Synchronization of chaotic system with uncertain variable parameters by linear coupling and pragmatical adaptive tracking

DOI 10.1007/s11071-012-0609-6 O R I G I N A L PA P E R

Synchronization of chaotic system with uncertain variable

parameters by linear coupling and pragmatical adaptive

tracking

Cheng-Hsiung Yang· Shih-Yu Li · Pu-Chien Tsen

Received: 5 March 2012 / Accepted: 10 September 2012 / Published online: 3 October 2012 © Springer Science+Business Media B.V. 2012

Abstract We study the synchronization of general chaotic systems which satisfy the Lipschitz condition only, with uncertain variable parameters by linear cou-pling and pragmatical adaptive tracking. The uncertain parameters of a system vary with time due to aging, environment, and disturbances. A sufficient condition is given for the asymptotical stability of common zero solution of error dynamics and parameter update dy-namics by the Ge–Yu–Chen pragmatical asymptotical stability theorem based on equal probability assump-tion. Numerical results are studied for a Lorenz sys-tem and a quantum cellular neural network oscillator to show the effectiveness of the proposed synchroniza-tion strategy.

C.-H. Yang (



Graduate Institute of Automation and Control, National Taiwan University of Science and Technology, 43 Section 4, Keelung Road, Taipei 106, Taiwan, Republic of China e-mail:[email protected]

S.-Y. Li

Institute of Electrical Control Engineering, National Chiao Tung University, Hsinchu, Taiwan, Republic of China S.-Y. Li

Brain Research Center, National Chiao Tung University, Hsinchu, Taiwan, Republic of China

P.-C. Tsen

Datan Power Plant, Taiwan Power Company, Taoyuan County 328, Taiwan, Republic of China

Keywords Quantum Cellular Neural Network (Quantum-CNN)· Lipschitz condition chaos

synchronization· Pragmatical asymptotical stability · Uncertain variable parameter

1 Introduction

The idea of synchronizing two identical chaotic sys-tems with different initial conditions was introduced by Pecora and Carroll [1]. Since then, there has been particular interest in chaotic synchronization, due to many potential applications in secure com-munication [2], and chemical and biological sys-tems [3,4]. There are many control methods to syn-chronize chaotic systems, such as linear coupling, for which the implementation is rather easy, adaptive con-trol, impulsive concon-trol, sliding mode concon-trol, and other methods [5]. Most of them are based on the exact knowledge of the system structure and parameters. But in practice, some or all of the system parameters are uncertain. Moreover, these parameters may change from time to time and become chaotic because of chaotic disturbances. For uncertain parameters, a lot of works have proceeded to solve this problem by adap-tive synchronization [6–12]. In the current scheme of adaptive synchronization [13–15], the traditional Lya-punov stability theorem and Barbalat lemma are used to prove that the error vector approaches zero as time approaches infinity. But the question, why the esti-mated parameters also approach the uncertain

parame-ters, has remained without answer. From the (Ge–Yu– Chen) GYC pragmatical asymptotical stability theo-rem [16–18], the question is strictly answered. In this paper, the synchronization of general chaotic systems which satisfy the Lipschitz condition only, with un-known parameters which are altered under some vari-able disturbances, by linear coupling and GYC prag-matical adaptive tracking, is studied first.

As numerical examples, the Lorenz system and recently developed quantum cellular neural network Quantum-CNN chaotic oscillator are used. GYC prag-matical adaptive tracking is used to track variable parameters in unidirectional coupled systems. Two Lorenz systems and two Quantum-CNN systems by GYC pragmatical adaptive tracking are given as sim-ulation examples. Quantum-CNN oscillator equations are derived from a Schrödinger equation taking into account quantum dots cellular automata structures to which in the last decade a wide interest has been de-voted with particular attention toward quantum com-puting [19–21].

This paper is organized as follows: In Sect.2, by the GYC pragmatical asymptotical stability theorem and by using Lipschitz conditions, theoretical analysis of synchronization is given. In Sect.3linear feedback controllers are used. By GYC pragmatical adaptive tracking, chaos synchronization of two Lorenz sys-tems and of two Quantum-CNN oscillator syssys-tems are achieved by numerical simulations. Conclusions are given in Sect.4. The GYC pragmatical asymptotical stability theorem is presented in theAppendix. Intu-itively, this theorem is different from the traditional Lyapunov stability theorem in that when the points in the neighborhood of zero solution initiating trajec-tories not approaching zero with time are “not too many,” i.e., in a subset of Lebesque measure 0 in math-ematical language, [22] we can neglect their existence, i.e., the zero solution is actually asymptotically stable.

