Adaptive Synchronization and Anti-Synchronization of

ISSN: 0256 - 307 X

中国物理快报

Chinese Physics Letters

Volume 30 Number 7 July 2013

A Series Journal of the Chinese Physical Society Distributed by IOP Publishing Online: http://iopscience.iop.org/0256-307X http://cpl.iphy.ac.cn

C HINESE P HYSICAL S OCIET Y Institute of Physics PUBLISHING

CHIN. PHYS. LETT. Vol. 30, No. 7 (2013) 070502

Adaptive Synchronization and Anti-Synchronization of a Chaotic Lorenz–Stenflo System with Fully Unknown Parameters M. Mossa Al-sawalha** Mathematics Department, Faculty of Science, University of Hail, Kingdom of Saudi Arabia

(Received 7 February 2013) An adaptive control scheme is developed to study the synchronization and the anti-synchronization behaviors between two identical Lorenz–Stenflo systems with unknown parameters. This adaptive controller is designed based on Lyapunov stability theory and an analytic expression of the controller with its adaptive laws of parameters is shown. Theoretical analysis and numerical simulations are shown to verify the results.

PACS: 05.45.−a, 47.52.+j, 89.75.−k

DOI: 10.1088/0256-307X/30/7/070502

Synchronization and anti-synchronization are important mechanisms for creating order in complex systems. Many nonlinear dynamical systems have been found to show a kind of behavior known as chaos, being characterized as chaotic systems by their extreme sensitivity to initial conditions and having noise-like behaviors. Because of these properties, it is difficult for chaotic systems to achieve synchronization and anti-synchronization. Since then, the synchronization and anti-synchronization of chaotic dynamical systems have attracted much attention due to the theoretical challenge and potential applications in secure communications, chemical reactions, biomedical science, social science, and many other fields.[1] A wide variety of approaches have been proposed for the synchronization or anti-synchronization of chaotic systems, which include generalized active control,[2] linear and nonlinear control,[3] adaptive control.[4] However, to our best knowledge, most of the methods synchronize or anti-synchronize only two systems with exactly knowing of their structure and parameters. In fact, in real physical systems or experimental situations some system parameters cannot be exactly known in advance. Thus, it is much more attractive and challenging to realize the synchronization or anti-synchronization of two different chaotic systems with unknown parameters. In this Letter, a novel adaptive synchronization and anti-synchronization scheme with fully unknown parameter is presented to achieve the synchronization and anti-synchronization behaviors between two identical chaotic Lorenz–Stenflo systems. Finally theoretical analysis and numerical simulations are shown to verify the results. The new set of four non-linear equations was derived by Stenflo[5,6] by studying the basic nonlinear equations of acoustic gravity waves, which have a structure similar to the famous Lorenz equation. The situation considered was the low-frequency, short wavelength acoustic gravity perturbation in the atmosphere. The model is two-dimensional in the sense that all the dynamical variables are considered to be functions of the variable 𝑧-coordinate and only one horizontal coordinate 𝑥. The original governing equations then reduce (under the assumption that 𝜕𝑦 = 0)

to 𝑑𝑡 ∇2 𝜓 = − 𝜕𝑥 𝜒 − Ω𝜙 + 𝜈∇4 𝜓, 𝑑𝑡 𝜙 = 4Ω𝜕𝑧2 𝜓 + 𝜈∇2 𝜙, 𝑑𝑡 𝜆 = 𝜔𝑔2 𝜕𝑥2 𝜓 + 𝑘∇2 𝜒,

(1)

with 𝑑𝑡 = 𝜕𝑡 + (𝜕𝑥 𝜓)𝜕𝑧 − (𝜕𝑧 𝜓)𝜕𝑥 and ∇2 = 𝜕𝑥2 + 𝜕𝑧2 . Here 𝜓, 𝜙 and 𝜒 represent velocity potential, derivative velocity in the 𝑦-direction and normalized density perturbation, respectively. Stenflo then made an expansion similar to Lorenz: 𝜓 = 𝜓𝐴 (𝑡) sin(𝑘𝑥 𝑥) sin(𝑘𝑧 𝑧), 𝜙 = 𝜙𝐴 (𝑡) sin(𝑘𝑥 𝑥) sin(𝑘𝑧 𝑧), 𝜒 = 𝜒𝐴 (𝑡) cos(𝑘𝑥 𝑥) sin(𝑘𝑧 𝑧) − 𝜒𝐵 (𝑡) sin(2𝑘𝑧 𝑧). (2) Substituting Eq. (2) into Eq. (1) leads to four coupled dynamical equations. The system under consideration deduced in the above manner can be written as 𝑥˙ = 𝑎(𝑦 − 𝑥) + 𝑐𝑤, 𝑦˙ = 𝑟𝑥 − 𝑥𝑧 − 𝑦, 𝑧˙ = 𝑥𝑦 − 𝑏𝑧, 𝑤˙ = −𝑥 − 𝑎𝑤.

(3)

Some properties of the set were analyzed by Refs. [7– 9]. Due to the presence of dissipative effects, it was observed that the system undergoes a period doubling route to chaos. The crucial values of the parameters are 𝑎 = 1, 𝑏 = 0.7, 𝑐 = 1.5, and 𝑟 = 26, for the existence of the chaotic attractor. It may be noted that similar non-linear equations can also arise out of plasma physics problems; for more detail see Refs. [10,11]. To formulate the adaptive synchronization and anti-synchronization with uncertain parameters problem, we consider nonlinear chaotic system as follows: 𝑥˙ = 𝑓 (𝑥) + 𝐹 (𝑥)𝛼,

