The generalized synchronization of a Quantum-CNN chaotic oscillator with different order systems

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The generalized synchronization of a Quantum-CNN

chaotic oscillator with diﬀerent order systems

Zheng-Ming Ge

*

_{, Cheng-Hsiung Yang}

Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, ROC Accepted 23 May 2006

Abstract

This paper presents a special kind of the generalized synchronization of different order systems, proved by Lyapunov asymptotical stability theorem. A sufficient condition is given for the asymptotical stability of the null solution of an error dynamics. The generalized synchronization developed may be applied to the design of secure communication. Finally, numerical results are studied for a Quantum-CNN oscillator synchronized with three different order systems respectively to show the effectiveness of the proposed synchronization strategy.

1. Introduction

In recent years, the synchronization of chaotic systems has been studied in various ﬁelds[1–4]. For a particular cha-otic system, a master, together with an identical or a diﬀerent system, a slave system, our goal is to synchronize them via coupling or other methods.

Among many kinds of synchronizations[5], the generalized synchronization is investigated[6–10]. It means there exists a functional relationship between the states of the master and those of the slave. In this paper, a special kind of generalized synchronizations

y¼ x þ F ðtÞ ð1Þ

is studied, where x, y are the state vectors of the master and the slave respectively, F(t) is a given vector function of time, which may take various forms, either regular or chaotic functions of time. The generalized synchronization developed may be applied to the design of secure communication. When F(t) = 0, it reduces to a complete synchronization[12–21]. As numerical examples, recently developed quantum cellular neural network (Quantum-CNN) chaotic oscillator is used to synchronize with three diﬀerent order systems respectively. Quantum-CNN oscillator equations are derived from a Schro¨dinger equation taking into account quantum dots cellular automata structures to which in the last decade a wide interest has been devoted, with particular attention towards quantum computing[11].

This paper is organized as follows. In Section2, by the Lyapunov asymptotical stability theorem, a generalized syn-chronization scheme is given. In Section3, various feedback controllers are designed for the synchronization of the

* _{Corresponding author. Tel.: +886 35712121; fax: +886 35720634.}

E-mail address:zmg@cc.nctu.edu.tw(Z.-M. Ge).

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Quantum-CNN oscillator with a Lorenz system and with a Chen system respectively. Numerical simulations are also given in Section3. Finally, some concluding remarks are given in Section4.

2. Generalized synchronization scheme

There are two identical nonlinear dynamical systems, and the master system controls the slave system. The master system is given here

_x¼ Ax þ f ðxÞ ð2Þ

where x = (x1, x2, . . . , xn)T2 Rn denotes a state vector, A is a n· n coeﬃcient matrix, and f is a nonlinear vector

function.

The slave system is given here

_y¼ By þ hðyÞ þ uðtÞ ð3Þ

where y = (y1, y2, . . . , yna)T2 Rnadenotes a state vector, a is a positive integer, 1 6 a 6 n, B is a (n a) · (n a)

coef-ﬁcient matrix, h is a nonlinear vector function, and u(t) = (u1(t), u2(t), . . . , una(t))T2 Rnais a control input vector.

Our goal is to design a controller u(t) so that the state vector of the slave system(3)asymptotically approaches the state vector of the master system(2)plus a given vector function F(t) = (F1(t), F2(t), . . . , Fna(t))T, and ﬁnally the

syn-chronization will be accomplished in the sense that the limit of the error vector e(t) = (e1, e2, . . . , ena)Tapproaches zero:

lim

t!1e¼ 0 ð4Þ

where

e¼ x y þ F ðtÞ ð5Þ

From Eq.(5)we have

_e¼ _x _y þ _F ðtÞ ð6Þ

_e¼ ðAna BÞe þ f ðxÞ hðyÞ þ _F ðtÞ uðtÞ ð7Þ

A Lyapunov function V(e) is chosen as a positive deﬁnite function VðeÞ ¼1

2e

T

e ð8Þ

Its derivative along any solution of Eq.(7)is _

VðeÞ ¼ eT_fðA

na BÞ½xi yi þ f ðxÞ hðyÞ þ _F ðtÞ uðtÞg; i¼ 1; 2; . . . ; n a ð9Þ

where [xi yi] is a (n a) · 1 column matrix, u(t) is chosen so that, _V ¼ eTCnae, Cna is a diagonal negative deﬁnite

matrix, and _V is a negative deﬁnite function of e. By the Lyapunov theorem of asymptotical stability, we have lim

t!1e¼ 0 ð10Þ

The generalized synchronization is obtained[22–31].

