð8Þ
Now define a Lyapunov function V ðeÞ ¼ 12 eT e; then clearly V(e) is positive definite. Then V_ ¼ eT ðP 1 þ P 2 Þe where 1 1 0 0 a1 þa2 0 z21 y21 a1 2d 11 d 12 d 21 d 13 d 31 2 C B z B 2 0 0C ð9Þ P 1 ¼ @ a1 þa d 12 d 21 a3 2d 22 d 23 d 32 A; P 2 ¼ @ 21 A: 2 y1 0 0 d 13 d 31 d 32 d 23 a4 2d 33 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 2 2 Eigen values of the matrix P2 are 0; 2 y 1 þ z1 ; 2 y 1 þ z1 . The eigenvalue of the matrix P2 which has maximum absopffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi lute value is 12 y 21 þ z21 . Let 12 y 21 þ z21 < M; where M = maxjxij, j yij, i = 1, 2, 3. We can always find such a bound for a chaotic system because for a chaotic system phase space must be bounded. Now we have V_ ¼ eT ðP 1 þ P 2 Þe 6 eT ðP 1 þ MIÞe ¼ eT Pe; 0
2 a1 þa þ d 12 þ d 21 2 a3 þ 2d 22 M d 32 þ d 23
a1 þ 2d 11 M B 2 P ¼ @ a1 þa þ d 12 þ d 21 2 d 13 þ d 31
1 d 13 þ d 31 C d 23 þ d 32 A: a4 þ 2d 33 M
ð10Þ
30 ’uni1.dat’ using 1:2
20
10 Error e1 (t)
where
0
-10
-20
-30 0
10
20
30
40
50
60
70
time t
Fig. 1. Time evolution of the error e1(t) for a = 0.0 in the unified chaotic system, i.e., in the Lorenz system.
888
A. Tarai (Poria) et al. / Chaos, Solitons and Fractals 40 (2009) 885–892
For V(e) to become a Lyapunov function by definition V_ (e) must be negative definite. Therefore, V(e) is a Lyapunov _ is negative definite. function for the error dynamical system for those values of the coupling parameters for which VðeÞ This holds only when P is a positive definite matrix. We know that a square matrix is positive definite if and only if it is symmetric and all its principal minors are positive. So for synchronization (i) dij’s must be so chosen that
40
30
20
Error e2 (t)
10
0
-10
-20
-30
-40 0
10
20
30
40
50
60
70
time t
Fig. 2. Time evolution of the error e2(t) for a = 0.0 in the unified chaotic system, i.e., in the Lorenz system.
40
30
20
Error e3 (t)
10
0
-10
-20
-30
-40 0
10
20
30
40
50
60
70
time t
Fig. 3. Time evolution of the error e3(t) for a = 0.0 in the unified chaotic system, i.e., in the Lorenz system.
A. Tarai (Poria) et al. / Chaos, Solitons and Fractals 40 (2009) 885–892
a1 þ 2d 11 M ¼ 25a þ 10 þ 2d 11 M > 0; (ii) n a þa o2 1 2 fa1 þ 2d 11 Mgfa3 þ 2d 22 Mg þ d 12 þ d 21 > 0 2 and (iii) det(P) > 0.
30
20
Error e1 (t)
10
0
-10
-20
-30 0
10
20
30
40
50
60
70
time t
Fig. 4. Time evolution of the error e1(t) for a = 0.8 in the unified chaotic system, i.e., in the Lu system. 40
30
20
Error e2 (t)
10
0
-10
-20
-30
-40 0
10
20
30
40
50
60
70
time t
Fig. 5. Time evolution of the error e2(t) for a = 0.8 in the unified chaotic system, i.e., in the Lu system.
889
890
A. Tarai (Poria) et al. / Chaos, Solitons and Fractals 40 (2009) 885–892
If P is positive definite then by Lyapunov stability theory limt!1ke(t)k = 0, and the systems (3) and (4) synchronize.
4. Results and discussion In this paper, we generalised the result of linearly bidirectionally coupled chaotic systems for the case of generalised linear coupling. We have discussed the identical synchronization of two bidirectionally coupled unified chaotic systems, 30
20
10
Error e3 (t)
0
-10
-20
-30
-40 0
10
20
30
40
50
60
70
time t
Fig. 6. Time evolution of the error e3(t) for a = 0.8 in the unified chaotic system, i.e., in the Lu system. 35 30 25 20
Error e1 (t)
15 10 5 0 -5 -10 -15 -20 0
10
20
30
40
50
60
70
Time t
Fig. 7. Time evolution of the error e1(t) for a = 1.0 in the unified chaotic system, i.e., in the Chen system.
