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Controllability and stability of the perturbed Chen chaotic dynamical system
Applied Mathematics and Computation 171 (2005) 927–947 www.elsevier.com/locate/amc
Controllability and stability of the perturbed Chen chaotic dynamical system Tidarut Plienpanich a, Piyapong Niamsup Yongwimon Lenbury b
a,*
,
a
b
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
Abstract In this paper, we study perturbed Chen chaotic dynamical system. Firstly, we study the sufficient conditions of parameters which guarantee that the equilibrium points of perturbed Chen chaotic dynamical system are asymptotically stable. Secondly, we study methods for controlling chaos such as feedback control and bounded feedback control that suppress the chaotic behavior to unstable equilibrium points. Finally, we present chaos synchronization of perturbed Chen chaotic dynamical system by using active control and adaptive control. 2005 Elsevier Inc. All rights reserved. Keywords: Perturbed Chen chaotic dynamical system; Controlling chaos; Synchronization
*
Corresponding author. E-mail addresses: [email protected] (T. Plienpanich), piyapongniamsup@hotmail. com, [email protected] (P. Niamsup), [email protected] (Y. Lenbury). 0096-3003/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.01.099
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1. Introduction In the recent years, controlling chaos and synchronization of the dynamical systems have attracted many researchers. Controlling chaos and chaos synchronization have focused on the nonlinear systems such as Chen chaotic dynamical system. Various control algorithms have been proposed to control chaotic systems. The existing control algorithms can be classified mainly into two categories: feedback and nonfeedback. In this paper, we only focus on feedback control. Linear feedback control and bounded feedback control are proposed to control chaos of the system to the equilibrium points. In [4], YassenÕs studied the optimal control of Chen chaotic dynamical system presented by x_ ¼ aðy xÞ; y_ ¼ ðc aÞx xz þ cy; z_ ¼ xy bz; where x, y, z are state variables and a, b, c are real positive constants. In [1], AgizaÕs studied the different methods to control chaotic behavior of the coupled dynamos system, where the mathematical model equations for this system are x_ ¼ lx þ yðz þ aÞ; y_ ¼ ly þ xðz aÞ; z_ ¼ 1 xy; where x, y, z are state variables and l, a are positive constants. In [2], Agiza and YassenÕs studied synchronization of Rossler and Chen chaotic dynamical systems using active control. In [3], Wang, Guan and WenÕs paper studied adaptive synchronization for Chen chaotic system with fully unknown parameters. The objectives of this paper are as follows. Firstly, to give sufficient conditions of parameters that make equilibrium points of perturbed Chen chaotic dynamical system to be asymptotically stable. Secondly, to apply linear feedback control and bounded feedback control for controlling chaos of the perturbed Chen chaotic dynamical system, described by x_ ¼ aðy xÞ; y_ ¼ ðc aÞx xz þ cy;
ð1:1Þ
2
z_ ¼ xy bz þ dx ; where x, y, z are the state variables and a, b, c, d are positive real constants.
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Thirdly, to study synchronization of perturbed Chen chaotic dynamical system using active control. Finally, to present adaptive synchronization for perturbed Chen chaotic dynamical system when the parameters of the drive system are fully unknown and different from those of the response system.
2. Stability of the perturbed Chen chaotic dynamical system We will study the perturbed Chen chaotic dynamical system that is described by system of ordinary differential equations (1.1). The equilibrium points of the system (1.1) are E1 ¼ ð0; 0; 0Þ; E2 ¼ ðb; b; cÞ; qffiffiffiffiffiffi bc and c = 2c a. where b ¼ 1þd
E3 ¼ ðb; b; cÞ;
Proposition 2.1. The equilibrium point E1 = (0, 0, 0) is (i) asymptotically stable if a > 2c and ac < b2 < 2ac. (ii) unstable if 2c > a.
Proof. The Jacobian matrix of the system (1.1) at the equilibrium point E1 = (0, 0, 0) is given by 2 3 a a 0 6 7 7 J1 ¼ 6 4 c a c 0 5: 0
0
b
The characteristic equation of the Jacobian J1 has the form k3 þ a1 k2 þ a2 k þ a3 ¼ 0; where a1 ¼ a þ b c; a2 ¼ bða cÞ þ aða 2cÞ; a3 ¼ abða 2cÞ; a1 a2 a3 ¼ ðab þ a2 Þða 2cÞ þ aðb2 acÞ þ cð2ac b2 Þ þ bc2 : We see that a1 and a1a2 a3 satisfy the Routh–Hurwitz criteria when a > 2c and ac < b2 < 2ac, thus the equilibrium point E1 = (0, 0, 0) is asymptotically stable. h
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Similarly, we have the following results: Proposition 2.2. The equilibrium point E2 = (b, b, c) is (i) asymptotically stable if 32 c < a < 2c, b > 6c and 13 < d < 1. (ii) unstable if b < c < a and a < 43 c. Proposition 2.3. The equilibrium point E3 = (b, b, c) is (i) asymptotically stable if 32 c < a < 2c, b > 6c and 13 < d < 1. (ii) unstable if b < c < a and a < 43 c.
3. Controlling chaos In this section, the chaos of system (1.1) is controlled to one of three equilibrium points of the system. Feedback and bounded feedback control are applied to achieve this goal. We shall study in the case when equilibrium points of (1.1) are unstable. For this purpose, we assume that b < c < a and a < 43 c. 3.1. Feedback control The goal of linear feedback control is to control the chaotic behavior of the system (1.1) to one of three unstable equilibrium points (E1, E2 or E3). We assume that the controlled system is given by x_ ¼ aðy xÞ þ u1 ; y_ ¼ ðc aÞx xz þ cy þ u2 ; z_ ¼ xy bz þ dx2 þ u3 ; where u1, u2 and u3 are controllers that satisfy the following control law x_ ¼ aðy xÞ k 11 ðx xÞ; y_ ¼ ðc aÞx xz þ cy k 22 ðy y Þ; z_ ¼ xy bz þ dx2 k 33 ðz zÞ;
ð3:1Þ
where E ¼ ðx; y ; zÞ is an equilibrium point of (1.1). 3.1.1. Stability of the equilibrium point E1 = (0, 0, 0) In this case E = E1 and the controlled system (3.1) is in the form of x_ ¼ aðy xÞ k 11 x; y_ ¼ ðc aÞx xz þ cy k 22 y; 2
z_ ¼ xy bz þ dx k 33 z:
ð3:2Þ
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Theorem 3.1.1. The equilibrium point E1 = (0, 0, 0) is asymptotically stable if k11 = 0, k33 > 0 and k22 > 3c. 3.1.2. Stability of the equilibrium point E2 = (b, b, c) In this case E = E2 and the controlled system (3.1) is in the form of x_ ¼ aðy xÞ k 11 ðx bÞ; y_ ¼ ðc aÞx xz þ cy k 22 ðy bÞ;
ð3:3Þ
2
z_ ¼ xy bz þ dx k 33 ðz cÞ: Theorem 3.1.2. The equilibrium point E2 = (b, b, c) is asymptotically stable if k11, k33 > 0 and k22 > 2c. 3.1.3. Stability of the equilibrium point E3 = (b, b, c) In this case E = E3 and the controlled system (3.1) is in the form of x_ ¼ aðy xÞ k 11 ðx þ bÞ; y_ ¼ ðc aÞx xz þ cy k 22 ðy þ bÞ;
ð3:4Þ
2
z_ ¼ xy bz þ dx k 33 ðz cÞ: Theorem 3.1.3. The equilibrium point E3 = (b, b, c) is asymptotically stable if k11, k33 > 0 and k22 > 2c. 3.1.4. Numerical simulation Numerical experiments are carried out to investigate controlled systems by using fourth-order Runge–Kutta method with time step 0.001. The parameters a, b, c and d are chosen as a = 35, b = 3, c = 28 and d = 2 to ensure the existence of chaos in the absence of control. The initial states are taken as x = 0.1, y = 0.2 and z = 0.3. The control is active at t = 10. The equilibrium point E1 = (0, 0, 0) of the system (1.1) is stabilized for k11 = 0, k22 = 85 and k33 = 5. Fig. 1 show the behavior of the states x, ypand z ffiffiffiffiffi of the controlled system (3.2) with time. The ffiffiffiffiffi p equilibrium point E2 ¼ ð 21; 21; 21Þ of the system (1.1) is stabilized for k11 = 1, k22 = 60 and k33 = 5. Fig. 2 show the behavior of the states x, y and z of pthe system (3.3) with time. The equilibrium point E3 ¼ ffiffiffiffiffi controlled pffiffiffiffiffi ð 21; 21; 21Þ of the system (1.1) is stabilized for k11 = 1, k22 = 60 and k33 = 5. Fig. 3 shows the behavior of the states x, y and z of the controlled system (3.4) with time. 3.2. Bounded feedback control In this case, we control chaos with bounded controller that vanishes after the stabilization is achieved.