3. Numerical results of the generalized chaos synchronization of the Quantum-CNN oscillator with diﬀerent order systems Case I. A complete synchronization as a special case of the generalized synchronization

For a two-cell Quantum-CNN, the following diﬀerential equations are obtained[11]:

d dtx1¼ 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p sin x2 d dtx2¼ x1ðx1 x2Þ þ 2a1 xffiffiffiffiffiffiffi_1x1 2 1 p cos x2 d dtx3¼ 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 p sin x4 d dtx4¼ x2ðx3 x1Þ þ 2a2 x3 ffiffiffiffiffiffiffi 1x2 3 p cos x4 8 > > > > > > < > > > > > > : ð11Þ

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where x1, x3are polarizations, x2, x4are quantum phase displacements, a1and a2are proportional to the inter-dot

en-ergy inside each cell and x1and x2are parameters that weigh eﬀects on the cell of the diﬀerence of the polarization of

neighboring cells, like the cloning templates in traditional CNNs. Let a1= 19.4, a2= 13.1, x1= 9.529, x2= 7.94.

A Lorenz system is described by

d dty1¼ rðy2 y1Þ d dty2¼ cy1 y2 y1y3 d dty3¼ y1y2 by3 8 > < > : ð12Þ where r = 10, c = 28, b¼3 8.

Take a = 1. In order to lead (y1, y2, y3) to (x1+ F1(t), x2+ F2(t), x3+ F3(t)), we add u1, u2, and u3to the ﬁrst, the

sec-ond, and the third equations of Eq.(12)respectively.

d dty1¼ rðy2 y1Þ þ u1 d dty2¼ cy1 y2 y1y3þ u2 d dty3¼ y1y2 by3þ u3 8 > < > : ð13Þ

Subtracting Eq.(13)from the ﬁrst three equations of Eq.(11), we obtain an error dynamics. The initial values of the Quantum-CNN system and the Lorenz system 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) =0.42, and y3(0) =0.11. lim t!1ei¼ limt!1ðxi yiÞ ¼ 0; i¼ 1; 2; 3 ð14Þ _e1¼ 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q sin x2 rðy2 y1Þ u1 _e2¼ x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 cy1þ y2þ y1y3 u2 ð15Þ _e3¼ 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 y1y2þ by3 u3 where e1= x1 y1, e2= x2 y2, e3= x3 y3.

Choose a Lyapunov function in the form of the positive deﬁnite function: Vðe1; e2; e3Þ ¼ 1 2ðe 2 1þ e 2 2þ e 2 3Þ ð16Þ

Its time derivative along any solution of Eq.(15)is _ V ¼ e1 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q sin x2 rðy2 y1Þ u1 þ e2 x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 cy1þ y2þ y1y3 u2 ! þ e3 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 y1y2þ by3 u3 ð17Þ Choose u1¼ 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q sin x2 ry2 rx1 u2¼ x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 cy1þ y1y3 x2 u3¼ 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 y1y2þ bx3

Eq.(17)can be rewritten as _ V ¼ re2 1 e 2 2 ce 2 3<0 ð18Þ

which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. This means that the complete chaos synchronization of the different order systems, the Quantum-CNN system and the Lorenz system, can be achieved. The numerical results are shown inFig. 1. After 10 s, the motion trajectories enter a chaotic attractor.

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Case II. A sine function synchronization We have

lim

t!1ei¼ limt!1ðxi yiþ Fisin xtÞ ¼ 0; i¼ 1; 2; 3 ð19Þ

where _e¼ _x _y þ F x cos xt.

Let F1= F2= F3= F, Eq.(7)becomes

_e1¼ 2a1

ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1

q

sin x2 rðy2 y1Þ u1þ F x cos xt

_e2¼ x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 cy1þ y2þ y1y3 u2þ F x cos xt ð20Þ _e3¼ 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 y1y2þ by3 u3þ F x cos xt

where e1= x1 y1+ F sin xt, e2= x2 y2+ F sin xt, e3= x3 y3+ F sin xt. F and x are taken as F = 0.7, x = 1.

Choose a Lyapunov function in the form of the positive deﬁnite function: Vðe1; e2; e3Þ ¼ 1 2ðe 2 1þ e 2 2þ e 2 3Þ ð21Þ 0 5 10 15 20 25 30 -20 -15 -10 -5 0 5 10 15 x1 y1 Time (sec) X1 Y1 0 5 10 15 20 25 30 -30 -25 -20 -15 -10 -5 0 5 10 15 x2 y2 X2 Y2 Time (sec) 0 5 10 15 20 25 30 -10 0 10 20 30 40 50 x3 y3 X₃ Y3 Time (sec) 0 5 10 15 20 25 30 -50 -40 -30 -20 -10 0 10 20 30 e1 e2 e3 Time (sec) e₁ e2 e₃

a

b

c

_d

Fig. 1. Time histories of the master states, of the slave states, and of the synchronization errors for the Quantum-CNN system and the Lorenz system.