A. Tarai (Poria) et al. / Chaos, Solitons and Fractals 40 (2009) 885–892
891
which are coupled linearly. We developed the restrictions on the coupling coefficients to guarantee the global asymptotical stability of the synchronized state. Asymptotical stability analysis was done by applying Lyapunov stability theory. These conditions on dij’s are not necessary but sufficient. Numerical simulations are done by fourth order Runge– Kutta method. We assume that the initial values of the error as e1(0) = 9.99, e2(t) = 20, and e3(t) = 30 in all simulations. In Figs. 1–3, we draw the time evolution of the error of the unified chaotic system for a = 0, i.e., for Lorenz system taking the coupling coefficients d11 = 1.0, d22 = 0.5, d33 = 0.3 and all other dij = 0. Pictures show that error is going to zero asymptotically, i.e., the two chaotic systems synchronize. In Figs. 4–6, we draw the time evolution of the error
40
30
Error e2 (t)
20
10
0
-10
-20
-30 0
10
20
30
40
50
60
70
time t
Fig. 8. Time evolution of the error e2(t) for a = 1.0 in the unified chaotic system, i.e., in the Chen system.
20
10
Error e3 (t)
0
-10
-20
-30
-40 0
10
20
30
40
50
60
70
Time t
Fig. 9. Time evolution of the error e3(t) for a = 1.0 in the unified chaotic system, i.e., in the Chen system.
892
A. Tarai (Poria) et al. / Chaos, Solitons and Fractals 40 (2009) 885–892
of the unified chaotic system for a = 0.8, i.e., for Lu system taking the coupling coefficients d11 = 1.0, d22 = 1.2, d33 = 1.7 and all other dij = 0. Clearly the figure shows the synchronization. In Fig. 7–9, we draw the time evolution of the error of the unified chaotic system for a = 1.0, i.e., for Chen system taking the coupling coefficients d11 = 1.0, d22 = 1., d33 = 1.2 and all other dij = 0. Figures show that error is going to zero asymptotically, i.e., the two chaotic systems synchronize identically.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64(8):821–4. Yu Y, Zhang S. Controlling uncertain Lu system using backstepping design. Chaos, Solitons & Fractals 2003;15:897. Yang Tao, Yang Lin-Bao, Yang Chun-Mei. Impulsive control of Lorenz system. Physica D 1997;110:18. Fuh Chyun-Chau, Tung Pi-Cheng. Controlling chaos using differential geometric method. Phys Rev Lett 1995;75(16):2952. Yu Xinghuo, Song Yanxing. Chaos synchronization vis controlling partial state of chaotic systems. Int J Bifurcat Chaos 2001;11:1737. Chen S, Lu¨ J. Synchronization of an uncertain unified chaotic system via adaptive control. Chaos, Solitons & Fractals 2002;14(4):643. Park JH. Controlling chaotic systems via nonlinear feedback control. Chaos, Solitons & Fractals 2005;23:1049. Park JH. Stability criterion for synchronization of linearly coupled unified system. Chaos, Solitons & Fractals 2005;23:1319. Rosenblum MG, Pikovsky AS, Kurths J. From phase to lag synchronization in coupled chaotic oscillators. Phys Rev Lett 1996;76:1804. Zhou Jin, Lu Jun-an, Wu Xiaoqun. Linearly and nonlinearly bidirectionally coupled synchronization of hyperchaotic systems. Chaos, Solitons & Fractals 2007;31:230–5. Lu Jun-an, Wu Xiaoqun. A unified chaotic system with continuous periodic switch. Chaos, Solitons & Fractals 2004;20:245C251. Lorenz EN. Deterministic non-periodic flows. J Atmos Sci 1963;20:130. Chen GR, Ueta T. Yet another chaotic attractor. Int J Bifurcat Chaos 1999;9:1465. Lu Jinhu, Chen Guanrong, Cheng Diazhan, Celikovsky Sergej. Bridge the gap between the Lorenz system and the Chen system. Int J Bifurcat Chaos 2002;12:2917C26.