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60 z
x, y,z
40
20
0 x, y
–20
–40 0
2
4
6
8
10
12
14
16
18
20
t
Fig. 1. The time responses for the states x, y and z of the controlled system (3.2) before and after control activation with time. The control is activated at t = 10, k11 = 0, k22 = 85 and k33 = 5.
80
60
40
x, y,z
z 20
0 x, y –20
–40 0
2
4
6
8
10
12
14
16
18
20
t
Fig. 2. The time responses for the states x, y and z of the controlled system (3.3) before and after control activation with time. The control is activated at t = 10, k11 = 1, k22 = 60 and k33 = 5.
3.2.1. Stability of the equilibrium point E1 = (0, 0, 0) In order to stabilize this equilibrium point by bounded feedback control, the control is chosen for system (1.1) as follows:
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80
60
x, y,z
40
z
20
0 x, y –20
–40 0
2
4
6
8
10
12
14
16
18
20
t Fig. 3. The time responses for the states x, y and z of the controlled system (3.4) before and after control activation with time. The control is activated at t = 10, k11 = 1, k22 = 60 and k33 = 5.
x_ ¼ aðy xÞ; y_ ¼ ðc aÞx xz þ cy þ uðtÞ;
ð3:5Þ
2
z_ ¼ xy bz þ dx ; where u(t) = k(a(x + y)), k > 0. Theorem 3.2.1. The equilibrium point E1 = (0, 0, 0) is asymptotically stable if k > 2c a. 3.2.2. Stability of the equilibrium point E2 = (b, b, c) In order to stabilize this equilibrium point by bounded feedback control, the control is chosen for system (1.1) as follows: x_ ¼ aðy xÞ; y_ ¼ ðc aÞx xz þ cy þ uðtÞ;
ð3:6Þ
2
z_ ¼ xy bz þ dx ; where u(t) = k(a(y x)), k > 0. Theorem pffiffiffi 3.2.2. The equilibrium point E2 = (b, b, c) is asymptotically stable if k > 2.
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3.2.3. Stability of the equilibrium point E3 = (b, b, c) In order to stabilize this equilibrium point by bounded feedback control, the control is chosen for system (1.1) as follows: x_ ¼ aðy xÞ; y_ ¼ ðc aÞx xz þ cy þ uðtÞ;
ð3:7Þ
2
z_ ¼ xy bz þ dx ; where u(t) = k(a(y x)), k > 0. Theorem pffiffiffi 3.2.3. The equilibrium point E3 = (b, b, c) is asymptotically stable if k > 2. 3.2.4. Numerical simulation We will show a series of numerical experiments by using the fourth-order Runge–Kutta method with step size 0.001. The parameters a, b, c and d are chosen as a = 35, b = 3, c = 28 and d = 2 to ensure the existence of chaos in the absence of control. The control is active at t = 10 for all simulations. In the first numerical experiment, we intend to control the chaos to equilibrium point E1 = (0, 0, 0) of system (1.1). Figs. 4–6 show the time response of the states x, y and z of system (3.5) and the controller u(t) with time for k = 1.6. The initial condition are x = 0.1, y = 0.2 and z = 0.3. In the second numerical experiment, we intend to control the chaos to equilibrium point
20 x
15
x,u
10 5 0 –5 –10 –15
u(t) 0
2
4
6
8
10
12
14
16
18
20
t Fig. 4. The states x of the controlled system (3.5) and the control u(t) respond with time before and after control activation. The control is activated at t = 10, k = 1.6.
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25 20 y 15 10
y,u
5 0 –5 –10 u(t)
–15 –20 –25 0
2
4
6
8
10
12
14
16
18
20
t Fig. 5. The states y of the controlled system (3.5) and the control u(t) respond with time before and after control activation. The control is activated at t = 10, k = 1.6. 80 70 60 z
z,u
50 40 30 20 10
u(t)
0 –10
0
2
4
6
8
10
12
14
16
18
20
t
Fig. 6. The states z of the controlled system (3.5) and the control u(t) respond with time before and after control activation. The control is activated at t = 10, k = 1.6.
pffiffiffiffiffi pffiffiffiffiffi E2 ¼ ð 21; 21; 21Þ of system (1.1). Figs. 7–9 show the time response of the states x, y and z of system (3.6) and the controller u(t) with time for k = 2. The initial condition are x = 2.5, y = 2.5 and z = 3. In the third numerical experiment, pffiffiffiffiffi we pffiffiffiffiffiintend to control the chaos to equilibrium point E3 ¼ ð 21; 21; 21Þ of system (1.1). Fig. 10–12 show the time response of the states x, y and z of system (3.7) and the controller u(t) with time for k = 2. The initial condition are x = 2.5, y = 2.5 and z = 3.
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x
x,u
5 0 u(t)
–5 –10 –15 –20
0
2
4
6
8
10
12
14
16
18
20
t
Fig. 7. The states x of the controlled system (3.6) and the control u(t) respond with time before and after control activation. The control is activated at t = 10, k = 2.
20 15 10
y
y, u
5 0 u(t)
–5 –10 –15 –20 –25
0
2
4
6
8
10
12
14
16
18
20
t Fig. 8. The states y of the controlled system (3.6) and the control u(t) respond with time before and after control activation. The control is activated at t = 10, k = 2.