(4)

where 𝑥 ∈ Ω1 ⊂ 𝑅𝑛 is the state vector, 𝛼 ∈ 𝑅𝑚 is the parameter vector of the system, 𝑓 (𝑥) is an 𝑛 × 1 matrix, 𝐹 (𝑥) is an 𝑛 × 𝑚 matrix, and the elements 𝐹𝑖𝑗 (𝑥) in matrix 𝐹 (𝑥) satisfy 𝐹𝑖𝑗 (𝑥) ∈ 𝐿∞ for 𝑥 ∈ Ω1 ⊂ 𝑅𝑛 . On the other hand, the response system is assumed by

** Corresponding author. Email: [email protected] © 2013 Chinese Physical Society and IOP Publishing Ltd

070502-1

𝑦˙ = 𝑔(𝑦) + 𝐺(𝑦)𝛽 + 𝑢,

(5)


where 𝑦 ∈ Ω2 ⊂ 𝑅𝑛 is the state vector, 𝛽 ∈ 𝑅𝑞 is the parameter vector of the system, 𝑔(𝑦) is an 𝑛 × 1 matrix, 𝐺(𝑦) is an 𝑛 × 𝑞 matrix, 𝑢 ∈ 𝑅𝑛 is a control input vector, and the elements 𝐺𝑖𝑗 (𝑦) in matrix 𝐺(𝑦) satisfy 𝐺𝑖𝑗 (𝑦) ∈ 𝐿∞ for 𝑦 ∈ Ω2 ⊂ 𝑅𝑛 . To observe the adaptive synchronization, we set 𝑒 = 𝑦 − 𝑥 to be the synchronization error vector. Our goal is to design a controller 𝑢 such that the trajectory of the response system (5) with initial condition 𝑦0 can asymptotically approach the drive system (4) with initial condition 𝑥0 and finally implement the synchronization such that lim ‖𝑒‖ = lim ‖𝑦(𝑡, 𝑦0 ) − 𝑥(𝑡, 𝑥0 )‖ = 0,

𝑡→∞

lim ‖𝑒‖ = lim ‖𝑦(𝑡, 𝑦0 ) + 𝑥(𝑡, 𝑥0 )‖ = 0,

𝑡→∞

where ‖ · ‖ is the Euclidean norm. Theorem 1. If the nonlinear control is selected as 𝑢 = 𝑓 (𝑥) + 𝐹 (𝑥)˜ 𝛼 − 𝑔(𝑦) − 𝐺(𝑦)𝛽˜ − 𝑘𝑒,

(7)

and adaptive laws of parameters are taken as ˙ 𝛼 ˜˙ = −[𝐹 (𝑥)]𝑇 𝑒, 𝛽˜ = [𝐺(𝑦)]𝑇 𝑒,

(8)

then the response system (5) can synchronize the drive system (4) globally and asymptotically. Here 𝑘 > 0 is a constant, 𝛼 ˜ and 𝛽˜ are estimations of the unknown parameters 𝛼 and 𝛽, respectively. Proof . From Eqs. (4) and (5), we obtain the error dynamical system as follows: 𝑒˙ = 𝐹 (𝑥)(˜ 𝛼 − 𝛼) + 𝐺(𝑦)(𝛽˜ − 𝛽) − 𝑘𝑒.

𝑢 = −𝑓 (𝑥) − 𝐹 (𝑥)˜ 𝛼 − 𝑔(𝑦) − 𝐺(𝑦)𝛽˜ − 𝑘𝑒, (13) and adaptive laws of parameters are taken as ˙ 𝛼 ˜˙ = [𝐹 (𝑥)]𝑇 𝑒, 𝛽˜ = [𝐺(𝑦)]𝑇 𝑒,

(10)

˙ ˆ = 𝑒˙ 𝑇 𝑒 + (˜ 𝑉˙ (𝑒, 𝛼 ˆ , 𝛽) 𝛼 − 𝛼)𝑇 𝛼 ˜˙ + (𝛽˜ − 𝛽)𝑇 𝛽˜ = [𝐹 (𝑥)(˜ 𝛼 − 𝛼) − 𝐺(𝑦)(𝛽˜ − 𝛽) − 𝑘𝑒]𝑇 𝑒 − (˜ 𝛼 − 𝛼)𝑇 [𝐹 (𝑥)]𝑇 𝑒 + (𝛽˜ − 𝛽)𝑇 [𝐺(𝑦)]𝑇 𝑒 (11)

Since 𝑉 is positive definite, and 𝑉˙ is negative semidefinite, it follows that 𝑒, 𝛼 − 𝛼 ˆ , 𝛽 − 𝛽ˆ ∈ 𝐿∞ . From ∫︀𝑡 2 the fact that ‖𝑒‖ 𝑑𝑡 = 21 [𝑉 (0) − 𝑉 (𝑡)] ≤ 𝑉 𝑘(0) , we 0

can easily know that 𝑒 ∈ 𝐿2 . From Eq. (15) we have 𝑒˙ ∈ 𝐿∞ . Thus, by Barbalat’s lemma, we have lim 𝑒 =

12 10 8 6 4 2 0 -2 -4 -6 40

x1 x2

(a)

30 20

y1 y2

(b)

10 0 -10 z1 z2

(c)

30 20 10 0 -5 0

5

10

15

20

25

t

-20 8 (d) 6 4 2 0 -2 -4 30 0 5

w1 w2

10

15

20

25

30

t

Fig. 1. State trajectories when the drive system in Eq. (18) and the response system in Eq. (19) are synchronized.