Chaos, Solitons and Fractals 40 (2009) 885–892 www.elsevier.com/locate/chaos
Synchronization of generalised linearly bidirectionally coupled unified chaotic system Anindita Tarai (Poria) a, Swarup Poria b, Prasanta Chatterjee a
c,*
Department of Mathematics, Aligunj R.R.B. High School Midnapore (West), West Bengal, India b Department of Mathematics, Midnapore College, Midnapore (West), West Bengal, India c Department of Mathematics, Visva Bharati University, Santiniketan, India Accepted 15 August 2007
Abstract Several important properties of chaos synchronization of bidirectionally coupled systems remain still unexplored. This paper investigates identical synchronization scheme for generalised linearly bidirectionally coupled unified chaotic system. The Lyapunov stability theory is used to substantiate the results. The study of linearly bidirectionally coupled unified chaotic systems are done first and conditions on coupling parameters for synchronization are derived. Finally, numerical simulation results are presented to show the feasibility and effectiveness of the approach. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Since the pioneering work by Pecora and Carroll [1] in 1990, chaos synchronization has received much attention because of its fundamental importance in nonlinear dynamics and potential applications to laser dynamics, electronic circuits, chemical and biological systems, and secure communications. Experimental realization of chaos synchronization have been achieved with a magnetic ribbon, a heart, a thermal convection loop, a diode oscillator, a Belousov– Zhabotinski reaction diffusion chemical system, and many other experiments. Many chaos synchronization and control methods have been developed, such as backstepping design method [2], impulsive control method [3], differential geometric method [4], invariant manifold method [5], adaptive control method [6], linear and nonlinear feedback control method [7], active control method [8] and synchronization in unidirectionally and bidirectionally coupled systems [9]. Chaotic synchronization means that, given two coupled dynamical systems (which may or may not be identical), there exist a smooth and invertible map g which carries trajectories on the attractor of the first system to trajectories on the attractor of the second system with the property that if an orbit of the first system approaches a trajectory r1(t) on the attractor of the first system, then the corresponding orbit of the second system approaches the trajectory r2(t) = g(r1(t)). Now consider a pair of coupled dynamical systems x_1 ¼ f ðx1 ; x2 Þ;
*
x_2 ¼ gðx1 ; x2 Þ
Corresponding author. E-mail address: prasantachatterjee1@rediffmail.com (P. Chatterjee).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.08.039
ð1Þ
886
A. Tarai (Poria) et al. / Chaos, Solitons and Fractals 40 (2009) 885–892
where f, g: R2n ! R2n are smooth functions. Synchronization between the systems x1 and x2 is defined in the following way: the systems x1 and x2 synchronize if there exists a compact, diagonal like, smooth n manifold M with boundary which is invarient under the flow, inflowing, and locally attracting. M will be referred to as the synchronization manifold. The synchronization manifold M can be viewed as the graph of the function g and so the existence of the manifold M implies the existence of the function g. Synchronization manifold may have both positive and negative Lyapunov exponents. In order to ensure synchronization under perturbation, we require that the rate at which trajectories are attracted towards the manifold is greater than the expansion or contraction rates within the manifold. The generalize Lyapunov number measures the rate of contraction or expansion under the flow in the direction transversal to the manifold. A sufficient condition for synchronization is that this generalised Lyapunov number be negative. Given a pair of coupled dynamical systems, the coupling is said to be unidirectional if one of the equations is independent of the other [i.e., Eq. (1) can be written as X_ ¼ f ðXÞ and Y_ ¼ gðX; YÞ]. The coupling can be unidirectional in only part of the phase space. Physically, this means that in part of the phase space, the behavior of one system has no influence on the behavior of the other. If coupling is not unidirectional then it must be bidirectional. There have been very few results about bidirectionally coupled synchronization of chaotic systems. But most of the natural systems are bidirectionally coupled. Therefore, the study of bidirectionally coupled system is necessary. We attempt to begin such a study in the present paper. Recently, Zhou et al. [10] investigated linearly and nonlinearly bidirectionally coupled chaotic systems. In this paper, we consider the most general type of linear bidirectional coupling and we have taken unified chaotic systems [11] for the study because unified chaotic system contains the Lorenz system [12], Chen system [13] and Lu system [14] for special choice of parameter values. We derive the restrictions on coupling parameters for both the cases. Finally, the numerical simulation results are presented to show power of the method.