4. Synchronization To begin with, the definition of chaos synchronization is given as follows. For two nonlinear chaotic system: x_ ¼ f ðt; xÞ;
ð4:1Þ
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70 60 50
z,u
40 z
30 20 10 0
u(t) –10 –20 0
2
4
6
8
10
12
14
16
18
20
t
Fig. 9. The states z of the controlled system (3.6) and the control u(t) respond with time before and after control activation. The control is activated at t = 10, k = 2. 20 15 10 u(t)
x, u
5 0 –5
x –10 –15 –20
0
2
4
6
8
10
12
14
16
18
20
t Fig. 10. The states x of the controlled system (3.7) and the control u(t) respond with time before and after control activation. The control is activated at t = 10, k = 2.
y_ ¼ gðt; yÞ þ uðt; x; yÞ;
ð4:2Þ
where x, y 2 R , f, g 2 C ½R R ; R , u 2 C ½R R R ; R , r P 1, Rþ is the set of non-negative real numbers. Assume that (4.1) is the drive system, and (4.2) is the response system, u(t, x, y) is the control vector. Response system and drive system are said to be synchronic if for "x(t0), yðt0 Þ 2 Rn , n
r
lim kxðtÞ yðtÞk ¼ 0:
t!1
þ
n
n
r
þ
n
n
n
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25 20 15
y, u
10 u(t)
5 0 –5
y –10 –15 –20
0
2
4
6
8
10
12
14
16
18
20
t Fig. 11. The states y of the controlled system (3.7) and the control u(t) respond with time before and after control activation. The control is activated at t = 10, k = 2.
70 60 50
z, u
40 z
30 20 10
u(t)
0 –10 –20
0
2
4
6
8
10
12
14
16
18
20
t
Fig. 12. The states z of the controlled system (3.7) and the control u(t) respond with time before and after control activation. The control is activated at t = 10, k = 2.
4.1. Active control In this section, we will give some particular active control which ensures synchronization of drive system and response system of perturbed Chen chaotic dynamical system. System (1.1) has chaotic behavior at the parameters values
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a = 35, b = 3, c = 28 and d = 2. Our aim is to make synchronization of system (1.1) by using active control. The drive system is defined as follows, x_ 1 ¼ aðy 1 x1 Þ; y_ 1 ¼ ðc aÞx1 x1 z1 þ cy 1 ; z_ 1 ¼ x1 y 1 bz1 þ
ð4:3Þ
dx21
and the response system is given by x_ 2 ¼ aðy 2 x2 Þ þ l1 ðtÞ; y_ 2 ¼ ðc aÞx2 x2 z2 þ cy 2 þ l2 ðtÞ; z_ 2 ¼ x2 y 2 bz2 þ
dx22
ð4:4Þ
þ l3 ðtÞ:
We have introduced three control functions l1(t),l2(t) and l3(t) in (4.4). These functions are to be determined. Let the error states be x3 ¼ x2 x1 ; y3 ¼ y2 y1; z3 ¼ z2 z1 : Using this notation, we obtain the error system. x_ 3 ¼ aðy 3 x3 Þ þ l1 ðtÞ; y_ 3 ¼ ðc aÞx3 þ cy 3 x2 z2 þ x1 z1 þ l2 ðtÞ; z_ 3 ¼ bz3 x1 y 1 þ x2 y 2 þ
dx22
dx21
ð4:5Þ
þ l3 ðtÞ:
We define the active control functions l1(t), l2(t) and l3(t) as l1 ðtÞ ¼ V 1 ðtÞ; l2 ðtÞ ¼ x2 z2 x1 z1 þ V 2 ðtÞ; l3 ðtÞ ¼ x1 y 1 x2 y 2
dx22
þ
ð4:6Þ dx21
þ V 3 ðtÞ:
Hence, x_ 3 ¼ aðy 3 x3 Þ þ V 1 ðtÞ; y_ 3 ¼ ðc aÞx3 þ cy 3 þ V 2 ðtÞ; z_ 3 ¼ bz3 þ V 3 ðtÞ:
ð4:7Þ
The control inputs V1(t), V2(t) and V3(t) are functions of x3, y3 and z3 and are chosen as 2 3 2 3 V 1 ðtÞ x3 6 7 6 7 ð4:8Þ 4 V 2 ðtÞ 5 ¼ A4 y 3 5; V 3 ðtÞ z3
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where the matrix A is given by 2 3 a1 a 0 6 7 A ¼ 4 a c ð1 þ cÞ 0 5: 0 0 b1 With this particular choice of A, (4.7) has eigenvalues which are found to be 1, 1 and 1. The choice will lead to the error states x3, y3 and z3 converge to zero as time t tends to infinity and this implies that the synchronization of perturbed Chen system is achieved. 4.1.1. Numerical simulation Fourth-order Runge–Kutta method of differential equations (4.3) and (4.4) with time step size 0.001 are used in all numerical simulations. The parameters are selected in (4.3) as follow: a = 35, b = 3, c = 28 and d = 2 to ensure the chaotic behavior of perturbed Chen system. The initial value of the drive system are x1(0) = 0.5, y1(0) = 1 and z1(0) = 1 and the initial value of the response system are x2(0) = 10.5, y2(0) = 1 and z2(0) = 38. Then the initial value of the error system are x3(0) = 10, y3(0) = 0 and z3(0) = 37. Figs. 13–15 show the synchronization is occurred after applying active control at t = 5. 4.2. Adaptive control This section considers adaptive synchronization of perturbed Chen system. This approach can synchronize the chaotic systems with fully unmatched 20 15 10
x1, x 2
5 0 –5 –10 –15 –20
0
2
4
6
8
10
12
14
16
18
20
t
Fig. 13. The states x1, x2 of the coupled perturbed Chen system of equations with the active control activated.
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25 20 15
y1, y2
10 5 0 –5 –10 –15 –20 –25
0
2
4
6
8
10
12
14
16
18
20
t Fig. 14. The states y1, y2 of the coupled perturbed Chen system of equations with the active control activated.
80 70 60
z1, z2
50 40 30 20 10 0
0
2
4
6
8
10
12
14
16
18
20
t
Fig. 15. The states z1, z2 of the coupled perturbed Chen system of equations with the active control activated.