Proof . From Eqs. (4) and (5), we obtain the error dynamical system as follows: ˜ − 𝑘𝑒. 𝑒˙ = 𝐹 (𝑥)(𝛼 − 𝛼 ˜ ) + 𝐺(𝑦)(𝛽 − 𝛽) (15) ˜ if a Lyapunov function Let 𝛼 ˆ = 𝛼−𝛼 ˜ and 𝛽ˆ = 𝛽 − 𝛽, candidate is chosen as ˆ = 1 [𝑒𝑇 𝑒 + (𝛼 − 𝛼 𝑉 (𝑒, 𝛼 ˆ , 𝛽) ˜ )𝑇 (𝛼 − 𝛼 ˜) 2 ˜ 𝑇 (𝛽 − 𝛽)], ˜ + (𝛽 − 𝛽) (16) the time derivative of 𝑉 along the trajectory of the error dynamical system (15) is as follows: ˆ = 𝑒˙ 𝑇 𝑒 + (𝛼 − 𝛼 ˜ 𝑇 𝛽˜˙ 𝑉˙ (𝑒, 𝛼 ˆ , 𝛽) ˜ )𝑇 𝛼 ˜˙ + (𝛽 − 𝛽) ˜ − 𝑘𝑒]𝑇 𝑒 = [𝐹 (𝑥)(𝛼 − 𝛼 ˜ ) + 𝐺(𝑦)(𝛽 − 𝛽) ˜ 𝑇 [𝐺(𝑦)]𝑇 𝑒 − (𝛼 − 𝛼 ˜ )𝑇 [𝐹 (𝑥)]𝑇 𝑒 − (𝛽 − 𝛽)

𝑡→∞

0, i.e., lim ‖𝑒‖ = lim ‖𝑦(𝑡, 𝑦0 ) + 𝑥(𝑡, 𝑥0 )‖ = 0. Thus

= − 𝑘𝑒𝑇 𝑒 ≤ 0.

𝑡→∞

the response system (5) can synchronize the drive system (4) globally and asymptotically. This completes the proof. To observe the adaptive anti-synchronization, let 𝑒 = 𝑦 + 𝑥 be the anti-synchronization error vector.

(14)

then the response system (5) can anti-synchronize the drive system (4) globally and asymptotically, where 𝑘 > 0 is a constant, 𝛼 ˜ and 𝛽˜ are estimations of the unknown parameters 𝛼 and 𝛽, respectively.

the time derivative of 𝑉 along the trajectory of the error dynamical system (15) is as follows:

𝑡→∞

(12)

where ‖ · ‖ is the Euclidean norm. Theorem 2. If the nonlinear control is selected as

(9)

Let 𝛼 ˆ=𝛼 ˜ − 𝛼 and 𝛽ˆ = 𝛽˜ − 𝛽, if a Lyapunov function candidate is chosen as

= − 𝑘𝑒𝑇 𝑒 ≤ 0.

𝑡→∞

(6)

𝑡→∞

ˆ = 1 [𝑒𝑇 𝑒 + (˜ 𝛼 − 𝛼)𝑇 − (˜ 𝛼 − 𝛼) 𝑉 (𝑒, 𝛼 ˆ , 𝛽) 2 + (𝛽˜ − 𝛽)𝑇 (𝛽˜ − 𝛽)],

Our goal is to design a controller 𝑢 such that the trajectory of the response system (5) with initial condition 𝑦0 can asymptotically approach the drive system (4) with initial condition 𝑥0 and finally implement the anti-synchronization such that

(17)

Since 𝑉 is positive definite, and 𝑉˙ is negative semidefinite, it follows that 𝑒, 𝛼 ˜ − 𝛼, 𝛽˜ − 𝛽 ∈ 𝐿∞ . From ∫︀𝑡 2 the fact that ‖𝑒‖ 𝑑𝑡 = 21 [𝑉 (0) − 𝑉 (𝑡)] ≤ 𝑉 𝑘(0) , we

070502-2

0


can easily know that 𝑒 ∈ 𝐿2 . From Eq. (15) we have 𝑒˙ ∈ 𝐿∞ . Thus, by Barbalat’s lemma, we have lim 𝑒 =

and 𝑥˙2 = 𝑎(𝑦2 − 𝑥2 ) + 𝑐𝑤2 + 𝑢1 , 𝑦˙2 = 𝑟𝑥2 − 𝑥2 𝑧2 − 𝑦2 + 𝑢2 , 𝑧˙2 = 𝑥2 𝑦2 − 𝑏𝑧2 + 𝑢3 , 𝑤˙2 = −𝑥2 − 𝑎𝑤2 + 𝑢4 , (19)

𝑡→∞

0, namely, lim ‖𝑒‖ = lim ‖𝑦(𝑡, 𝑦0 ) + 𝑥(𝑡, 𝑥0 )‖ = 0. 𝑡→∞

𝑡→∞

Thus the response system (5) can anti-synchronize the drive system (4) globally and asymptotically. This completes the proof. 12

40

e1 e2 e3 e4

(a)

8 4

where 𝑢1 , 𝑢2 , 𝑢3 , and 𝑢4 are four control functions to be designed. In order to determine the control functions to realize the adaptive synchronization between systems in Eqs. (18) and (19), we subtract (19) from (18) and obtain

c r

a b

30 20

0

10

-4 -8

(b)

0

5

10

15

20

25

30

0 -5

0

5

10

t

15

20

25

t

Fig. 2. (a) Synchronization errors, 𝑒1 , 𝑒2 , 𝑒3 , and 𝑒4 , of the drive system in Eq. (18) and the response system in Eq. (19) with time 𝑡. (b) Changing parameters 𝑎, 𝑏, 𝑐, and 𝑟 of the drive system in Eq. (18) and the response system in Eq. (19) with time 𝑡. 10 (a) 5 0 -5 -10 -15 -20

0 x1 x2 y1

y2

0

5

10

15

20

25

-20 8 (d) 6 4 2 0 -2 -4 30 0 5

w1

10

t

15

w2

20

25

30

t

Fig. 3. State trajectories when the drive system in Eq. (18) and the response system in Eq. (19) are antisynchronized. 25 20 (a) 15 10 5 0 -5 -10 0 4

e1 e2 e3 e4

12 16 20 24 28

t

For the two systems (in Eqs. (18) and (19)) without controls (𝑢𝑖 = 0; 𝑖 = 1, 2, 3, and 4), the trajectories of the two systems will quickly separate from each other and become irrelevant if the initial condition (𝑥1 (0), 𝑦1 (0), 𝑧1 (0), 𝑤1 (0)) ̸= (𝑥2 (0), 𝑦2 (0), 𝑧2 (0), 𝑤2 (0)). However, when controls are applied, the two systems will approach synchronization for any initial conditions by appropriate control functions. For this end, we propose the following adaptive control law for the system in Eq. (20),