2. The unified chaotic system Lorenz [12] had found the first classical chaotic system in 1963. Chen and Ueta [13] have found a chotic system which is similar to Lorenz system but not topologically equivalent to Lorenz system in 1999. Recently, a chaotic system is presented by Lu et al. [14], which bridged the gap between the Lorenz and Chen systems. A new unified chaotic system with continuous periodic switch between the Lorenz and Chen system is presented by Lu and Wu [11] in 2004. The unified chaotic system can be described by the following system of differential equations: dx ¼ ð25a þ 10Þðy xÞ; dt dy ¼ ð28 35aÞx xz þ ð29a 1Þy; dt dz 8þa ¼ xy z; dt 3
ð2Þ
where a 2 [0,1]. When a = 0, 0.8, 1 this system represents the Lorenz chaotic system, Lu chaotic system and Chen chaotic system, respectively. Practically, unified system is chaotic for any a 2 [0, 1]. Consider two unified hyperchaotic systems: X_ ¼ AX þ f ðX Þ þ DðY X Þ; Y_ ¼ AY þ f ðY Þ þ DðX Y Þ; where X = (x1, y1, 0 a1 B A ¼ @ a2 0
T
ð3Þ ð4Þ T
z1) , Y = (x2, y2, z2) , f(X) = (0, x1z1, x1y1) and 1 1 0 d 11 d 12 d 13 a1 0 C C B a3 0 A; D ¼ @ d 21 d 22 d 23 A: 0 a4 d 31 d 32 d 33
ð5Þ
Here a1 = (25a + 10), a2 = (28 35a), a3 = (29a 1) and a4 ¼ 8þa and dij’s are the coupling coefficients. Notice that D 3 is the generalised linear bidirectional coupling coefficient matrix. These coefficients are constants for the case of linear coupling and are functions of state variable in the case of nonlinear coupling. Systems (3) and (4) asymptotically synchronizes if limt!1kY(t) X(t)k = 0, where k Æ k denotes the Euclidean norm. So our tusk is to find suitable coupling coefficients such that systems (3) and (4) synchronizes.
A. Tarai (Poria) et al. / Chaos, Solitons and Fractals 40 (2009) 885–892
887
3. Synchronization of linearly coupled unified systems In this case, the coupling coefficients dij’s are constants. Here our aim is to find the validity range of these coefficients for synchronization. Obviously the validity ranges are dependent on the parameter a of the unified system. Let e = Y(t) X(t). Now using systems (3) and (4) we find the dynamical system of the synchronization error as e_ ¼ Ae þ f ðY Þ f ðX Þ 2De:
ð6Þ
Let us write system (6) as e_ ¼ ðA þ B 2DÞe; where
0
0
B B ¼ @ z1 y1
0 0 x2
0
ð7Þ 1
C x2 A: 0
ð8Þ
Now define a Lyapunov function V ðeÞ ¼ 12 eT e; then clearly V(e) is positive definite. Then V_ ¼ eT ðP 1 þ P 2 Þe where 1 1 0 0 a1 þa2 0 z21 y21 a1 2d 11 d 12 d 21 d 13 d 31 2 C B z B 2 0 0C ð9Þ P 1 ¼ @ a1 þa d 12 d 21 a3 2d 22 d 23 d 32 A; P 2 ¼ @ 21 A: 2 y1 0 0 d 13 d 31 d 32 d 23 a4 2d 33 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 2 2 Eigen values of the matrix P2 are 0; 2 y 1 þ z1 ; 2 y 1 þ z1 . The eigenvalue of the matrix P2 which has maximum absopffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi lute value is 12 y 21 þ z21 . Let 12 y 21 þ z21 < M; where M = maxjxij, j yij, i = 1, 2, 3. We can always find such a bound for a chaotic system because for a chaotic system phase space must be bounded. Now we have V_ ¼ eT ðP 1 þ P 2 Þe 6 eT ðP 1 þ MIÞe ¼ eT Pe; 0
2 a1 þa þ d 12 þ d 21 2 a3 þ 2d 22 M d 32 þ d 23
a1 þ 2d 11 M B 2 P ¼ @ a1 þa þ d 12 þ d 21 2 d 13 þ d 31
1 d 13 þ d 31 C d 23 þ d 32 A: a4 þ 2d 33 M
ð10Þ
30 ’uni1.dat’ using 1:2
20
10 Error e1 (t)
where
0
-10
-20
-30 0
10
20
30
40
50
60
70
time t
Fig. 1. Time evolution of the error e1(t) for a = 0.0 in the unified chaotic system, i.e., in the Lorenz system.