2 Theoretical analyses

Consider a nonautonomous system in the form as fol-lows:

˙x = Ft, x, B(t ) (1)

The slave system is given by

˙y = Ft, y, ˆB(t )+ ˆK(x− y) (2)

where x ∈ Rn, y ∈ Rn, B ∈ RM is a vector of uncertain variable coefficients in F, ˆB ∈ RM is a vector of estimated coefficients in F , F : Ω1 ⊂

R₊ × Rn × RM → Rn satisfies Lipschitz condi-tionsF (t, x1, B)− F (t, x2, B) ≤ Gx1− x2 and F (t, x, B) − F (t, x, ˆB) ≤ GB − ˆB in Ω1 with Lipschitz constant G. ˆK= diag[ ˆK1, . . . , ˆKi, . . . , ˆKn],

ˆ

Ki: Ω2⊂ R+× Rn× Rn→ R (i = 1, . . . , n) is the estimated coupling strength entry. Ω1and Ω2are do-mains containing the origin. For given (t0, x0, y0, B0) ∈ Ω1∩Ω2, the solutions[xT(t, t0, x0, y0, B0), yT(t, t0,

x0, y0, B0)]T of Eqs. (1) and (2) exist for t≥ t0. If the synchronization can be accomplished when

t → ∞, the limit of the error vector e(t) = [e1, e2,

. . . , en]T must approach zero: lim

t→∞e= 0 (3)

where

e= x − y (4)

From Eqs. (1), (2), and (4), we have

˙e = ˙x − ˙y (5)

˙e = F (t, x, B) − F (t, x − e, ˆB) − ˆK(x− y) (6) A Lyapnuov function V (e, ˜B, ˜G)is chosen as a posi-tive definite function

V (e, ˜B, ˜G)=1 2e T_e+1 2 ˜B T ˜B +1 2 ˜G 2 (7) where ˜G= G − ˆG; ˆG is the estimated Lipschitz

con-stant, ˜B= B − ˆB.

When M= n, the time derivative of V along any solution of the differential equation system consisting of Eq. (6) and update differential equations for ˜Band

˜G is ˙V (e, ˜B, ˜G) = eT_{F (t, x, B)− F (t, x − e, B) + F (t, x − e, B)} − F (t, x − e, ˆB) − ˆKe+ ˜BT ˙˜B + ˜G ˙˜G = eT_{F (t, x, B)− F (t, x − e, B) − ˆKe}_{+ ˜G ˙˜G} + eT_{F (t, x− e, B) − F (t, x − e, ˆB)}_{+ ˜B}T ˙˜B (8) where ˜B= B − ˆB. By Lipschitz condition, ˙V (e, ˜B, ˜G) ≤ Ge2_{− e}T_Keˆ _{+ ˜G ˙˜G} + eT_{F (t, x− e, B) − F (t, x − e, ˆB)} + ˜BT ˙˜B ₍₉₎

Since

eTF (t, x− e, B) − F (t, x − e, ˆB)

≤ |e1| ·F1(t, x− e, B) − F1(t, x− e, ˆB) +··· + |en| ·Fn(t, x− e, B) − Fn(t, x− e, ˆB) (10)

by Schwarz inequality [23] and Lipschitz condition, it is obtained that |e1| ·F1(t, x− e, B) − F1(t, x− e, ˆB) +··· + |en| ·Fn(t, x− e, B) − Fn(t, x− e, ˆB) ≤ e ·F (t, x− e, B) − F (t, x − e, ˆB) ≤ Ge ·  ˜B (11) Therefore, ˙V (e, ˜B, ˜G) ≤ Ge2_{− e}T_Keˆ _{+ ˜G ˙˜G + Ge ·  ˜B} + ˜B1˙˜B1+ · · · + ˜Bn˙˜Bn (12) Choose ˙˜G = −eT_{e, K}ˆ _{= diag[ ˆG + G] (13)} and ˙˜B1= −G ˜B1e/ ˜B, . . . , ˙˜BN = −G ˜Bne/ ˜B (14) we have ˜BT ˙˜B = −G ˜B2 1+ · · · + ˜BN2  e/ ˜B = −G ˜B2_{· e/ ˜B} = −Ge ·  ˜B (15)

Introducing Eqs. (15), (13) in and (12), we get

˙V (e, ˜B, ˜G) ≤ Ge2_{− diag[ ˆG + G]e}2_{− ˜Ge}2 + Ge ·  ˜B − Ge ·  ˜B = −Ge2_{= −G}_e2 1+ · · · + e 2 n  (16)

˙V is a negative semidefinite of e, ˜B, ˜G, by the GYC

pragmatical asymptotical stability theorem (see the Appendix), the solution e= 0, ˜B = 0, ˜G = 0 is asymp-totically stable.

When M= n, on the right-hand side of Eq. (9), the other terms remain unchanged, and we want only to reduce last two terms

eTF (t, x− e, B) − F (t, x − e, ˆB)+ ˜BT ˙˜B (17) When M > n, we put

eT = eT = [e1, . . . , en, en+1, . . . , eM]T (18)

where en+1= en+2= · · · = eM= 0. The first term of

Eq. (17) becomes eTF (t, x− e, B) − F (t, x − e, ˆB) ≤ |e1| ·F1(t, x− e, B) − F1(t, x− e, ˆB) + · · · + |en| ·Fn(t, x− e, B) − Fn(t, x− e, ˆB) + |en+1| ·Fn+1(t, x− e, B) − Fn+1(t, x− e, ˆB) +··· + |eM| ·FM(t, x− e, B) − FM(t, x− e, ˆB) ≤ GeM ·  ˜B (19)

In Eq. (19), the last term is obtained by Schwarz in-equality. Similarly, we choose

˙˜B1= −G ˜B1e/ ˜B, . . . , ˙˜BM = −G ˜BMe/ ˜B (20) Then ˜BT ˙˜B = −G ˜B2 1+ · · · + ˜BM2  e/ ˜B = −G ˜B2_{e/ ˜B = −Ge ·  ˜B} (21) Introducing Eqs. (19), (21), in Eq. (9), we can also get lastly

˙V (e, ˜B, ˜G) ≤ −Ge2₁+ · · · + e2_n (22) By the same reasoning as when M= n, the solution

e= 0, ˜B = 0, ˜G = 0 is asymptotically stable.