parameters. The synchronization problem of perturbed Chen systems with fully unknown parameters will be studied in which the adaptive controller will be introduced. Let system (1.1) be the drive system. Suppose that the parameters of the system (1.1) are unknown or uncertain, then the response system is given by
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~x_ ¼ ^ að~y ~xÞ u1 ; ~y_ ¼ ð^c ^ aÞ~x ~x~z þ ^c~y u2 ;
ð4:9Þ
^ x u3 ; ~z_ ¼ ~x~y ^ b~z þ d~ 2
where ^ a; ^ b; ^c and d^ are parameters of the response system which need to be estimated. Suppose that u1 ¼ k 1 e x ; u2 ¼ k 2 e y ;
ð4:10Þ
u3 ¼ k 3 ez þ d~xex ; where ex ¼ ~x x, ey ¼ ~y y and ez ¼ ~z z and ^_ ¼ fa ¼ cð~y ~xÞqex þ c~xey ; a _ ^ b ¼ fb ¼ h~zez ; ^c_ ¼ fc ¼ bð~x þ ~y Þey ;
ð4:11Þ
_ d^ ¼ fd ¼ d~x2 ez ; where k1, k2, k3 P 0 and q, c, h, b, d > 0 are constants. Theorem 4.2.1. Suppose that M Cx > jxj; M Cy > jyj; M Cz > jzj, q, c, h, b, d are positive constants. When k1, k2 and k3P0 are properly chosen such that the following matrix inequality holds, 2 3 12 ðqa a þ c þ M Cz Þ 12 ðM Cy þ dM Cx Þ qðk 1 þ aÞ 6 7 k2 c 0 P ¼ 4 12 ðqa a þ c þ M Cz Þ 5>0 12 ðM Cy þ dM Cx Þ
0
k3 þ b ð4:12Þ
or equivalently if k1, k2 and k3 are chosen so that the following inequalities hold: 1 ðiÞ A ¼ qðk 1 þ aÞðk 2 cÞ ðqa a þ c þ M Cz Þ2 > 0; 4 1 ðiiÞ B ¼ Aðk 3 þ bÞ ðM Cy þ dM Cx Þ2 ðk 2 cÞ > 0 4
ð4:13Þ
then the two perturbed Chen systems (1.1) and (4.9) can be synchronized under the adaptive control of (4.10) and (4.11). Proof. It is easy to see from (1.1) and (4.9) that the error dynamics can be obtained as follow
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^ð~y ~xÞ aðy xÞ u1 ; e_ x ¼ a e_y ¼ ^ a~x þ ax þ ^c~x cx þ ^c~y cy ~x~z þ xz u2 ; ^ e_z ¼ b~z þ bz þ ~x~y xy u3 :
943
ð4:14Þ
a a, eb ¼ ^ b b, ec ¼ ^c c, ed ¼ d^ d. Choose the following LyapuLet ea ¼ ^ nov function: 1 1 1 1 1 qe2x þ e2y þ e2z þ e2a þ e2b þ e2c þ e2d V ðex ; ey ; ez Þ ¼ ð4:15Þ 2 c h b d in which the differentiation of V along trajectories of (4.14) gives 1 1 1 1 V_ ¼ qex e_ x þ ey e_ y þ ez e_ z þ ea e_ a þ ea e_ a þ eb e_ b þ ed e_ d c h b d ¼ qex ½^ að~y ~xÞ aðy xÞ u1 þ ey ½^ a~x þ ax þ ^c~x cx þ ^c~y cy ~x~z þ xz u2 1 1 1 1 b~z þ bz þ ~x~y xy u3 þ ea fa þ eb fb þ ec fc þ ed fd þ ez ½^ c h b d ¼ ½q^ að~y ~xÞ qað~y ~xÞ þ qað~y ~xÞ qaðy xÞex qu1 ex þ ½^ a~x þ a~x a~x þ axey þ ½~x~z þ ~xz ~xz þ xzey u2 ey þ ½^cð~x þ ~y Þ cð~x þ ~y Þ þ cð~x þ ~y Þ cðx þ yÞey þ ½^ b~z þ b~z b~z þ bzez þ ½~x~y ~xy þ ~xy xyez ^x2 d~x2 þ d~x2 dx2 ez u3 ez þ 1 ea fa þ 1 eb fb þ 1 ec fc þ 1 ed fd þ ½d~ c h b d ¼ qð~y ~xÞea ex þ qaðey ex Þex qu1 ex ~xea ey aex ey ~xey ez zex ey u2 ey þ ec ey ð~x þ ~y Þ þ cðex þ ey Þey ~zeb ez be2z þ ~xey ez þ yex ez 1 1 1 1 þ ~x2 ed ez þ dex ez ð~x þ xÞ u3 ez þ ea fa þ eb fb þ ec fc þ ed fd c h b d 2 ¼ qð~y ~xÞea ex þ qaðey ex Þex qk 1 ex ~xea ey aex ey ~xey ez zex ey k 2 e2y þ ec ey ð~x þ ~y Þ þ cðex þ ey Þey ~zeb ez be2z þ ~xey ez þ yex ez þ ~x2 ed ez 1 1 1 1 þ dex ez ð~x þ xÞ ðk 3 ez þ d~xex Þez þ ea fa þ eb fb þ ec fc þ ed fd c h b d 2 2 2 ¼ qðk 1 þ aÞex ðk 2 cÞey ðk 3 þ bÞez þ ðqa þ c a zÞex ey þ ðy þ xdÞex ez 1 1 þ ea fa þ ð~y ~xÞqex ~xey þ eb fb ~zez c h 1 1 2 þ ec fc þ ð~x þ ~y Þey ed fd þ ~x ez b d < qðk 1 þ aÞe2x ðk 2 cÞe2y ðk 3 þ bÞe2z þ ðqa þ c a M Cz Þjex ey j þ ðM Cy þ dM Cx Þjex ez j ¼ eT Pe;
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where e ¼ ½ jex j jey j jez j T , P is as in (4.12). Thus the differentiation of V(ex, ey, ez) is negative definite, which implies that the origin of error system (4.14) is asymptotically stable. Therefore, the response system (4.9) is synchronizing with the drive system (1.1). h
10 9 8 7
|ex|
6 5 4 3 2 1 0
0
2
4
6
8
10
12
14
16
18
20
18
20
t
Fig. 16. Synchronization errors: jexj.
18 16 14 12
|ey|
10 8 6 4 2 0
0
2
4
6
8
10
12
14
16
t
Fig. 17. Synchronization errors: jeyj.
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945
4.2.1. Numerical simulation The numerical simulations are carried out using the fourth-order Runge– Kutta method. The initial conditions of the drive and response systems are (0.5, 1, 5) and (10.5, 20, 38). The parameters of the drive system are a = 35, b = 3, c = 28 and d = 2. In order to choose the control parameters, M Cx > jxj, M Cy > jyj and M Cz > jzj must be estimated. Through simulations, we obtain M Cx 20, 35 30
|ez|
25 20 15 10 5 0 0
2
4
6
8
10
12
14
16
18
20
18
20
t
Fig. 18. Synchronization errors: jezj.
35 30 25
a
20 15 10 5 0 –5 0
2
4
6
8
10
12
14
t Fig. 19. Changing parameters: ^a.
16
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T. Plienpanich et al. / Appl. Math. Comput. 171 (2005) 927–947 10 9 8 7
b
6 5 4 3 2 1 0
0
2
4
6
8
10
12
14
16
18
20
16
18
20
t Fig. 20. Changing parameters: ^b.
30 25 20
c
15 10 5 0 –5 0
2
4
6
8
10
12
14
t Fig. 21. Changing parameters: ^c.
M Cy 25 and M Cz 70. Then we firstly choose q ¼ M 2Cy =ðabÞ. Then choose c = h = b = 1 and then choose k1 = 25, k2 = 88, k3 = 50 which satisfy (4.13) and the initial values of the parameters ^ a; ^ b; ^c and d^ are all chosen to be 0, the response system synchronizes with the drive system as shown in Figs. 16–18 and the changing parameters of ^ a; ^ b; ^c and d^ are shown in Figs. 19–22.
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15 10 5
d
0 –5 –10 –15 –20 –25 0
2
4
6
8
10
12
14
16
18
20
t ^ Fig. 22. Changing parameters: d.
References [1] H.N. Agiza, Controlling chaos for the dynamical system of coupled dynamos, Chaos, Solitons and Fractals 13 (2002) 341–352. [2] H.N. Agiza, M.T. Yassen, Synchronization of Rossler and Chen chaotic dynamical system, Physic Letter A 278 (2001) 191–197. [3] Y. Wang, Z.H. Guan, X. Wen, Adaptive synchronization for Chen chaotic system with fully unknown parameters, Chaos, Solitons and Fractals 19 (2004) 899–903. [4] M.T. Yassen, The optimal control of Chen chaotic dynamical system, Applied Mathematics and Computation 131 (2002) 171–180.