20 10

𝑢4 = 𝑒1 + 𝑎 ˜𝑒4 − 𝑒4 ,

a b

(b)

30

-10 0

(20)

where 𝑒1 = 𝑥2 − 𝑥1 , 𝑒2 = 𝑦2 − 𝑦1 , 𝑒3 = 𝑧2 − 𝑧1 , and 𝑒4 = 𝑤2 − 𝑤1 . Our goal is to find proper control functions 𝑢𝑖 (𝑖 = 1, 2, 3, and 4) and parameter update rule, such that the system in Eq. (19) globally synchronizes the system in Eq. (18) asymptotically, i.e., lim ‖𝑒‖ = 0, where 𝑒 = [𝑒1 , 𝑒2 , 𝑒3 ]𝑇 .

𝑢1 = − 𝑎 ˜𝑒2 + 𝑎 ˜𝑒1 − 𝑐˜𝑒4 − 𝑒1 , 𝑢2 = − 𝑟˜𝑒1 + 𝑒2 + 𝑥2 𝑧2 − 𝑥1 𝑧1 − 𝑒2 , 𝑢3 = − 𝑥2 𝑦2 + 𝑥1 𝑦1 + ˜𝑏𝑒3 − 𝑒3 ,

40

c r

0 8

= 𝑎𝑒2 − 𝑎𝑒1 + 𝑐𝑒4 + 𝑢1 , = 𝑟𝑒1 − 𝑥2 𝑧2 − 𝑒2 + 𝑥1 𝑧1 + 𝑢2 , = 𝑥2 𝑦2 − 𝑏𝑒3 − 𝑥1 𝑦1 + 𝑢3 , = − 𝑒1 − 𝑎𝑒4 + 𝑢4 ,

𝑡→∞

-10

-10 0

y2

10

10

-20

y1

20 (b)

20 (c)

𝑒˙1 𝑒˙2 𝑒˙3 𝑒˙4

30

(21)

and parameter update rule 4

8

12 16 20 24 28

˙ 𝑎 ˜˙ = (𝑒2 − 𝑒1 ) − 𝑒24 , ˜𝑏 = −𝑒23 , 𝑐˜˙ = 𝑒4 𝑒1 , 𝑟˜˙ = 𝑒1 𝑒2 ,

t

Fig. 4. (a) Anti-synchronization errors, 𝑒1 , 𝑒2 , 𝑒3 , and 𝑒4 , of the drive system in Eq. (18) and the response system in Eq. (19) with time 𝑡. (b) Changing parameters 𝑎, 𝑏, 𝑐, and 𝑟 of the drive system in Eq. (18) and the response system in Eq. (19) with time 𝑡.

In order to observe the synchronization behavior in two identical Lorenz–Stenflo systems via adaptive control, we assume that we have two Lorenz–Stenflo systems where the drive system is denoted by the subscript 1 and the response system having identical equations is denoted by the subscript 2. However, the initial condition on the drive system is different from that of the response system. The two Lorenz–Stenflo systems are described, respectively, by the following equations 𝑥˙1 = 𝑎(𝑦1 − 𝑥1 ) + 𝑐𝑤1 , 𝑦˙1 = 𝑟𝑥1 − 𝑥1 𝑧1 − 𝑦1 , 𝑧˙1 = 𝑥1 𝑦1 − 𝑏𝑧1 , 𝑤˙1 = −𝑥1 − 𝑎𝑤1 .

(22)

where 𝑎 ˜, ˜𝑏, 𝑐˜, and 𝑟˜ are the estimates of 𝑎, 𝑏, 𝑐, and 𝑟, respectively. Applying control law in Eq. (21) to Eq. (20) yields the resulting error dynamics as follows: 𝑒˙1 = 𝑎 ˆ𝑒2 − 𝑎 ˆ𝑒1 + 𝑐ˆ𝑒4 − 𝑒1 , 𝑒˙2 = 𝑟ˆ𝑒1 − 𝑒2 , ˆ 𝑒˙3 = −𝑏𝑒3 − 𝑒3 , 𝑒˙4 = −ˆ 𝑎𝑒4 − 𝑒4 , (23) where 𝑎 ˆ = 𝑎−𝑎 ˜, ˆ𝑏 = 𝑏 − ˜𝑏, 𝑐ˆ = 𝑐 − 𝑐˜, and 𝑟ˆ = 𝑟 − 𝑟˜. Considering the following Lyapunov function 𝑉 =

1 𝑇 (𝑒 𝑒 + 𝑎 ˆ2 + ˆ𝑏2 + 𝑐ˆ2 + 𝑟ˆ2 ), 2

(24)

the time derivative of 𝑉 along the solution of error dynamical system in Eq. (22) gives that (18) 070502-3

𝑉˙ = −𝑒𝑇 𝑒 < 0, as long as 𝑒 ̸= 0.