When M < n, we put

Fi(t, x− e, B) − Fi(t, x− e, ˆB) = 0

i= M + 1, . . . , n (23)

since BM+1, . . . , Bndo not exist,

˜BM₊₁= · · · = ˜Bn= 0 (24)

 ˜B2_{= ˜B}2

1+ · · · + ˜BM2 + ˜BM2+1+ · · · + ˜Bn2 (25)

Then by the Schwarz inequality,

eTF (t, x− e, B) − F (t, x − e, ˆB) ≤ |e1| ·F1(t, x− e, B) − F1(t, x− e, ˆB) + · · · + |eM| ·FM(t, x− e, B) − FM(t, x− e, ˆB) +|eM+1| ·FM+1(t, x− e, B) − FM+1(t, x− e, ˆB) +··· + |en| ·Fn(t, x− e, B) − Fn(t, x− e, ˆB) ≤ Ge ·  ˜B (26)

Similarly, choose ˙˜B1= −G ˜B1e/ ˜B, . . . , ˙˜BM = −G ˜BMe/ ˜B, ˙˜BM+1 = −G ˜BM+1e/ ˜B, . . . , ˙˜Bn = −G ˜Bne/ ˜B (27) ˜BT ˙˜B = −G ˜B2 1+ · · · + ˜B 2 n  e/ ˜B = −G ˜B2_{e/ ˜B = −Ge ·  ˜B} (28) Introducing Eqs. (26), (28) in Eq. (9), we can also get lastly

˙V (e, ˜B, ˜G) ≤ −Ge2₁+ · · · + e2_n= −GeTe (29) By the same reasoning as the case M= n, the solution

e= 0, ˜B = 0, ˜G = 0 is asymptotically stable.

Remark In the current scheme of adaptive synchro-nization [13–15], the traditional Lyapunov stability theorem and Barbalat lemma are used to prove the error vector approaches zero, as time approaches in-finity. But the question, why the estimated parameters also approach uncertain parameters, remains no an-swer. By GYC pragmatical asymptotical stability the-orem, the question can be answered strictly. Moreover, the asymptotical stability is global; see theAppendix.

3 Numerical examples

Case I Periodic parameters for Lorenz system, M= n The master Lorenz system with uncertain variable pa-rameters is ⎧ ⎪ ⎨ ⎪ ⎩ ˙x1= −A1(t )(x1− x2) ˙x2= A2(t )x1− x2− x1x3 ˙x3= x1x2− A3(t )x3 (30)

where A1(t ), A2(t )and A3(t )are uncertain parame-ters. In simulation, we take

A1(t )= σ (1 + d1sin 1t )

A2(t )= γ (1 + d2sin 2t )

A3(t )= b(1 + d3sin 3t )

(31)

where σ , γ , b, d1, d2, d3, 1, 2, and 3are positive constants.

By Eq. (2), the slave Lorenz system is

⎧ ⎪ ⎨ ⎪ ⎩ ˆy1= − ˆA1(t )(y1− y2)+ ( ˆG + G)(x1− y1) ˆy2= ˆA2(t )y1− y2− y1y3+ ( ˆG + G)(x2− y2) ˆy3= y1y2− ˆA3(t )y3+ ( ˆG + G)(x3− y3) (32) where ˆK= ˆG + G. ˆG is the estimated value of G.

Take σ= 10, γ = 28, b = 8/3, d1= 0.05, d2= 0.01,

d3= 0.1, 1= 9, 2= 15, 3= 18, and the initial condition is[x₀T yT₀ AˆT₀ ˆG0]T = [111 000 000 0]T.

Subtracting Eq. (32) from Eq. (30), we obtain an error dynamics.

˙e1= −A1(t )(x1− x2)+ ˆA1(t )(y1− y2) − ( ˆG + G)(x1− y1) ˙e2= A2(t )x1− x2− x1x3− ˆA2(t )y1+ y2 + y1y3− ( ˆG + G)(x2− y2) ˙e3= x1x2− A3(t )x3− y1y2+ ˆA3(t )y3 − ( ˆG + G)(x3− y3) (33) where e1= x1− y1, e2= x2− y2, e3= x3− y3. Our aim is lim t→∞ei= limt→∞(xi− yi)= 0, i = 1, 2, 3 (34)

Let adaptive law be

˙˜G = ˙G − ˙ˆG = − ˙ˆG = −eT_{e (35)}

since G is constant, ˙G= 0. Define

˜ A(t )= ˜A1(t ) A˜2(t ) A˜3(t ) T (36) ˜ A1(t )= A1(t )− ˆA1(t ) ˜ A2(t )= A2(t )− ˆA2(t ) ˜ A3(t )= A3(t )− ˆA3(t ) (37) then ˙˜A1(t )= σ d11cos 1t− ˙ˆA1(t ) ˙˜A2(t )= γ d22cos 2t− ˙ˆA2(t ) ˙˜A3(t )= bd33cos 3t− ˙ˆA3(t ) (38) Choose ˙˜A1(t ), ˙˜A2(t ), and ˙˜A3(t )as ˙˜A1= −G ˜A1e/ ˜A ˙˜A2= −G ˜A2e/ ˜A (39) ˙˜A3= −G ˜A3e/ ˜A