Controllability and stability of the perturbed Chen chaotic dynamical system Tidarut Plienpanich a, Piyapong Niamsup Yongwimon Lenbury b
a,*
,
a
b
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
Abstract In this paper, we study perturbed Chen chaotic dynamical system. Firstly, we study the sufficient conditions of parameters which guarantee that the equilibrium points of perturbed Chen chaotic dynamical system are asymptotically stable. Secondly, we study methods for controlling chaos such as feedback control and bounded feedback control that suppress the chaotic behavior to unstable equilibrium points. Finally, we present chaos synchronization of perturbed Chen chaotic dynamical system by using active control and adaptive control. 2005 Elsevier Inc. All rights reserved. Keywords: Perturbed Chen chaotic dynamical system; Controlling chaos; Synchronization
*
Corresponding author. E-mail addresses: [email protected] (T. Plienpanich), piyapongniamsup@hotmail. com, [email protected] (P. Niamsup), [email protected] (Y. Lenbury). 0096-3003/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.01.099
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1. Introduction In the recent years, controlling chaos and synchronization of the dynamical systems have attracted many researchers. Controlling chaos and chaos synchronization have focused on the nonlinear systems such as Chen chaotic dynamical system. Various control algorithms have been proposed to control chaotic systems. The existing control algorithms can be classified mainly into two categories: feedback and nonfeedback. In this paper, we only focus on feedback control. Linear feedback control and bounded feedback control are proposed to control chaos of the system to the equilibrium points. In [4], YassenÕs studied the optimal control of Chen chaotic dynamical system presented by x_ ¼ aðy xÞ; y_ ¼ ðc aÞx xz þ cy; z_ ¼ xy bz; where x, y, z are state variables and a, b, c are real positive constants. In [1], AgizaÕs studied the different methods to control chaotic behavior of the coupled dynamos system, where the mathematical model equations for this system are x_ ¼ lx þ yðz þ aÞ; y_ ¼ ly þ xðz aÞ; z_ ¼ 1 xy; where x, y, z are state variables and l, a are positive constants. In [2], Agiza and YassenÕs studied synchronization of Rossler and Chen chaotic dynamical systems using active control. In [3], Wang, Guan and WenÕs paper studied adaptive synchronization for Chen chaotic system with fully unknown parameters. The objectives of this paper are as follows. Firstly, to give sufficient conditions of parameters that make equilibrium points of perturbed Chen chaotic dynamical system to be asymptotically stable. Secondly, to apply linear feedback control and bounded feedback control for controlling chaos of the perturbed Chen chaotic dynamical system, described by x_ ¼ aðy xÞ; y_ ¼ ðc aÞx xz þ cy;
ð1:1Þ
2
z_ ¼ xy bz þ dx ; where x, y, z are the state variables and a, b, c, d are positive real constants.
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Thirdly, to study synchronization of perturbed Chen chaotic dynamical system using active control. Finally, to present adaptive synchronization for perturbed Chen chaotic dynamical system when the parameters of the drive system are fully unknown and different from those of the response system.
2. Stability of the perturbed Chen chaotic dynamical system We will study the perturbed Chen chaotic dynamical system that is described by system of ordinary differential equations (1.1). The equilibrium points of the system (1.1) are E1 ¼ ð0; 0; 0Þ; E2 ¼ ðb; b; cÞ; qffiffiffiffiffiffi bc and c = 2c a. where b ¼ 1þd
E3 ¼ ðb; b; cÞ;
Proposition 2.1. The equilibrium point E1 = (0, 0, 0) is (i) asymptotically stable if a > 2c and ac < b2 < 2ac. (ii) unstable if 2c > a.
Proof. The Jacobian matrix of the system (1.1) at the equilibrium point E1 = (0, 0, 0) is given by 2 3 a a 0 6 7 7 J1 ¼ 6 4 c a c 0 5: 0
0
b
The characteristic equation of the Jacobian J1 has the form k3 þ a1 k2 þ a2 k þ a3 ¼ 0; where a1 ¼ a þ b c; a2 ¼ bða cÞ þ aða 2cÞ; a3 ¼ abða 2cÞ; a1 a2 a3 ¼ ðab þ a2 Þða 2cÞ þ aðb2 acÞ þ cð2ac b2 Þ þ bc2 : We see that a1 and a1a2 a3 satisfy the Routh–Hurwitz criteria when a > 2c and ac < b2 < 2ac, thus the equilibrium point E1 = (0, 0, 0) is asymptotically stable. h
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Similarly, we have the following results: Proposition 2.2. The equilibrium point E2 = (b, b, c) is (i) asymptotically stable if 32 c < a < 2c, b > 6c and 13 < d < 1. (ii) unstable if b < c < a and a < 43 c. Proposition 2.3. The equilibrium point E3 = (b, b, c) is (i) asymptotically stable if 32 c < a < 2c, b > 6c and 13 < d < 1. (ii) unstable if b < c < a and a < 43 c.
3. Controlling chaos In this section, the chaos of system (1.1) is controlled to one of three equilibrium points of the system. Feedback and bounded feedback control are applied to achieve this goal. We shall study in the case when equilibrium points of (1.1) are unstable. For this purpose, we assume that b < c < a and a < 43 c. 3.1. Feedback control The goal of linear feedback control is to control the chaotic behavior of the system (1.1) to one of three unstable equilibrium points (E1, E2 or E3). We assume that the controlled system is given by x_ ¼ aðy xÞ þ u1 ; y_ ¼ ðc aÞx xz þ cy þ u2 ; z_ ¼ xy bz þ dx2 þ u3 ; where u1, u2 and u3 are controllers that satisfy the following control law x_ ¼ aðy xÞ k 11 ðx xÞ; y_ ¼ ðc aÞx xz þ cy k 22 ðy y Þ; z_ ¼ xy bz þ dx2 k 33 ðz zÞ;
ð3:1Þ
where E ¼ ðx; y ; zÞ is an equilibrium point of (1.1). 3.1.1. Stability of the equilibrium point E1 = (0, 0, 0) In this case E = E1 and the controlled system (3.1) is in the form of x_ ¼ aðy xÞ k 11 x; y_ ¼ ðc aÞx xz þ cy k 22 y; 2
z_ ¼ xy bz þ dx k 33 z:
ð3:2Þ
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Theorem 3.1.1. The equilibrium point E1 = (0, 0, 0) is asymptotically stable if k11 = 0, k33 > 0 and k22 > 3c. 3.1.2. Stability of the equilibrium point E2 = (b, b, c) In this case E = E2 and the controlled system (3.1) is in the form of x_ ¼ aðy xÞ k 11 ðx bÞ; y_ ¼ ðc aÞx xz þ cy k 22 ðy bÞ;
ð3:3Þ
2
z_ ¼ xy bz þ dx k 33 ðz cÞ: Theorem 3.1.2. The equilibrium point E2 = (b, b, c) is asymptotically stable if k11, k33 > 0 and k22 > 2c. 3.1.3. Stability of the equilibrium point E3 = (b, b, c) In this case E = E3 and the controlled system (3.1) is in the form of x_ ¼ aðy xÞ k 11 ðx þ bÞ; y_ ¼ ðc aÞx xz þ cy k 22 ðy þ bÞ;
ð3:4Þ
2
z_ ¼ xy bz þ dx k 33 ðz cÞ: Theorem 3.1.3. The equilibrium point E3 = (b, b, c) is asymptotically stable if k11, k33 > 0 and k22 > 2c. 3.1.4. Numerical simulation Numerical experiments are carried out to investigate controlled systems by using fourth-order Runge–Kutta method with time step 0.001. The parameters a, b, c and d are chosen as a = 35, b = 3, c = 28 and d = 2 to ensure the existence of chaos in the absence of control. The initial states are taken as x = 0.1, y = 0.2 and z = 0.3. The control is active at t = 10. The equilibrium point E1 = (0, 0, 0) of the system (1.1) is stabilized for k11 = 0, k22 = 85 and k33 = 5. Fig. 1 show the behavior of the states x, ypand z ffiffiffiffiffi of the controlled system (3.2) with time. The ffiffiffiffiffi p equilibrium point E2 ¼ ð 21; 21; 21Þ of the system (1.1) is stabilized for k11 = 1, k22 = 60 and k33 = 5. Fig. 2 show the behavior of the states x, y and z of pthe system (3.3) with time. The equilibrium point E3 ¼ ffiffiffiffiffi controlled pffiffiffiffiffi ð 21; 21; 21Þ of the system (1.1) is stabilized for k11 = 1, k22 = 60 and k33 = 5. Fig. 3 shows the behavior of the states x, y and z of the controlled system (3.4) with time. 3.2. Bounded feedback control In this case, we control chaos with bounded controller that vanishes after the stabilization is achieved.