(25)


Therefore, the response system in Eq. (19) can globally synchronize the drive system in Eq. (18) asymptotically. In order to observe the anti-synchronization behavior in two Lorenz–Stenflo systems via adaptive control. We add Eq. (19) to Eq. (18) and obtain that 𝑒˙1 𝑒˙2 𝑒˙3 𝑒˙4

= 𝑎𝑒2 − 𝑎𝑒1 + 𝑐𝑒4 + 𝑢1 , = 𝑟𝑒1 − 𝑥2 𝑧2 − 𝑒2 − 𝑥1 𝑧1 + 𝑢2 , = 𝑥2 𝑦2 − 𝑏𝑒3 + 𝑥1 𝑦1 + 𝑢3 , = − 𝑒1 − 𝑎𝑒4 + 𝑢4 ,

(26)

where 𝑒1 = 𝑥2 + 𝑥1 , 𝑒2 = 𝑦2 + 𝑦1 , 𝑒3 = 𝑧2 + 𝑧1 , and 𝑒4 = 𝑤2 + 𝑤1 . Our goal is to find proper control functions 𝑢𝑖 (𝑖=1, 2, 3, and 4) and parameter update rule, such that the system in Eq. (19) globally anti-synchronizes the system in Eq. (18) asymptotically, i.e., lim ‖𝑒‖ = 0, where 𝑡→∞

𝑒 = [𝑒1 , 𝑒2 , 𝑒3 ]𝑇 . For the two systems (in Eqs. (18) and (19)) without controls (𝑢𝑖 = 0, 𝑖 = 1, 2, 3, and 4), the trajectories of the two systems will quickly separate from each other and become irrelevant if the initial condition (𝑥1 (0), 𝑦1 (0), 𝑧1 (0), 𝑤1 (0)) ̸= (𝑥2 (0), 𝑦2 (0), 𝑧2 (0), 𝑤2 (0)). However, when controls are applied, the two systems will approach antisynchronization for any initial conditions by appropriate control functions. To this end, we propose the following adaptive control law for the system in Eq. (26), 𝑢1 = − 𝑎 ˜𝑒2 + 𝑎 ˜𝑒1 − 𝑐˜𝑒4 − 𝑒1 , 𝑢2 = − 𝑟˜𝑒1 + 𝑒2 + 𝑥2 𝑧2 + 𝑥1 𝑧1 − 𝑒2 , 𝑢3 = − 𝑥2 𝑦2 − 𝑥1 𝑦1 + ˜𝑏𝑒3 − 𝑒3 , 𝑢4 = 𝑒1 + 𝑎 ˜𝑒4 − 𝑒4 ,

(27)

and parameter update rule ˙ 𝑎 ˜˙ =(𝑒2 − 𝑒1 ) − 𝑒24 , ˜𝑏 = −𝑒23 , 𝑐˜˙ = 𝑒4 𝑒1 , 𝑟˜˙ = 𝑒1 𝑒2 ,

(28)

where 𝑎 ˜, ˜𝑏, 𝑐˜, and 𝑟˜ are the estimates of 𝑎, 𝑏, 𝑐, and 𝑟, respectively. Applying control law in Eq. (27) to Eq. (26) yields the resulting error dynamics as follows: 𝑒˙1 = 𝑎 ˆ𝑒2 − 𝑎 ˆ𝑒1 + 𝑐ˆ𝑒4 − 𝑒1 , 𝑒˙2 = 𝑟ˆ𝑒1 − 𝑒2 , ˆ 𝑒˙3 = −𝑏𝑒3 − 𝑒3 , 𝑒˙4 = −ˆ 𝑎𝑒4 − 𝑒4 , (29) where 𝑎 ˆ = 𝑎−𝑎 ˜, ˆ𝑏 = 𝑏 − ˜𝑏, 𝑐ˆ = 𝑐 − 𝑐˜, and 𝑟ˆ = 𝑟 − 𝑟˜. Considering the following Lyapunov function, 𝑉 =

1 𝑇 (𝑒 𝑒 + 𝑎 ˆ2 + ˆ𝑏2 + 𝑐ˆ2 + 𝑟ˆ2 ), 2

(30)

the time derivative of 𝑉 along the solution of error dynamical system in Eq. (28) gives that 𝑉˙ = −𝑒𝑇 𝑒 < 0, as long as 𝑒 ̸= 0.

(31)

Therefore, the response system in Eq. (19) can globally anti-synchronize the drive system in Eq. (18) asymptotically. To verify and demonstrate the effectiveness of the proposed method, we discuss the simulation result for Lorenz–Stenflo systems. In the numerical simulations, the fourthorder Runge–Kutta method is used to solve the systems with time step size 0.001. For this numerical simulation, we employ the initial conditions, (𝑥1 (0), 𝑦1 (0), 𝑧1 (0), 𝑤1 (0)) = (2, −2, 5, 7) and (𝑥2 (0), 𝑦2 (0), 𝑧2 (0), 𝑤2 (0)) = (4, −3, −1, 6). Hence, the error system has the initial values 𝑒1 (0) = 2, 𝑒2 (0) = −1, 𝑒3 (0) = −6, and 𝑒4 (0) = −1 in the case of synchronization and 𝑒1 (0) = 6, 𝑒2 (0) = −5, 𝑒3 (0) = 4, and 𝑒4 (0) = 13 in the case of anti-synchronization. The four unknown parameters are chosen as 𝑎 = 1, 𝑏 = 0.7, 𝑐 = 1.5, and 𝑟 = 26 in the simulations such that the systems exhibits chaotic behavior. Synchronization of the systems in Eqs. (18) and (19) via adaptive control laws (Eqs. (21) and (22)) with the initial estimated parameters 𝑎 ˜(0) = 5, ˜𝑏(0) = −1, 𝑐˜(0) = 3, and 𝑟˜(0) = −3 are shown in Figs. 1 and 2. Figures 1(a)–1(c) display state trajectories of the drive system in Eq. (18) and the response system in Eq. (19). Figure 2(a) displays the synchronization errors between systems in Eqs. (18) and (19). Figure 2(b) shows that the estimates 𝑎 ˜(𝑡), ˜𝑏(𝑡), 𝑐˜(𝑡) and 𝑟˜(𝑡) of the unknown parameters converge to 𝑎 = 1, 𝑏 = 0.7, 𝑐 = 1.5, and 𝑟 = 26 as 𝑡 → ∞. Anti-synchronization of the systems in Eqs. (18) and (19) via adaptive control laws (Eqs. (27) and (28)) with the initial estimated parameters 𝑎 ˜(0) = 5, ˜𝑏(0) = −1, 𝑐˜(0) = 3, and 𝑟˜(0) = −3 are shown in Figs. 3 and 4. Figures 1(a)–1(c) display state trajectories of the drive system in Eq. (18) and the response system in Eq. (19). Figure 4(a) displays the anti-synchronization errors between the systems in Eqs. (18) and (19). Figure 4(b) shows that the estimates 𝑎 ˜(𝑡), ˜𝑏(𝑡), 𝑐˜(𝑡) and 𝑟˜(𝑡) of the unknown parameters converge to 𝑎 = 1, 𝑏 = 0.7, 𝑐 = 1.5 and 𝑟 = 26 as 𝑡 → ∞.