Choose a Lyapunov function is given in the form of positive definite function:

V (e1, e2, e3, ˜A1, ˜A2, ˜A3, ˜G) =1

2



e2₁+ e2₂+ e2₃+ ˜A₁2+ ˜A2₂+ ˜A2₃+ ˜G2 (40) Its time derivative along any solution of Eqs. (33), (35), and (39) is ˙V = e1  −A1(t )(x1− x2)+ ˆA1(t )(y1− y2) − ( ˆG + G)(x1− y1)  + e2  A2(t )x1− x2− x1x3 − ˆA2(t )y1+ y2+ y1y3− ( ˆG + G)(x2− y2)  + e3  x1x2− A3(t )x3− y1y2+ ˆA3(t )y3 − ( ˆG + G)(x3− y3) 

+ ˜A1˙˜A1+ ˜A2˙˜A2+ ˜A3˙˜A3 − ˜G ˙ˆG ˙V = e1  −A1(t )(x1− x2)+ A1(t )(y1− y2) − ( ˆG + G)(x1− y1)  + e2  A2(t )x1− x2− x1x3 − A2(t )y1+ y2+ y1y3− ( ˆG + G)(x2− y2)  + e3  x1x2− A3(t )x3− y1y2+ A3(t )y3 − ( ˆG + G)(x3− y3)  + ˜A1(y1− y2)e1− ˜A2y1e2 − ˜A3y3e3− Ge ˜A21+ ˜A22+ ˜A23  / ˜A − ˜G ˙ˆG

˙V ≤ Ge2_{− ( ˆG + G)e}2_{+ Ge ˜A} − Ge ˜A2₁+ ˜A2₂+ ˜A2₃/ ˜A − ˜G ˙ˆG

˙V can be rewritten as ˙V ≤ −Ge2

(41)

˙V is negative semidefinite function of e, ˜A, ˜G. The

Lyapunov asymptotical stability theorem is not satis-fied. We cannot obtain that the common origin of er-ror dynamics (33), adaptive laws (35), and parameter dynamics (39) is asymptotically stable. Now, D is a 4-manifold, n= 7 and the number of error state variables

p= 3. When ei = 0 (i = 1, 2, 3) and ˜Ai, ˜Gtake

arbi-trary values, ˙V = 0, so X is a 4-manifold, m = n−p =

7− 3 = 4. m + 1 < n is satisfied. By GYC pragmat-ical asymptotpragmat-ical stability theorem, error vector e proaches zero and the estimated parameters also ap-proach the uncertain parameters. The GYC pragmati-cal generalized synchronization is obtained. The equi-librium point ei = ˜Ai= ˜G = 0 (i = 1, 2, 3) is

totically stable. Moreover, the result is global asymp-totically stable (see theAppendix). The numerical re-sults are shown in Figs. 1, 2 and3. The chaos syn-chronization is accomplished. The coupling strength required is K= 2G = 38.26.

Fig. 1 Phase portrait for Lorenz with σ= 10, γ = 28, b = 8/3

Fig. 2 Phase portrait for Eq. (30) with A1(t ) = σ (1+ d1sin 1t ), A2(t )= γ (1 + d2sin 2t ) and A3(t )= b(1+ d3sin 3t )

Case II Exponentially increasing and decreasing pa-rameters for Quantum-CNN system, M = n For a two-cell Quantum-CNN, the following differential equations are obtained [1–3]:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˙x1= −2a1  1− x₁2sin x2 ˙x2= −ω1(x1− x3)+ 2a1 x1  1− x₁2 cos x2 ˙x3= −2a2  1− x₃2sin x4 ˙x4= −ω2(x3− x1)+ 2a2 x3  1− x₃2 cos x4 (42)

where x1, x3 are polarizations, x2, x4 are quantum phase displacements, a1 and a2 are proportional to the interdot energy inside each cell, and ω1 and ω2 are parameters that weigh effects on the cell of the difference of the polarization of neighboring cells, like the cloning templates in traditional CNNs. When

a1= 6.8, a2= 4.3, ω1= 4.7, and ω2= 3.9, the sys-tem is chaotic.

The master Quantum-CNN system with uncertain variable parameters is ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˙x1= −2A1(t )  1− x₁2sin x2 ˙x2= −A3(t )(x1− x3)+ 2A1(t ) x1  1− x₁2 cos x2 ˙x3= −2A2(t )  1− x₃2sin x4 ˙x4= −A4(t )(x3− x1)+ 2A2(t ) x3  1− x₃2 cos x4 (43)

where A1(t ), A2(t ), A3(t ), and A4(t )are uncertain pa-rameters. In simulation, we take

A1(t )= a1  1+ c1  1− e−b1t A2(t )= a2  1+ c2  1− e−b2t A3(t )= ω1  1+ c3  1− e−b3t A4(t )= ω2  1+ c4  1− e−b4t (44)

where b1, b2, b3, b4, c1, c2, c3, and c4are constants. Take b1= 0.05, b2= 0.004, b3= 0.004, b4= 0.005,

c1= −0.25, c2= 0.15, c3= −0.2, and c4= 0.1. By Eq. (2), the slave Quantum-CNN system is