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T. Plienpanich et al. / Appl. Math. Comput. 171 (2005) 927–947 80
60 z
x, y,z
40
20
0 x, y
–20
–40 0
2
4
6
8
10
12
14
16
18
20
t
Fig. 1. The time responses for the states x, y and z of the controlled system (3.2) before and after control activation with time. The control is activated at t = 10, k11 = 0, k22 = 85 and k33 = 5.
80
60
40
x, y,z
z 20
0 x, y –20
–40 0
2
4
6
8
10
12
14
16
18
20
t
Fig. 2. The time responses for the states x, y and z of the controlled system (3.3) before and after control activation with time. The control is activated at t = 10, k11 = 1, k22 = 60 and k33 = 5.
3.2.1. Stability of the equilibrium point E1 = (0, 0, 0) In order to stabilize this equilibrium point by bounded feedback control, the control is chosen for system (1.1) as follows:
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933
80
60
x, y,z
40
z
20
0 x, y –20
–40 0
2
4
6
8
10
12
14
16
18
20
t Fig. 3. The time responses for the states x, y and z of the controlled system (3.4) before and after control activation with time. The control is activated at t = 10, k11 = 1, k22 = 60 and k33 = 5.
x_ ¼ aðy xÞ; y_ ¼ ðc aÞx xz þ cy þ uðtÞ;
ð3:5Þ
2
z_ ¼ xy bz þ dx ; where u(t) = k(a(x + y)), k > 0. Theorem 3.2.1. The equilibrium point E1 = (0, 0, 0) is asymptotically stable if k > 2c a. 3.2.2. Stability of the equilibrium point E2 = (b, b, c) In order to stabilize this equilibrium point by bounded feedback control, the control is chosen for system (1.1) as follows: x_ ¼ aðy xÞ; y_ ¼ ðc aÞx xz þ cy þ uðtÞ;
ð3:6Þ
2
z_ ¼ xy bz þ dx ; where u(t) = k(a(y x)), k > 0. Theorem pffiffiffi 3.2.2. The equilibrium point E2 = (b, b, c) is asymptotically stable if k > 2.
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3.2.3. Stability of the equilibrium point E3 = (b, b, c) In order to stabilize this equilibrium point by bounded feedback control, the control is chosen for system (1.1) as follows: x_ ¼ aðy xÞ; y_ ¼ ðc aÞx xz þ cy þ uðtÞ;
ð3:7Þ
2
z_ ¼ xy bz þ dx ; where u(t) = k(a(y x)), k > 0. Theorem pffiffiffi 3.2.3. The equilibrium point E3 = (b, b, c) is asymptotically stable if k > 2. 3.2.4. Numerical simulation We will show a series of numerical experiments by using the fourth-order Runge–Kutta method with step size 0.001. The parameters a, b, c and d are chosen as a = 35, b = 3, c = 28 and d = 2 to ensure the existence of chaos in the absence of control. The control is active at t = 10 for all simulations. In the first numerical experiment, we intend to control the chaos to equilibrium point E1 = (0, 0, 0) of system (1.1). Figs. 4–6 show the time response of the states x, y and z of system (3.5) and the controller u(t) with time for k = 1.6. The initial condition are x = 0.1, y = 0.2 and z = 0.3. In the second numerical experiment, we intend to control the chaos to equilibrium point
20 x
15
x,u
10 5 0 –5 –10 –15
u(t) 0
2
4
6
8
10
12
14
16
18
20
t Fig. 4. The states x of the controlled system (3.5) and the control u(t) respond with time before and after control activation. The control is activated at t = 10, k = 1.6.
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935
25 20 y 15 10
y,u
5 0 –5 –10 u(t)
–15 –20 –25 0
2
4
6
8
10
12
14
16
18
20
t Fig. 5. The states y of the controlled system (3.5) and the control u(t) respond with time before and after control activation. The control is activated at t = 10, k = 1.6. 80 70 60 z
z,u
50 40 30 20 10
u(t)
0 –10
0
2
4
6
8
10
12
14
16
18
20
t
Fig. 6. The states z of the controlled system (3.5) and the control u(t) respond with time before and after control activation. The control is activated at t = 10, k = 1.6.
pffiffiffiffiffi pffiffiffiffiffi E2 ¼ ð 21; 21; 21Þ of system (1.1). Figs. 7–9 show the time response of the states x, y and z of system (3.6) and the controller u(t) with time for k = 2. The initial condition are x = 2.5, y = 2.5 and z = 3. In the third numerical experiment, pffiffiffiffiffi we pffiffiffiffiffiintend to control the chaos to equilibrium point E3 ¼ ð 21; 21; 21Þ of system (1.1). Fig. 10–12 show the time response of the states x, y and z of system (3.7) and the controller u(t) with time for k = 2. The initial condition are x = 2.5, y = 2.5 and z = 3.
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T. Plienpanich et al. / Appl. Math. Comput. 171 (2005) 927–947 20 15 10
x
x,u
5 0 u(t)
–5 –10 –15 –20
0
2
4
6
8
10
12
14
16
18
20
t
Fig. 7. The states x of the controlled system (3.6) and the control u(t) respond with time before and after control activation. The control is activated at t = 10, k = 2.
20 15 10
y
y, u
5 0 u(t)
–5 –10 –15 –20 –25
0
2
4
6
8
10
12
14
16
18
20
t Fig. 8. The states y of the controlled system (3.6) and the control u(t) respond with time before and after control activation. The control is activated at t = 10, k = 2.