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Its time derivative along any solution of Eq.(20)is _ V ¼ e1 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q

sin x2 rðy2 y1Þ u1þ F x cos xt

þ e2 x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 cy1þ y2þ y1y3 u2þ F x cos xt ! þ e3 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 y1y2þ by3 u3þ F x cos xt ð22Þ Choose u1¼ 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q

sin x2 ry2þ rx1þ F ðx cos xt þ sin xtÞ

u2¼ x1ðx1 x3Þ þ 2a1

x1

p cos x2 cy1þ y1y3þ x2þ F ðx cos xt þ sin xtÞ

u3¼ 2a2

q

sin x4 y1y2þ bx3þ F ðx cos xt þ sin xtÞ

which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. This means that the sine function synchronization of the different order systems, the Quantum-CNN system and the Lorenz system, can be achieved. The numerical results are shown inFig. 2. After 10 s, the motion trajectories enter a chaotic attractor.

Case III. A Chen system state synchronization The goal system for synchronization is a Chen system

d dtz1¼ aðz2 z1Þ d dtz2¼ ðc aÞz1 z1z3 cz2 d dtz3¼ z1z2 bz3 8 > < > : ð24Þ where a = 10, b¼3

8, c = 28. The initial values of the states of the Chen system is taken as z1(0) = 0.5, z2(0) = 0.5, and

z3(0) = 0.5. The chaotic vector state z = (z1, z1, z1) is chosen as F(t). Then

lim t!1ei¼ limt!1ðxi yiþ z1Þ ¼ 0; i¼ 1; 2; 3 ð25Þ Eq.(7)becomes _e¼ _x _y þ _z _e1¼ 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q

sin x2 rðy2 y1Þ u1þ aðz2 z1Þ

_e2¼ x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 cy1þ y2þ y1y3 u2þ aðz2 z1Þ ð26Þ _e3¼ 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 y1y2þ by3 u3þ aðz2 z1Þ where e1= x1 y1+ z1, e2= x2 y2+ z1, e3= x3 y3+ z1.

Choose a Lyapunov function in the form of the positive deﬁnite function: Vðe1; e2; e3Þ ¼ 1 2ðe 2 1þ e 2 2þ e 2 3Þ ð27Þ

Its time derivative along any solution of Eq.(26)is _ V ¼ e1ð2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q

sin x2 rðy2 y1Þ u1þ aðz2 z1ÞÞ þ e2ðx1ðx1 x3Þ þ 2a1

x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 cy1 þ y₂þ y₁y₃ u2þ aðz2 z1ÞÞ þ e3 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 y1y2þ by3 u3þ aðz2 z1Þ ð28Þ

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0 5 10 15 20 25 30 -20 -15 -10 -5 0 5 10 15 x1 y1 Time (sec) x1 y1 0 5 10 15 20 25 30 -30 -25 -20 -15 -10 -5 0 5 10 15 x2 y2 x2 y2 Time (sec) 0 5 10 15 20 25 30 -10 0 10 20 30 40 50 x3 y3 x3 y3 Time (sec) 0 5 10 15 20 25 30 -50 -40 -30 -20 -10 0 10 20 30 x1-y1 x2-y2 x3-y3 Fsinwt x1-y1 x2-y2 x3-y3 Fsinwt Time (sec)

a

_b

c

_d

0 5 10 15 20 25 30 -50 -40 -30 -20 -10 0 10 20 30 e1 e2 e3 e1 e2 e3 Time (sec)

e

Fig. 2. Time histories of the master states, of the slave states, and of the sine function synchronization errors for the Quantum-CNN system and the Lorenz system, where ei= xi yi+ F sin xt, i = 1, 2, 3.

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Choose u1¼ 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q sin x2 ry2þ rx1þ aðz2 z1Þ þ z1 u2¼ x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 cy1þ y1y3 x2þ aðz2 z1Þ þ z1 u3¼ 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 y1y2þ bx3þ aðz2 z1Þ þ z1

which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. This means that the Chen system state synchronization of the different order systems, the Quantum-CNN system and the Lorenz system, can be achieved. The numerical results are shown inFig. 3. After 10 s, the motion trajectories enter a chaotic attractor.