References [1] Pecora L M and Carroll T L 1990 Phys. Rev. Lett. 64 821 [2] Al-Sawalha M M and Noorani M S M 2008 Chin. Phys. Lett. 25 2743 [3] Al-Sawalha M M and Noorani M S M 2009 Chaos Solitons Fractals 42 170 [4] Al-Sawalha M M and Noorani M S M 2009 Chin. Phys. Lett. 26 092901 [5] Banerjee S, Saha P and Chowdhury A R 2004 Int. J. NonLinear Mech. 39 25 [6] Stenflo L 1996 Phys. Scr. 53 83 [7] Liu Z 2000 Phys. Scr. 61 526 [8] Zhou C T 1997 J. Math. Phys. 38 5225 [9] Banerjee S, Saha P and Chowdhury A R 2001 Phys. Scr. 63 177 [10] Liu Z 2000 Phys. Scr. 61 530 [11] Banerjee S, Saha P and Chowdhury A R 2004 Int. J. NonLinear Mech. 39 25

070502-4

Chinese Physics Letters Volume 30

Number 7

July 2013

GENERAL 070201 Dynamical Decomposition of Markov Processes without Detailed Balance AO Ping, CHEN Tian-Qi, SHI Jiang-Hong 070202 Multisymplectic Scheme for the Improved Boussinesq Equation CAI Jia-Xiang, QIN Zhi-Lin, BAI Chuan-Zhi 070301 Infrared Spectra of PH3 and NF3 : An Algebraic Approach Joydeep Choudhury, Nirmal Kumar Sarkar, Ramendu Bhattacharjee 070302 Deterministic Three-Copy Entanglement Concentration of Photons through Direct Sum Extension and Auxiliary Degrees of Freedom ZHAO Jie, LI Wen-Dong, GU Yong-Jian 070401 Canonical Ensemble Model for the Black Hole Quantum Tunneling Radiation ZHANG Jing-Yi 070501 Brownian Motion and the Temperament of Living Cells Roumen Tsekov, Marga C. Lensen 070502 Adaptive Synchronization and Anti-Synchronization of a Chaotic Lorenz–Stenflo System with Fully Unknown Parameters M. Mossa Al-sawalha 070503 Cusp Bursting and Slow-Fast Analysis with Two Slow Parameters in Photosensitive Belousov–Zhabotinsky Reaction LI Xiang-Hong, BI Qin-Sheng 070504 Stochastic Resonance for a Time-Delayed Metapopulation System Driven by Multiplicative and Additive Noises WANG Kang-Kang, LIU Xian-Bin 070505 Function Projective Synchronization for Two Gyroscopes under Specific Constraints MIN Fu-Hong 070506 Enhancement of the Neuronal Dynamic Range by Proper Intensities of Channel Noise WANG Lei, ZHANG Pu-Ming, LIANG Pei-Ji, QIU Yi-Hong

THE PHYSICS OF ELEMENTARY PARTICLES AND FIELDS 071201 Enhanced Electron-Positron Pair Production of a Vacuum in a Strong Laser Pulse Field by Frequency Variation LI Zi-Liang, SANG Hai-Bo, XIE Bai-Song

NUCLEAR PHYSICS 072801 Extraction of Mechanical-Reactivity Influences from Neutron Noise Spectra at the IBR-2 Reactor M. Dima, Yu. Pepelyshev

ATOMIC AND MOLECULAR PHYSICS 073101 Theoretical Calculation of Vector Correlations of the Reaction D′ (2 S)+DS(X1 Σ+ )→S(1 D)+D2 WEI Qiang 073201 The Dynamics of Rubidium Atoms in THz Laser Fields JIA Guang-Rui, ZHAO Yue-Jin, ZHANG Xian-Zhou, LIU Yu-Fang, YU Kun 073202 Effect of Electron Initial Longitudinal Velocity on Low-Energy Structure in Above-Threshold Ionization Spectra WU Ming-Yan, WANG Yan-Lan, LIU Xiao-Jun, LI Wei-Dong, HAO Xiao-Lei, CHEN Jing

073401 Electronic Excitation of H2 by Electron Impact Using Multichannel Static-Exchange-Optical Method WANG Yuan-Cheng, MA Jia, ZHOU Ya-Jun 073402 Observation of Photoassociation Spectra of Ultracold 174 Yb Atoms at 1S0 –3P1 Inter-Combination Line ¨ Bao-Long, HE Ling-Xiang LONG Yun, XIONG Zhuan-Xian, ZHANG Xi, ZHANG Meng-Jiao, LU 073701 Manipulation of Ions in Microscopic Surface-Electrode Ion Traps WAN Wei, CHEN Liang, WU Hao-Yu, XIE Yi, ZHOU Fei, FENG Mang