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˙y1= −2ˆa1  1− y₁2sin y2 + ( ˆG + G)(x1− y1) ˙y2= − ˆω1(y1− y3)+ 2ˆa1 y1  1− y₁2 cos y2 + ( ˆG + G)(x2− y2) ˙y3= −2ˆa2  1− y₃2sin y4+ ( ˆG + G)(x3− y3) ˙y4= − ˆω2(y3− y1)+ 2ˆa2 y3  1− y₃2 cos y4 + ( ˆG + G)(x4− y4) (45)

where ˆK= ˆG + G. ˆG is the estimated value of G. The

initial values are taken as x1(0)= 0.8, x2(0)= −0.77,

x3(0)= −0.72, x4(0)= 0.57, y1(0)= −0.2, y2(0)= 0.41, y3(0) = 0.25, y4(0) = −0.81 and [ˆa10 ˆa20 ˆω10 ˆω20 ˆG0]T = [0 0 0 0 0]T. The error dy-namic is ˙e1= −2A1(t )  1− x2₁sin x2+ 2ˆa1  1− y₁2sin y2 − ( ˆG + G)e1 ˙e2= −A3(t )(x1− x3)+ 2A1(t ) x1  1− x₁2 cos x2 + ˆω1(y1− y3)− 2ˆa1 y1  1− y₁2 cos y2 − ( ˆG + G)e2 ˙e3= −2A2(t )  1− x2₃sin x4+ 2ˆa2  1− y₃2sin y4 − ( ˆG + G)e3 ˙e4= −A4(t )(x3− x1)+ 2A2(t ) x3  1− x₃2 cos x4 + ˆω2(y3− y1)− 2ˆa2 y3  1− y₃2 cos y4 − ( ˆG + G)e4 (46) where e1= x1− y1, e2= x2− y2, e3= x3− y3, e4− x4− y4. Our aim is lim t→∞ei= limt→∞(xi− yi)= 0, i = 1, 2, 3, 4 (47)

Let adaptive law be

˙˜G = ˙G − ˙ˆG = − ˙ˆG = −eT_e

(48) since G is constant, ˙G= 0. Define

˜a1= A1(t )− ˆa1, ˜a2= A2(t )− ˆa2 ˜ω1= A3(t )− ˆω1, ˜ω2= A4(t )− ˆω2 (49) then ˙˜a1= a1b1c1e−b1t− ˙ˆa1 ˙˜a2= a2b2c2e−b2t− ˙ˆa2 ˙˜ω1= ω1b3c3e−b3t− ˙ˆω1 ˙˜ω2= ω2b4c4e−b4t− ˙ˆω2 (50) Let ˜ A= [˜a1 ˜a2 ˜ω1 ˜ω2] (51)

Choose ˙˜a1, ˙˜a2, ˙˜ω1, and ˙˜ω2as ˙˜a1= −G˜a1e/ ˜A ˙˜ω1= −G ˜ω1e/ ˜A ˙˜a2= −G˜a2e/ ˜A and ˙˜ω2= −G ˜ω2e/ ˜A

A Lyapunov function is given in the form of posi-tive definite function:

V (e1, e2, e3, e4,˜a1,˜a2,˜ω1,˜ω2, ˜G) =1 2  e₁2+ e2₂+ e2₃+ e2₄+ ˜a₁2+ ˜a₂2+ ˜ω2₁+ ˜ω2₂+ ˜G2 (53) Its time derivative along any solution of Eqs. (46), (48), and (52) is ˙V = e1 −2A1(t )  1− x₁2sin x2+ 2ˆa1  1− y₁2sin y2 − ( ˆG + G)e1  + e2  −A3(t )(x1− x3) + 2A1(t ) x1  1− x₁2 cos x2+ ˆω1(y1− y3) − 2ˆa1 y1  1− y₁2 cos y2− ( ˆG + G)e2  + e3 −2A2(t )  1− x₃2sin x4 + 2ˆa2  1− y₃2sin y4− ( ˆG + G)e3  + e4  −A4(t )(x3− x1)+ 2A2(t ) x3  1− x₃2 cos x4 + ˆω2(y3− y1)− 2ˆa2 y3  1− y2 3 cos y4 − ( ˆG + G)e4 

+ ˜a1˙˜a1+ ˜a2˙˜a2+ ˜ω1˙˜ω1+ ˜ω2˙˜ω2 − ˜G ˙ˆG ˙V = e1 −2A1(t )  1− x₁2sin x2 + 2A1(t )  1− y₁2sin y2− ( ˆG + G)e1  + e2  −A3(t )(x1− x3)+ 2A1(t ) x1  1− x₁2 cos x2 + A3(t )(y1− y3)− 2A1(t ) y1  1− y₁2 cos y2 − ( ˆG + G)e2  + e3 −2A2(t )  1− x₃2sin x4 + 2A2(t )  1− y₃2sin y4− ( ˆG + G)e3  + e4  −A4(t )(x3− x1)+ 2A2(t ) x3  1− x₃2 cos x4 + A4(t )(y3− y1)− 2A2(t ) y3  1− y₃2 cos y4 − ( ˆG + G)e4  + ˜a1  2  1− y₁2sin y2e1 − 2y1  1− y2₁ cos y2e2  + ˜ω1  (y1− y3)e2  + ˜a2  2  1− y₃2sin y4e3− 2y3  1− y₃2 cos y4e4  + ˜ω2  (y3− y1)e4  − Ge˜a2 1+ ˜a 2 2+ ˜ω 2 1+ ˜ω 2 2  / ˜A − ˜G ˙ˆG