Case IV. A Chen system states synchronization We have

lim

t!1ei¼ limt!1ðxi yiþ ziÞ ¼ 0; i¼ 1; 2; 3 ð30Þ

and

_e¼ _x _y þ _z

where _z = (_z1, _z2, _z3). Eq.(7)becomes

_e1¼ 2a1

q

_e2¼ x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 cy1þ y2þ y1y3 u2þ ðc aÞz1 z1z2 cz2 ð31Þ _e3¼ 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 y1y2þ by3 u3þ z1z2 bz3 where e1= x1 y1+ z1, e2= x2 y2+ z2, e3= x3 y3+ z3.

Choose a Lyapunov function in the form of the positive deﬁnite function: Vðel; e2; e3Þ ¼ 1 2ðe 2 lþ e 2 2þ e 2 3Þ ð32Þ

Its derivative along any solution of Eq.(31)is _ V ¼ e1 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q

þ e2 x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 cy1þ y2þ y1y3 u2þ ðc aÞz1 z1z3 cz2 ! þ e3 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 y1y2þ by3 u3þ z1z2 bz3 ð33Þ Choose u1¼ 2a1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 q sin x2 ry2þ rx1þ aðz2 z1Þ þ z1 u2¼ x1ðx1 x3Þ þ 2a1 x1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 1 p cos x2 cy1þ y1y3þ x2þ ðc aÞz1 z1z3þ z2ð1 cÞ u3¼ 2a2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 3 q sin x4 y1y2þ bx3þ z1 z2þ z3ð1 bÞ

which is negative definite. The Lyapunov asymptotical stability theorem is satisfied. This means that the Chen system states synchronization of the different order systems, the Quantum-CNN system and the Lorenz system, can be achieved. The numerical results are shown inFig. 4. After 10 second, the motion trajectories enter a chaotic attractor.

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0 5 10 15 20 25 30 -25 -20 -15 -10 -5 0 5 10 15 20 25 x1 y1 x1 y1 Time (sec) 0 5 10 15 20 25 30 -30 -20 -10 0 10 20 30 x2 y2 Time (sec) x2 y2 0 5 10 15 20 25 30 -30 -20 -10 0 10 20 30 40 50 x3 y3 x3 y3 Time (sec)

a

b

c

d

0 5 10 15 20 25 30 -50 -40 -30 -20 -10 0 10 20 30 z1 z1 Time (sec) 0 5 10 15 20 25 30 -80 -60 -40 -20 0 20 40 60 e1 e2 e3 e1 e2 e3 Time (sec) x1-y1 x2-y2 x3-y3 x1-y1 x2-y2 x3-y3

e

Fig. 3. Time histories of the master states, of the slave states, and of the Chen system state synchronization errors for the Quantum-CNN system and the Lorenz system, where ei= xi yi+ z1, i = 1, 2, 3.

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0 5 10 15 20 25 30 -25 -20 -15 -10 -5 0 5 10 15 20 25 x1 y1 x₁ y1 Time (sec)

a

0 5 10 15 20 25 30 -30 -20 -10 0 10 20 30 x2 y2 x₂ y₂ Time (sec)

b

0 5 10 15 20 25 30 -10 0 10 20 30 40 50 x3 y3 Time (sec) x3 y₃

c

0 5 10 15 20 25 30 -30 -20 -10 0 10 20 30 x1-y1 z1 x1-y1 z₁ Time (sec)

d

0 5 10 15 20 25 30 -40 -30 -20 -10 0 10 20 30 40 x2-y2 z2 x₂_{- y}₂ z2 Time (sec)

e

0 5 10 15 20 25 30 -60 -40 -20 0 20 40 60 x3-y3 z3 x₃-y₃ z₃ Time (sec)

f

0 5 10 15 20 25 30 -30 -20 -10 0 10 20 30 40 50 60 e1 e2 e3 e1 e2 e3 Time (sec)

g

Fig. 4. Time histories of the master states, of the slave states, and of the Chen system states synchronization errors for the Quantum-CNN system and the Lorenz system, where ei= xi yi+ zi, i = 1, 2, 3.

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4. Conclusions

The generalized chaos synchronization of the diﬀerent order systems are investigated by using the Lyapunov asymp-totical stability theorem. Two diﬀerent chaotic dynamical systems, the Quantum-CNN system and the Lorenz system, are chosen for the complete synchronization as a special case and for the given regular time function synchronization. The Chen system is chosen for the given chaotic time function synchronizations. The generalized synchronization of chaos system can be used to increase the security of communication.

Acknowledgement

This research was supported by the National Science Council, Republic of China, under Grant Number NSC 94-2212-E-009-013.

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