FUNDAMENTAL AREAS OF PHENOMENOLOGY(INCLUDING APPLICATIONS) 074201 Enhancement of Photodetector Responsivity and Response Speed Using Cascaded-Cavity Structure with Subwavelength Metallic Slit DU Ming-Di, SUN Jun-Qiang, QIN Yi, LIAO Jian-Fei 074202 High Power Quasi-Continuous-Wave Diode-End-Pumped Nd:YAG Slab Amplifier at 1319 nm ZHENG Jian-Kui, BO Yong, XIE Shi-Yong, ZUO Jun-Wei, WANG Peng-Yuan, GUO Ya-Ding, LIU Biao-Long, PENG Qin-Jun, CUI Da-Fu, LEI Wen-Qiang, XU Zu-Yan 074203 Ptychographical Imaging Algorithm with a Single Random Phase Encoding SHI Yi-Shi, WANG Ya-Li, LI Tuo, GAO Qian-Kun, WAN Hao, ZHANG San-Guo, WU Zhi-Bo 074204 Tuning Properties of External Cavity Violet Semiconductor Laser LV Xue-Qin, CHEN Shao-Wei, ZHANG Jiang-Yong, YING Lei-Ying, ZHANG Bao-Ping 074205 End-Output Coupling Efficiency Measurement of Silicon Wire Waveguides Based on Correlated Photon Pair Generation LV Ning, ZHANG Wei, GUO Yuan, ZHOU Qiang, HUANG Yi-Dong, PENG Jiang-De 074206 Gain Improvement of Fiber Parametric Amplifier via the Introduction of Standard Single-Mode Fiber for Phase Matching ZHU Hong-Na, LUO Bin, PAN Wei, YAN Lian-Shan, ZHAO Jian-Peng, WANG Ze-Yong, GAO Xiao-Rong 074207 End-Pumped Slab Yb:YAG Crystal Emitting 1030 nm Laser at Room Temperature XU Liu, ZHANG Heng-Li, MAO Ye-Fei, DENG Bo, HE Jing-Liang, XIN Jian-Guo 074208 Modeling 2D Gyromagnetic Photonic Crystals by Modified FDTD Method LI Qing-Bo, WU Rui-Xin, YANG Yan, SUN Hui-Ling 074209 Phonon Lifetime Measurement by Stimulated Brillouin Scattering Slow Light Technique in Optical Fiber CHEN Wei, MENG Zhou, ZHOU Hui-Juan 074301 Numerical Solution of Range-Dependent Acoustic Propagation QIN Ji-Xing, LUO Wen-Yu, ZHANG Ren-He, YANG Chun-Mei 074302 Modeling of Shock Wave Generated from a Strong Focused Ultrasound Transducer CHEN Tao, QIU Yuan-Yuan, FAN Ting-Bo, ZHANG Dong 074701 Surface Tension Gradients on Mixing Processes after Coalescence of Binary Equal-Sized Droplets LIU Dong, GUO Yin-Cheng, LIN Wen-Yi 074702 An Immersed Boundary-Lattice Boltzmann Simulation of Particle Hydrodynamic Focusing in a Straight Microchannel SUN Dong-Ke, JIANG Di, XIANG Nan, CHEN Ke, NI Zhong-Hua

PHYSICS OF GASES, PLASMAS, AND ELECTRIC DISCHARGES 075201 Observation of Electron Fishbone-Like Instabilities in EAST Heavy Impurity Ohmic Plasma XU Li-Qing, HU Li-Qun, EAST team 075202 A Polymer-Rich Re-deposition Technique for Non-volatile Etching By-products in Reactive Ion Etching Systems A. Limcharoen, C. Pakpum, P. Limsuwan

CONDENSED MATTER: STRUCTURE, MECHANICAL AND THERMAL PROPERTIES 076101 A Band-Gap Energy Model of the Quaternary Alloy Inx Gay Al1−x−y N using Modified Simplified Coherent Potential Approximation ZHAO Chuan-Zhen,ZHANG Rong, LIU Bin, LI Ming, XIU Xiang-Qian, XIE Zi-Li, ZHENG You-Dou 076102 Low-Dose 1 MeV Electron Irradiation-Induced Enhancement in the Photoluminescence Emission of Ga-Rich InGaN Multiple Quantum Wells ZHANG Xiao-Fu, LI Yu-Dong, GUO Qi, LU Wu 076201 The Anomalous Temperature Effect on the Ductility of Nanocrystalline Cu Films Adhered to Flexible Substrates HU Kun, CAO Zhen-Hua, WANG Lei, SHE Qian-Wei, MENG Xiang-Kang 076202 The Material Behavior and Fracture Mechanism of a Frangible Bullet Composite LI Jian, RONG Ji-Li, ZHANG Yu-Ning, XU Tian-Fu, LI Bin 076401 Harnessing Light and Single Masks to Create Multiple Patterns in a Ternary Blend with Photoinduced Reaction PAN Jun-Xing, ZHANG Jin-Jun, WANG Bao-Feng, WU Hai-Shun, SUN Min-Na 076402 Partial Order in Potts Models on the Generalized Decorated Square Lattice QIN Ming-Pu, CHEN Jing, CHEN Qiao-Ni, XIE Zhi-Yuan, KONG Xin, ZHAO Hui-Hai, Bruce Normand, XIANG Tao