4. Synchronization To begin with, the definition of chaos synchronization is given as follows. For two nonlinear chaotic system: x_ ¼ f ðt; xÞ;
ð4:1Þ
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937
70 60 50
z,u
40 z
30 20 10 0
u(t) –10 –20 0
2
4
6
8
10
12
14
16
18
20
t
Fig. 9. The states z of the controlled system (3.6) and the control u(t) respond with time before and after control activation. The control is activated at t = 10, k = 2. 20 15 10 u(t)
x, u
5 0 –5
x –10 –15 –20
0
2
4
6
8
10
12
14
16
18
20
t Fig. 10. The states x of the controlled system (3.7) and the control u(t) respond with time before and after control activation. The control is activated at t = 10, k = 2.
y_ ¼ gðt; yÞ þ uðt; x; yÞ;
ð4:2Þ
where x, y 2 R , f, g 2 C ½R R ; R , u 2 C ½R R R ; R , r P 1, Rþ is the set of non-negative real numbers. Assume that (4.1) is the drive system, and (4.2) is the response system, u(t, x, y) is the control vector. Response system and drive system are said to be synchronic if for "x(t0), yðt0 Þ 2 Rn , n
r
lim kxðtÞ yðtÞk ¼ 0:
t!1
þ
n
n
r
þ
n
n
n
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25 20 15
y, u
10 u(t)
5 0 –5
y –10 –15 –20
0
2
4
6
8
10
12
14
16
18
20
t Fig. 11. The states y of the controlled system (3.7) and the control u(t) respond with time before and after control activation. The control is activated at t = 10, k = 2.
70 60 50
z, u
40 z
30 20 10
u(t)
0 –10 –20
0
2
4
6
8
10
12
14
16
18
20
t
Fig. 12. The states z of the controlled system (3.7) and the control u(t) respond with time before and after control activation. The control is activated at t = 10, k = 2.
4.1. Active control In this section, we will give some particular active control which ensures synchronization of drive system and response system of perturbed Chen chaotic dynamical system. System (1.1) has chaotic behavior at the parameters values
T. Plienpanich et al. / Appl. Math. Comput. 171 (2005) 927–947
939
a = 35, b = 3, c = 28 and d = 2. Our aim is to make synchronization of system (1.1) by using active control. The drive system is defined as follows, x_ 1 ¼ aðy 1 x1 Þ; y_ 1 ¼ ðc aÞx1 x1 z1 þ cy 1 ; z_ 1 ¼ x1 y 1 bz1 þ
ð4:3Þ
dx21
and the response system is given by x_ 2 ¼ aðy 2 x2 Þ þ l1 ðtÞ; y_ 2 ¼ ðc aÞx2 x2 z2 þ cy 2 þ l2 ðtÞ; z_ 2 ¼ x2 y 2 bz2 þ
dx22
ð4:4Þ
þ l3 ðtÞ:
We have introduced three control functions l1(t),l2(t) and l3(t) in (4.4). These functions are to be determined. Let the error states be x3 ¼ x2 x1 ; y3 ¼ y2 y1; z3 ¼ z2 z1 : Using this notation, we obtain the error system. x_ 3 ¼ aðy 3 x3 Þ þ l1 ðtÞ; y_ 3 ¼ ðc aÞx3 þ cy 3 x2 z2 þ x1 z1 þ l2 ðtÞ; z_ 3 ¼ bz3 x1 y 1 þ x2 y 2 þ
dx22
dx21
ð4:5Þ
þ l3 ðtÞ:
We define the active control functions l1(t), l2(t) and l3(t) as l1 ðtÞ ¼ V 1 ðtÞ; l2 ðtÞ ¼ x2 z2 x1 z1 þ V 2 ðtÞ; l3 ðtÞ ¼ x1 y 1 x2 y 2
dx22
þ
ð4:6Þ dx21
þ V 3 ðtÞ:
Hence, x_ 3 ¼ aðy 3 x3 Þ þ V 1 ðtÞ; y_ 3 ¼ ðc aÞx3 þ cy 3 þ V 2 ðtÞ; z_ 3 ¼ bz3 þ V 3 ðtÞ:
ð4:7Þ
The control inputs V1(t), V2(t) and V3(t) are functions of x3, y3 and z3 and are chosen as 2 3 2 3 V 1 ðtÞ x3 6 7 6 7 ð4:8Þ 4 V 2 ðtÞ 5 ¼ A4 y 3 5; V 3 ðtÞ z3
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where the matrix A is given by 2 3 a1 a 0 6 7 A ¼ 4 a c ð1 þ cÞ 0 5: 0 0 b1 With this particular choice of A, (4.7) has eigenvalues which are found to be 1, 1 and 1. The choice will lead to the error states x3, y3 and z3 converge to zero as time t tends to infinity and this implies that the synchronization of perturbed Chen system is achieved. 4.1.1. Numerical simulation Fourth-order Runge–Kutta method of differential equations (4.3) and (4.4) with time step size 0.001 are used in all numerical simulations. The parameters are selected in (4.3) as follow: a = 35, b = 3, c = 28 and d = 2 to ensure the chaotic behavior of perturbed Chen system. The initial value of the drive system are x1(0) = 0.5, y1(0) = 1 and z1(0) = 1 and the initial value of the response system are x2(0) = 10.5, y2(0) = 1 and z2(0) = 38. Then the initial value of the error system are x3(0) = 10, y3(0) = 0 and z3(0) = 37. Figs. 13–15 show the synchronization is occurred after applying active control at t = 5. 4.2. Adaptive control This section considers adaptive synchronization of perturbed Chen system. This approach can synchronize the chaotic systems with fully unmatched 20 15 10
x1, x 2
5 0 –5 –10 –15 –20
0
2
4
6
8
10
12
14
16
18
20
t
Fig. 13. The states x1, x2 of the coupled perturbed Chen system of equations with the active control activated.
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25 20 15
y1, y2
10 5 0 –5 –10 –15 –20 –25
0
2
4
6
8
10
12
14
16
18
20
t Fig. 14. The states y1, y2 of the coupled perturbed Chen system of equations with the active control activated.
80 70 60
z1, z2
50 40 30 20 10 0
0
2
4
6
8
10
12
14
16
18
20
t
Fig. 15. The states z1, z2 of the coupled perturbed Chen system of equations with the active control activated.