888
A. Tarai (Poria) et al. / Chaos, Solitons and Fractals 40 (2009) 885–892
For V(e) to become a Lyapunov function by definition V_ (e) must be negative definite. Therefore, V(e) is a Lyapunov _ is negative definite. function for the error dynamical system for those values of the coupling parameters for which VðeÞ This holds only when P is a positive definite matrix. We know that a square matrix is positive definite if and only if it is symmetric and all its principal minors are positive. So for synchronization (i) dij’s must be so chosen that
40
30
20
Error e2 (t)
10
0
-10
-20
-30
-40 0
10
20
30
40
50
60
70
time t
Fig. 2. Time evolution of the error e2(t) for a = 0.0 in the unified chaotic system, i.e., in the Lorenz system.
40
30
20
Error e3 (t)
10
0
-10
-20
-30
-40 0
10
20
30
40
50
60
70
time t
Fig. 3. Time evolution of the error e3(t) for a = 0.0 in the unified chaotic system, i.e., in the Lorenz system.
A. Tarai (Poria) et al. / Chaos, Solitons and Fractals 40 (2009) 885–892
a1 þ 2d 11 M ¼ 25a þ 10 þ 2d 11 M > 0; (ii) n a þa o2 1 2 fa1 þ 2d 11 Mgfa3 þ 2d 22 Mg þ d 12 þ d 21 > 0 2 and (iii) det(P) > 0.
30
20
Error e1 (t)
10
0
-10
-20
-30 0
10
20
30
40
50
60
70
time t
Fig. 4. Time evolution of the error e1(t) for a = 0.8 in the unified chaotic system, i.e., in the Lu system. 40
30
20
Error e2 (t)
10
0
-10
-20
-30
-40 0
10
20
30
40
50
60
70
time t
Fig. 5. Time evolution of the error e2(t) for a = 0.8 in the unified chaotic system, i.e., in the Lu system.
889
890
A. Tarai (Poria) et al. / Chaos, Solitons and Fractals 40 (2009) 885–892
If P is positive definite then by Lyapunov stability theory limt!1ke(t)k = 0, and the systems (3) and (4) synchronize.
4. Results and discussion In this paper, we generalised the result of linearly bidirectionally coupled chaotic systems for the case of generalised linear coupling. We have discussed the identical synchronization of two bidirectionally coupled unified chaotic systems, 30
20
10
Error e3 (t)
0
-10
-20
-30
-40 0
10
20
30
40
50
60
70
time t
Fig. 6. Time evolution of the error e3(t) for a = 0.8 in the unified chaotic system, i.e., in the Lu system. 35 30 25 20
Error e1 (t)
15 10 5 0 -5 -10 -15 -20 0
10
20
30
40
50
60
70
Time t
Fig. 7. Time evolution of the error e1(t) for a = 1.0 in the unified chaotic system, i.e., in the Chen system.
A. Tarai (Poria) et al. / Chaos, Solitons and Fractals 40 (2009) 885–892
891
which are coupled linearly. We developed the restrictions on the coupling coefficients to guarantee the global asymptotical stability of the synchronized state. Asymptotical stability analysis was done by applying Lyapunov stability theory. These conditions on dij’s are not necessary but sufficient. Numerical simulations are done by fourth order Runge– Kutta method. We assume that the initial values of the error as e1(0) = 9.99, e2(t) = 20, and e3(t) = 30 in all simulations. In Figs. 1–3, we draw the time evolution of the error of the unified chaotic system for a = 0, i.e., for Lorenz system taking the coupling coefficients d11 = 1.0, d22 = 0.5, d33 = 0.3 and all other dij = 0. Pictures show that error is going to zero asymptotically, i.e., the two chaotic systems synchronize. In Figs. 4–6, we draw the time evolution of the error
40
30
Error e2 (t)
20
10
0
-10
-20
-30 0
10
20
30
40
50
60
70
time t
Fig. 8. Time evolution of the error e2(t) for a = 1.0 in the unified chaotic system, i.e., in the Chen system.
20
10
Error e3 (t)
0
-10
-20
-30
-40 0
10
20
30
40
50
60
70
Time t
Fig. 9. Time evolution of the error e3(t) for a = 1.0 in the unified chaotic system, i.e., in the Chen system.
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of the unified chaotic system for a = 0.8, i.e., for Lu system taking the coupling coefficients d11 = 1.0, d22 = 1.2, d33 = 1.7 and all other dij = 0. Clearly the figure shows the synchronization. In Fig. 7–9, we draw the time evolution of the error of the unified chaotic system for a = 1.0, i.e., for Chen system taking the coupling coefficients d11 = 1.0, d22 = 1., d33 = 1.2 and all other dij = 0. Figures show that error is going to zero asymptotically, i.e., the two chaotic systems synchronize identically.
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