CONDENSED MATTER: ELECTRONIC STRUCTURE, ELECTRICAL, MAGNETIC, AND OPTICAL PROPERTIES 077101 The Structural, Electronic and Elastic Properties, and the Raman Spectra of Orthorhombic CaSnO3 through First Principles Calculations A. Yangthaisong 077102 A Density Functional Study of the Gold Cages MAu16 (M = Si, Ge, and Sn) TANG Chun-Mei, ZHU Wei-Hua, ZHANG Ai-Mei, ZHANG Kai-Xiao, LIU Ming-Yi 077201 Enhanced Performance and Stability in Polymer Photovoltaic Cells Using Ultraviolet-Treated PEDOT:PSS XU Xue-Jian, YANG Li-Ying, TIAN Hui, QIN Wen-Jing, YIN Shou-Gen, ZHANG Fengling 077301 The Effect of Intraband Transitions on the Optical Spectra of Metallic Carbon Nanotubes T. Movlarooy 077302 The Nuclear Dark State under Dynamical Nuclear Polarization YU Hong-Yi, LUO Yu, YAO Wang 077303 Room-Temperature Multi-Peak NDR in nc-Si Quantum-Dot Stacking MOS Structures for Multiple Value Memory and Logic QIAN Xin-Ye, CHEN Kun-Ji, HUANG Jian, WANG Yue-Fei, FANG Zhong-Hui, XU Jun, HUANG Xin-Fan 077304 Quantum Size and Doping Concentration Effects on the Current-Voltage Characteristics in GaN Resonant Tunneling Diodes Hassen Dakhlaoui 077305 The Unconventional Transport Properties of Dirac Fermions in Graphyne LIN Xin, WANG Hai-Long, PAN Hui, XU Huai-Zhe 077306 The C–V and G/ω–V Electrical Characteristics of 60 Co γ-Ray Irradiated Al/Si3 N4 /p-Si (MIS) Structures S. Zeyrek, A. Turan, M. M. B¨ ulb¨ ul 077307 The Effect of Multiple Interface States and nc-Si Dots in a Nc-Si Floating Gate MOS Structure Measured by their G–V Characteristics SHI Yong, MA Zhong-Yuan, CHEN Kun-Ji, JIANG Xiao-Fan, LI Wei, HUANG Xin-Fan, XU Ling, XU Jun, FENG Duan 077308 Fano-Resonance of a Planar Metamaterial HUANG Wan-Xia

077402 An Insight into the Structural, Electronic and Transport Characteristics of XIn2 S4 (X = Zn, Hg) Thiospinels using a Highly Accurate All-Electron FP-LAPW+Lo Method Masood Yousaf, M. A. Saeed, Ahmad Radzi Mat Isa, H. A. Rahnamaye Aliabad, M. R. Sahar 077403 The Optical Study of Single Crystalline Cs0.8 (Fe1.05 Se)2 with High N´ eel Temperature YUAN Rui-Hua, DONG Tao, WANG Nan-Lin 077404 The Finite Temperature Effect on Josephson Junction between an s-Wave Superconductor and an s± -Wave Superconductor WANG Da, LU Hong-Yan, WANG Qiang-Hua 077501 Synthesis and Characterization of Alkaline-Earth Metal (Ca, Sr, and Ba) Doped Nanodimensional LaMnO3 Rare-Earth Manganites Asma Khalid, Saadat Anwar Siddiqi, Affia Aslam 077502 A First-Principles Investigation of the Carrier Doping Effect on the Magnetic Properties of Defective Graphene LEI Shu-Lai, LI Bin, HUANG Jing, LI Qun-Xiang, YANG Jin-Long 077503 Room-Temperature d 0 Ferromagnetism in Nitrogen-Doped In2 O3 Films SUN Shao-Hua, WU Ping, XING Peng-Fei 077701 Wafer-Scale Flexible Surface Acoustic Wave Devices Based on an AlN/Si Structure ZHANG Cang-Hai, YANG Yi, ZHOU Chang-Jian, SHU Yi, TIAN He, WANG Zhe, XUE Qing-Tang, REN Tian-Ling 077801 Magnetic-Field-Induced Stress-Birefringence in Laminate Composites of Terfenol-D and Polycarbonate LUO Xiao-Bin, WU Dong, ZHANG Ning 077802 The Fano-Like Resonance in Self-Assembled Trimer Clusters ZHANG Mei, LI Liang-Sheng, ZHENG Ning, SHI Qing-Fan

CROSS-DISCIPLINARY PHYSICS AND RELATED AREAS OF SCIENCE AND TECHNOLOGY 078101 Overcoming Decomposition with Order-Reversed Quenching Obtained by Flash Melting SI Ping-Zhan, XIAO Xiao-Fei, FENG He, YU Sen-Jiang, GE Hong-Liang 078501 A Low Specific on-Resistance SOI Trench MOSFET with a Non-Depleted Embedded p-Island FAN Jie, ZHANG Bo, LUO Xiao-Rong, LI Zhao-Ji 078502 High-Efficiency InGaN/GaN Nanorod Arrays by Temperature Dependent Photoluminescence WANG Wen-Jie, CHEN Peng, YU Zhi-Guo, LIU Bin, XIE Zi-Li, XIU Xiang-Qian, WU Zhen-Long, XU Feng, XU Zhou, HUA Xue-Mei, ZHAO Hong, HAN Ping, SHI Yi, ZHANG Rong, ZHENG You-Dou 078503 A Distributed Phase Shifter Using Bi1.5 Zn1.0 Nb1.5 O7 /Ba0.5 Sr0.5 TiO3 Thin Films LI Ru-Guan, JIANG Shu-Wen, GAO Li-Bin, LI Yan-Rong 078801 Optimization of Metal Coverage on the Emitter in n-Type Interdigitated Back Contact Solar Cells Using a PC2D Simulation ZHANG Wei, CHEN Chen, JIA Rui, Janssen G. J. M., ZHANG Dai-Sheng, XING Zhao, Bronsveld P. C. P., Weeber A. W., JIN Zhi, LIU Xin-Yu

COMMENTS AND REPLIES 079901 Comment on “Improvement of Controlled Bidirectional Quantum Direct Communication Using a GHZ State” [Chin. Phys. Lett. 30 (2013) 040305] LIU Zhi-Hao, CHEN Han-Wu 079902 Reply to the Comment on “Improvement of Controlled Bidirectional Quantum Direct Communication Using a GHZ State” [Chin. Phys. Lett. 30 (2013) 040305] YE Tian-Yu, JIANG Li-Zhen 079903 Comment on “Cryptanalysis and Improvement of a Quantum Network System of QSS-QDC Using χ-Type Entangled States” [Chin. Phys. Lett. 29 (2012) 110305] LIU Zhi-Hao, CHEN Han-Wu

079904 Reply to the Comment on “Cryptanalysis and Improvement of a Quantum Network System of QSS-QDC Using χ-Type Entangled States” [Chin. Phys. Lett. 29 (2012) 110305] GAO Gan, FANG Ming, CHENG Mu-Tian