˙V ≤ Ge2_{− ( ˆG + G)e}2_{+ Ge ˜A} − Ge˜a2 1+ ˜a 2 2+ ˜ω 2 1+ ˜ω 2 2  / ˜A − ˜G ˙ˆG ˙V can be rewritten as ˙V ≤ −Ge2₁+ e₂2+ e₃2+ e2₄ (54)

˙V is a negative semidefinite function of ei,˜aj, ˜ωj,

˜G (i = 1,2,3,4; j = 1,2). The Lyapunov

asymptoti-cal stability theorem is not satisfied. We cannot obtain that the common origin of error dynamics (46), adap-tive laws (48), and parameter dynamics (52) is asymp-totically stable. Now, D is a 5-manifold, n= 9 and the number of error state variables p= 4. When ei= 0

(i= 1, 2, 3, 4) and ˜aj,˜ωj, ˜G (j = 1, 2) take arbitrary

values, ˙V = 0, so X is a 5-manifold, m = n − p =

9− 4 = 5. m + 1 < n is satisfied. By the GYC prag-matical asymptotical stability theorem, error vector e approaches zero and the estimated parameters also ap-proach the uncertain parameters. The GYC pragmati-cal generalized synchronization is obtained. The equi-librium point ei = ˜aj = ˜ωj = ˜G = 0 (i = 1, 2, 3, 4;

j = 1, 2) is asymptotically stable. Moreover, the

re-sult is global asymptotically stable (see theAppendix). The numerical results are shown in Figs.4,5 and6. The chaos synchronization is accomplished. The cou-pling strength required is K= 2G = 5.54.

Case III Periodically and exponentially increasing and decreasing parameters for Quantum-CNN system,

un-Fig. 4 Phase portrait for chaotic system (42)

certain variable parameters is

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˙x1= −2A1(t )  1− x₁2sin x2 ˙x2= −A3(t )(x1− x3)+ 2A1(t ) x1  1− x₁2 cos x2 ˙x3= −2A2(t )  1− x₃2sin x4 ˙x4= −A4(t )(x3− x1)+ 2A2(t ) x3  1− x₃2 cos x4 (55)

where A1(t ), A2(t ), A3(t ), and A4(t )are uncertain pa-rameters. In simulation, we take

A1(t )= a1  1+ c1  1− e−b1t_{sin } 1t  A2(t )= a2  1+ c2  1− e−b2t_{sin } 2t  A3(t )= ω1  1+ c3  1− e−b3t_{sin } 3t  A4(t )= ω2  1+ c4  1− e−b4t_{sin } 4t  (56) where b1, b2, b3, b4, c1, c2, c3, c4, 1, 2, 3, and

4are constants. Take b1= 0.001, b2= 0.002, b3= 0.004, b4= 0.005, c1= −0.25, c2= 0.15, c3= −0.2,

c4= 0.1, 1= 5, 2= 1, 3= 3, and 4= 6. Sys-tem (55) is chaotic.

By Eq. (2), the slave Quantum-CNN system is

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˙y1= −2ˆa1  1− y₁2sin y2− ( ˆG + G)(y1− x1) ˙y2= − ˆω1(y1− y3)+ 2ˆa1 y1  1− y₁2 cos y2 − ( ˆG + G)(y2− x2) ˙y3= −2ˆa2  1− y₃2sin y4− ( ˆG + G)(y3− x3) ˙y4= −A4(t )(y3− y1)+ 2ˆa2

y3  1− y₃2 cos y4 − ( ˆG + G)(y4− x4) (57)

Fig. 5 Phase portrait for chaotic system (43)

where ˆK= ˆG + G. ˆG is the estimated value of G. The

error dynamic is ˙e1= −2A1(t )  1− x₁2sin x2+ 2ˆa1  1− y₁2sin y2 − ( ˆG + G)e1 ˙e2= −A3(t )(x1− x3)+ 2A1(t ) x1  1− x₁2 cos x2 + ˆω1(y1− y3)− 2ˆa1 y1  1− y₁2 cos y2 − ( ˆG + G)e2 ˙e3= −2A2(t )  1− x₃2sin x4+ 2ˆa2  1− y₃2sin y4 − ( ˆG + G)e3 ˙e4= −A4(t )(x3− x1)+ 2A2(t ) x3  1− x₃2 cos x4 + A4(t )(y3− y1) − 2ˆa2 y3  1− y₃2 cos y4− ( ˆG + G)e4 (58) where e1= x1− y1, e2= x2− y2, e3= x3− y3, e4− x4− y4. Our aim is lim t→∞ei= limt→∞(xi− yi)= 0, i = 1, 2, 3, 4 (59)