parameters. The synchronization problem of perturbed Chen systems with fully unknown parameters will be studied in which the adaptive controller will be introduced. Let system (1.1) be the drive system. Suppose that the parameters of the system (1.1) are unknown or uncertain, then the response system is given by
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~x_ ¼ ^ að~y ~xÞ u1 ; ~y_ ¼ ð^c ^ aÞ~x ~x~z þ ^c~y u2 ;
ð4:9Þ
^ x u3 ; ~z_ ¼ ~x~y ^ b~z þ d~ 2
where ^ a; ^ b; ^c and d^ are parameters of the response system which need to be estimated. Suppose that u1 ¼ k 1 e x ; u2 ¼ k 2 e y ;
ð4:10Þ
u3 ¼ k 3 ez þ d~xex ; where ex ¼ ~x x, ey ¼ ~y y and ez ¼ ~z z and ^_ ¼ fa ¼ cð~y ~xÞqex þ c~xey ; a _ ^ b ¼ fb ¼ h~zez ; ^c_ ¼ fc ¼ bð~x þ ~y Þey ;
ð4:11Þ
_ d^ ¼ fd ¼ d~x2 ez ; where k1, k2, k3 P 0 and q, c, h, b, d > 0 are constants. Theorem 4.2.1. Suppose that M Cx > jxj; M Cy > jyj; M Cz > jzj, q, c, h, b, d are positive constants. When k1, k2 and k3P0 are properly chosen such that the following matrix inequality holds, 2 3 12 ðqa a þ c þ M Cz Þ 12 ðM Cy þ dM Cx Þ qðk 1 þ aÞ 6 7 k2 c 0 P ¼ 4 12 ðqa a þ c þ M Cz Þ 5>0 12 ðM Cy þ dM Cx Þ
0
k3 þ b ð4:12Þ
or equivalently if k1, k2 and k3 are chosen so that the following inequalities hold: 1 ðiÞ A ¼ qðk 1 þ aÞðk 2 cÞ ðqa a þ c þ M Cz Þ2 > 0; 4 1 ðiiÞ B ¼ Aðk 3 þ bÞ ðM Cy þ dM Cx Þ2 ðk 2 cÞ > 0 4
ð4:13Þ
then the two perturbed Chen systems (1.1) and (4.9) can be synchronized under the adaptive control of (4.10) and (4.11). Proof. It is easy to see from (1.1) and (4.9) that the error dynamics can be obtained as follow
T. Plienpanich et al. / Appl. Math. Comput. 171 (2005) 927–947
^ð~y ~xÞ aðy xÞ u1 ; e_ x ¼ a e_y ¼ ^ a~x þ ax þ ^c~x cx þ ^c~y cy ~x~z þ xz u2 ; ^ e_z ¼ b~z þ bz þ ~x~y xy u3 :
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ð4:14Þ
a a, eb ¼ ^ b b, ec ¼ ^c c, ed ¼ d^ d. Choose the following LyapuLet ea ¼ ^ nov function: 1 1 1 1 1 qe2x þ e2y þ e2z þ e2a þ e2b þ e2c þ e2d V ðex ; ey ; ez Þ ¼ ð4:15Þ 2 c h b d in which the differentiation of V along trajectories of (4.14) gives 1 1 1 1 V_ ¼ qex e_ x þ ey e_ y þ ez e_ z þ ea e_ a þ ea e_ a þ eb e_ b þ ed e_ d c h b d ¼ qex ½^ að~y ~xÞ aðy xÞ u1 þ ey ½^ a~x þ ax þ ^c~x cx þ ^c~y cy ~x~z þ xz u2 1 1 1 1 b~z þ bz þ ~x~y xy u3 þ ea fa þ eb fb þ ec fc þ ed fd þ ez ½^ c h b d ¼ ½q^ að~y ~xÞ qað~y ~xÞ þ qað~y ~xÞ qaðy xÞex qu1 ex þ ½^ a~x þ a~x a~x þ axey þ ½~x~z þ ~xz ~xz þ xzey u2 ey þ ½^cð~x þ ~y Þ cð~x þ ~y Þ þ cð~x þ ~y Þ cðx þ yÞey þ ½^ b~z þ b~z b~z þ bzez þ ½~x~y ~xy þ ~xy xyez ^x2 d~x2 þ d~x2 dx2 ez u3 ez þ 1 ea fa þ 1 eb fb þ 1 ec fc þ 1 ed fd þ ½d~ c h b d ¼ qð~y ~xÞea ex þ qaðey ex Þex qu1 ex ~xea ey aex ey ~xey ez zex ey u2 ey þ ec ey ð~x þ ~y Þ þ cðex þ ey Þey ~zeb ez be2z þ ~xey ez þ yex ez 1 1 1 1 þ ~x2 ed ez þ dex ez ð~x þ xÞ u3 ez þ ea fa þ eb fb þ ec fc þ ed fd c h b d 2 ¼ qð~y ~xÞea ex þ qaðey ex Þex qk 1 ex ~xea ey aex ey ~xey ez zex ey k 2 e2y þ ec ey ð~x þ ~y Þ þ cðex þ ey Þey ~zeb ez be2z þ ~xey ez þ yex ez þ ~x2 ed ez 1 1 1 1 þ dex ez ð~x þ xÞ ðk 3 ez þ d~xex Þez þ ea fa þ eb fb þ ec fc þ ed fd c h b d 2 2 2 ¼ qðk 1 þ aÞex ðk 2 cÞey ðk 3 þ bÞez þ ðqa þ c a zÞex ey þ ðy þ xdÞex ez 1 1 þ ea fa þ ð~y ~xÞqex ~xey þ eb fb ~zez c h 1 1 2 þ ec fc þ ð~x þ ~y Þey ed fd þ ~x ez b d < qðk 1 þ aÞe2x ðk 2 cÞe2y ðk 3 þ bÞe2z þ ðqa þ c a M Cz Þjex ey j þ ðM Cy þ dM Cx Þjex ez j ¼ eT Pe;
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where e ¼ ½ jex j jey j jez j T , P is as in (4.12). Thus the differentiation of V(ex, ey, ez) is negative definite, which implies that the origin of error system (4.14) is asymptotically stable. Therefore, the response system (4.9) is synchronizing with the drive system (1.1). h
10 9 8 7
|ex|
6 5 4 3 2 1 0
0
2
4
6
8
10
12
14
16
18
20
18
20
t
Fig. 16. Synchronization errors: jexj.
18 16 14 12
|ey|
10 8 6 4 2 0
0
2
4
6
8
10
12
14
16
t
Fig. 17. Synchronization errors: jeyj.
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4.2.1. Numerical simulation The numerical simulations are carried out using the fourth-order Runge– Kutta method. The initial conditions of the drive and response systems are (0.5, 1, 5) and (10.5, 20, 38). The parameters of the drive system are a = 35, b = 3, c = 28 and d = 2. In order to choose the control parameters, M Cx > jxj, M Cy > jyj and M Cz > jzj must be estimated. Through simulations, we obtain M Cx 20, 35 30
|ez|
25 20 15 10 5 0 0
2
4
6
8
10
12
14
16
18
20
18
20
t
Fig. 18. Synchronization errors: jezj.
35 30 25
a
20 15 10 5 0 –5 0
2
4
6
8
10
12
14
t Fig. 19. Changing parameters: ^a.
16
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T. Plienpanich et al. / Appl. Math. Comput. 171 (2005) 927–947 10 9 8 7
b
6 5 4 3 2 1 0
0
2
4
6
8
10
12
14
16
18
20
16
18
20
t Fig. 20. Changing parameters: ^b.
30 25 20
c
15 10 5 0 –5 0
2
4
6
8
10
12
14
t Fig. 21. Changing parameters: ^c.
M Cy 25 and M Cz 70. Then we firstly choose q ¼ M 2Cy =ðabÞ. Then choose c = h = b = 1 and then choose k1 = 25, k2 = 88, k3 = 50 which satisfy (4.13) and the initial values of the parameters ^ a; ^ b; ^c and d^ are all chosen to be 0, the response system synchronizes with the drive system as shown in Figs. 16–18 and the changing parameters of ^ a; ^ b; ^c and d^ are shown in Figs. 19–22.
T. Plienpanich et al. / Appl. Math. Comput. 171 (2005) 927–947
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15 10 5
d
0 –5 –10 –15 –20 –25 0
2
4
6
8
10
12
14
16
18
20
t ^ Fig. 22. Changing parameters: d.
References [1] H.N. Agiza, Controlling chaos for the dynamical system of coupled dynamos, Chaos, Solitons and Fractals 13 (2002) 341–352. [2] H.N. Agiza, M.T. Yassen, Synchronization of Rossler and Chen chaotic dynamical system, Physic Letter A 278 (2001) 191–197. [3] Y. Wang, Z.H. Guan, X. Wen, Adaptive synchronization for Chen chaotic system with fully unknown parameters, Chaos, Solitons and Fractals 19 (2004) 899–903. [4] M.T. Yassen, The optimal control of Chen chaotic dynamical system, Applied Mathematics and Computation 131 (2002) 171–180.