Let adaptive law be

˙˜G = ˙G − ˙ˆG = − ˙ˆG = −eT_e

(60) since G is constant, ˙G= 0. Define

˜a1= A1(t )− ˆa1, ˜a2= A2(t )− ˆa2 ˜ω1= A3(t )− ˆω1 (61) then ˙˜a1= a1b1c1e−b1tsin 1t − a1c11e−b1tcos 1t− ˙ˆa1 ˙˜ω1= ω1b3c3e−b3tsin 3t − ω1c33e−b3tcos 3t− ˙ˆω1 ˙˜a2= a2b2c2e−b2tsin 2t − a2c22e−b2tcos 2t− ˙ˆa2 (62)

Let

˜

A= [˜a1 ˜a2 ˜ω1] (63)

Choose ˙˜a1, ˙˜a2, and ˙˜ω1as ˙˜a1= −G˜a1e/ ˜A ˙˜ω1= −G ˜ω1e/ ˜A and ˙˜a2= −G˜a2e/ ˜A

(64)

A Lyapunov function is given in the form of a positive definite function: V (e1, e2, e3, e4,˜a1,˜a2,˜ω1, ˜G) =1 2  e2₁+ e2₂+ e2₃+ e2₄+ ˜a₁2+ ˜a₂2+ ˜ω2₁+ ˜G2 (65) Its time derivative along any solution of Eqs. (58), (60), and (64) is ˙V = e1  −2A1(t )  1− x₁2sin x2+ 2ˆa1  1− y₁2sin y2 − ( ˆG + G)e1  + e2  −A3(t )(x1− x3) + 2A1(t ) x1  1− x₁2 cos x2+ ˆω1(y1− y3) − 2ˆa1 y1  1− y₁2 cos y2− ( ˆG + G)e2  + e3  −2A2(t )  1− x₃2sin x4 + 2ˆa2  1− y₃2sin y4 − ( ˆG + G)e3  + e4  −A4(t )(x3− x1) + 2A2(t ) x3  1− x₃2 cos x4+ A4(t )(y3− y1) − 2ˆa2 y3  1− y₃2 cos y4− ( ˆG + G)e4 

+ ˜a1˙˜a1+ ˜a2˙˜a2+ ˜ω1˙˜ω1− ˜G ˙ˆG ˙V = e1  −2A1(t )  1− x₁2sin x2 + 2A1(t )  1− y₁2sin y2 − ( ˆG + G)e1  + e2  −A3(t )(x1− x3) + 2A1(t ) x1  1− x₁2 cos x2+ A3(t )(y1− y3) − 2A1(t ) y1  1− y₁2 cos y2− ( ˆG + G)e2  + e3  −2A2(t )  1− x₃2sin x4 + 2A2(t )  1− y₃2sin y4− ( ˆG + G)e3  + e4  −A4(t )(x3− x1)+ 2A2(t ) x3  1− x₃2 cos x4 + A4(t )(y3− y1)− 2A2(t ) y3  1− y₃2 cos y4 − ( ˆG + G)e4  + ˜a1  2  1− y₁2sin y2e1 − 2y1  1− y2₁ cos y2e2  + ˜ω1  (y1− y3)e2  + ˜a2  2  1− y₃2sin y4e3− 2y3  1− y₃2 cos y4e4  − Ge˜a2 1+ ˜a22+ ˜ω21   ˜A − ˜G ˙ˆG ˙V ≤ Ge2_{− ( ˆG + G)e}2_{+ Ge ˜A}

− Ge˜a2 1+ ˜a 2 2+ ˜ω 2 1   ˜A − ˜G ˙ˆG ˙V can be rewritten as ˙V ≤ −Ge2₁+ e₂2+ e₃2+ e2₄ (66)

˙V is a negative semidefinite function of ei, ˜aj, ˜ω1, ˜G (i = 1,2,3,4; j = 1,2). The Lyapunov

asymptoti-cal stability theorem is not satisfied. We cannot obtain that the common origin of error dynamics (58), adap-tive laws (60) and parameter dynamics (64) is asymp-totically stable. Now, D is a 4-manifold, n= 8 and the number of error state variables p= 4. When ei = 0

(i= 1, 2, 3, 4) and ˜aj, ˜ω1, ˜G (j = 1, 2) take arbitrary values, ˙V = 0, so X is a 4-manifold, m = n − p =

8− 4 = 4. m + 1 < n is satisfied. By GYC pragmat-ical asymptotpragmat-ical stability theorem, error vector e proaches zero and the estimated parameters also ap-proach the uncertain parameters. The GYC pragmati-cal generalized synchronization is obtained. The equi-librium point ei = ˜aj = ˜ω1= ˜G = 0 (i = 1, 2, 3, 4;

j = 1, 2) is asymptotically stable. Moreover, the

re-sult is global asymptotically stable (see theAppendix). The numerical results are shown in Figs.7and8. The chaos synchronization is accomplished. The coupling strength required is K= 2G = 6.34.

4 Conclusions

Using the Lipschitz condition, the synchronization of Lorenz chaotic systems and of Quantum-CNN chaotic oscillator systems with uncertain variable parame-ters by linear coupling and GYC pragmatical adap-tive tracking is implemented by the GYC pragmat-ical asymptotpragmat-ical stability theorem. Tracking uncer-tain variable parameters is firstly studied in this

pa-Fig. 7 Phase portrait for chaotic system (54)

per. This is more reasonable, because system param-eters always vary due to aging, environment, and dis-turbances. Two Lorenz systems are synchronization in one case: with oscillating parameters. Two Quantum-CNN systems are the synchronization in two cases: (1) with exponentially increasing and decreasing pa-rameters (2) with periodically and exponentially in-creasing and dein-creasing parameters. The computer simulation results imply that the present scheme is very satisfactory.

Acknowledgements This research was supported by the Na-tional Science Council, Republic of China, under Grant Number NSC 98-2218-E-011-010.

Appendix: GYC pragmatical asymptotical stability theorem

The stability for many problems in real dynamical systems is actual asymptotical stability, although may

not be mathematical asymptotical stability. The math-ematical asymptotical stability demands that trajecto-ries from all initial states in the neighborhood of zero solution must approach the origin as t→ ∞. If there are only a small part or even a few of the initial states from which the trajectories do not approach the ori-gin as t→ ∞, the zero solution is not mathematically asymptotically stable. However, when the probability of occurrence of an event is zero, it means the event does not occur actually. If the probability of occur-rence of the event that the trajectories from the initial states are that they do not approach zero when t→ ∞, i.e., these trajectories are not asymptotical stale for the zero solution is zero, the stability of the zero solution is actual asymptotical stability though it is not mathe-matical asymptotical stability. In order to analyze the asymptotical stability of the equilibrium point of such systems, the GYC pragmatical asymptotical stability theorem is used.

Let X and Y be two manifolds of dimensions m and

from X to Y ; then ϕ(X) is subset of Lebesque measure 0 of Y [22]. For an autonomous system

˙x = f (x1, . . . , xn) (67)

where x= [x1, . . . , xn]T is a state vector, the func-tion f = [f1, . . . , fn]Tis defined on D⊂ Rn, an n-manifold.

Let x = 0 be an equilibrium point for the sys-tem (67). Then

f (0)= 0 (68)

For a nonautonomous system,

˙x = f (x1, . . . , xn+1) (69)

where x= [x1, . . . , xn+1]T, the function f = [f1, . . . ,

fn]T is defined on D⊂ Rn_×R

+, here t= xn₊₁⊂ R₊. The equilibrium point is

f (0, xn+1)= 0 (70)

Definition The equilibrium point for the system is pragmatically asymptotically stable provided that with initial points on C which is a subset of Lebesque mea-sure 0 of D, the behaviors of the corresponding trajec-tories cannot be determined, while with initial points on D− C, the corresponding trajectories behave as that agree with traditional asymptotical stability [19, 20].

Theorem Let V = [x1, x2, . . . , xn]T : D → R+ be positive definite and analytic on D, where x1, x2,

. . . , xnare all space coordinates such that the

deriva-tive of V through Eqs. (67) or (69), ˙V, is negative semidefinite of[x1, x2, . . . , xn]T.

For the autonomous system, let X be the m-manifold consisting of the point set for which∀x = 0,

˙V (x) = 0 and D is a n-manifold. If m + 1 < n, then

the equilibrium point of the system is pragmatically asymptotically stable.

For the nonautonomous system, let X be the

m+ 1-manifold consisting of the point set for which

∀x = 0, ˙V (x1, x2, . . . , xn) = 0 and D is an

n+ 1-manifold. If m + 1 + 1 < n + 1, i.e., m + 1 < n,

then the equilibrium point of the system is pragmat-ically asymptotpragmat-ically stable. Therefore, for both the autonomous and nonautonomous system, the formula

m+ 1 < n is universal. So, the following proof is only

for the autonomous system. The proof for the nonau-tonomous system is similar.

Proof Since every point of X can be passed by a tra-jectory of Eq. (67), which is one-dimensional, the col-lection of these trajectories, C, is a (m+ 1)-manifold [16,17].

If m+ 1 < n, then the collection C is a subset of Lebesque measure 0 of D. By the above definition, the equilibrium point of the system is pragmatically asymptotically stable.

If an initial point is ergodicly chosen in D, the prob-ability of that the initial point falls on the collection

C is zero. Here, equal probability is assumed for ev-ery point chosen as an initial point in the neighbor-hood of the equilibrium point. Hence, the event that the initial point is chosen from collection C does not occur actually. Therefore, under the equal probabil-ity assumption, pragmatical asymptotical stabilprobabil-ity be-comes actual asymptotical stability. When the initial point falls on D− C, ˙V (x) < 0, the corresponding tra-jectories behave as that agree with traditional asymp-totical stability because by the existence and unique-ness of the solution of initial-value problem, these tra-jectories never meet C.

For Eq. (7), the Lyapunov function is a positive def-inite function of n variables, i.e., p error state vari-ables and n− p = m differences between unknown and estimated parameters, while ˙V = eTCeis a neg-ative semidefinite function of n variables. Since the number of error state variables is always more than one, p > 1, m+ 1 < n is always satisfied, by pragmat-ical asymptotpragmat-ical stability theorem we have

lim

t→∞e= 0 (71)

and the estimated parameters approach the uncertain parameters. The pragmatical adaptive control theorem is obtained. Therefore, the equilibrium point of the system is pragmatically asymptotically stable. Under the equal probability assumption, it is actually asymp-totically stable for both error state variables and

pa-rameter variables. 

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