Synchronization of different chaotic systems and electronic circuitanalysis

J.H. Park1, T.H. Lee1, D.H. Ji2, H.Y. Jung3, S.M. Lee41Department of Electrical Engineering, Yeungnam University, Kyongsan, Republic of Korea.

2Digital Media and Communications, Samsung Electronics, Suwon, Republic of Korea.3Department of Information and Communication Engineering, Yeungnam University, Republic of Korea.

4Department of Electronic Engineering, Daegu University, Gyungsan, Republic of Korea.

Abstract— This paper investigates the problem of synchro-nization between the Chen-Lee and Lorenz chaotic systems.Based on the Lyapunov stability theory and active controlmethod, an effective controller is designed for asymptoticstability of the null solution of an error dynamics betweenmaster and slave chaotic systems. In order to verify theeffectiveness of proposed control scheme, the computer sim-ulation via Matlab software is applied to the Chen-Leeand Lorenz chaotic systems. Then, the realization modelof the Chen-Lee and Lorenz chaotic systems is revised forelectronic circuit simulation. Finally, the circuit simulationvia NI (National Instruments) Multisim is performed toconfirm the efficiency of our results.

Keywords: Chen-Lee chaotic system, Lorenz chaotic system,Chaos synchronization, Lyapunov method, Circuit analysis.

1. IntroductionSynchronization of one system with another is very impor-

tant process in the control of complex physical, chemical andbiological systems as well as engineering. Therefore, manyresearchers have focused on this topic and developed severalefficient synchronization techniques for various dynamicsystems including chaotic systems, which are very sensitiveto variations in the parameters and initial conditions. SincePecora and Carroll [1] introduced the concept of synchro-nization in chaotic systems, the study of chaos synchro-nization has received increasing interest from scientists andengineers. It makes very big issue in nonlinear society. Up todate, not only various applications of chaos synchronization[2-7] but also diverse methods for control of the chaossynchronization have been introduced [8-10]. Originally,the chaos synchronization refers to the state in which themaster (or drive) and the slave (or response) systems haveprecisely identical trajectories for time to infinity. We usuallyregard such a synchronization as complete synchronizationor identical synchronization.During the last several years, for more practical and realapplications, the investigation of synchronization betweendifferent chaotic systems has been researched. For example,Yassen [11] studied the synchronization problem of differentunified chaotic systems such as Lorenz-Chen, Lorenz-Lü

and Lü-Chen. Park [12] proposed a method to synchronizebetween Genesio-Tesi and Rössler chaotic systems. In moredevelopment point of view, Huang [13] studied the chaossynchronization between hyperchaotic Lorenz system andhyperchaotic Lü system which has more complicated chaoticbehavior and more than four Lyapunov exponent. By theway, when the mathematical model of chaotic system isimplemented to electronic circuit, some adjustment due todifference of time scale between the mathematical model andelectronic circuit model is sure to conduct. However, this isnot easy job. Hence, that’s why the numerical simulationsare only provided to verify their synchronization algorithmswithout circuit simulation in most of literatures.In addition, most of chaos circuit analysis dealt with onlyChua’s circuit and single chaos system which are a sim-ple electronic circuit that exhibits chaotic behavior [14-17]. Sometimes, even though the circuit analysis for chaossynchronization is conducted, these researches only dealwith two identical chaotic systems not different chaoticsystems. For example, Cuomo et al. [18] and Lian et al.[19] presented a solution to the synchronization problemfor identical Lorenz systems. They used transformed Lorenzequation because of some errors between theoretical systemand practical system. In [20], Du et al. investigated thesynchronization of Qi hyperchaotic master and slave systemswith parameters mismatch using high order differentiator.Also, Xiao et al. [21] studied the synchronization problembetween two identical Van der Pol oscillators using adaptivecontrol method.As is well-known, some difference between theoretical sys-tem parameters and practical system parameters exists. So, itis difficult and significant to materialize theoretical system toreal one. In addition, the electrical circuit simulation of dif-ferent chaotic systems have more complicated problems suchas readjustment of time range or difference of limitation inpower supply and electronic device and so on. Therefore, inthis paper, the synchronization scheme between the revisedpractical Chen-Lee chaotic master system and the revisedpractical Lorenz chaotic slave system will be showed byapplying our control law via NI Multisim. To the best ofauthors’ knowledge, this is the first circuit analysis betweendifferent chaotic systems.

This paper is organized as follows. In Section 2, systemdescription is given. In Section 3, the theoretical synchro-nization scheme between Chen-Lee and Lorenz chaoticsystems is illustrated. In Section 4, a numerical simulationvia Matlab is given to demonstrate the effectiveness of theproposed control method. In Section 5, the electronic circuitimplementations are presented to show real applications ofthe method. Finally, some conclusions are given in Section6.

2. System descriptionConsider the following master (drive) and slave (response)

chaotic systems

x(t) = f(t, x), (1)y(t) = g(t, y) + u(t, x, y), (2)

where x(t) = (x1, x2, . . . , xn)T ∈ Rn and y(t) =

(y1, y2, . . . , yn)T ∈ Rn are master and slave state vectors,

respectively, f : R × Rn → Rn and g : R × Rn → Rn

are continuous nonlinear vector functions and u(t, x, y) =(u1, u2, . . . , un)

T ∈ Rn is the control input for synchroniza-tion between master (1) and slave system (2).As previously stated, we deal with the Chen-Lee master sys-tem and Lorenz slave system for synchronization problem.

Now let us consider following Chen-Lee master chaoticsystem

x1(t) = ax1(t)− x2(t)x3(t)

x2(t) = −bx2(t) + x1(t)x3(t)

x3(t) = −cx3(t) +1

3x1(t)x2(t), (3)

where a = 5, b = 10, c = 3.8.In order to see chaotic motion of the system (3), let us takean initial condition x(0) = (−5,−7,−10)T . Then, Fig. 1shows chaotic behavior of Chen-Lee system.Next, the Lorenz chaotic systems as slave system is givenas follows

y1(t) = a1(y2(t)− y1(t)) + u1(t)

y2(t) = b1y1(t)− y1(t)y3(t)− y2(t) + u2(t)

y3(t) = y1(t)y2(t)− c1y3(t) + u3(t), (4)

where a1 = 10, b1 = 28, c1 = 8/3.The chaotic behavior of system (4) with an initial conditiony(0) = (0,−1,−1)T is presented in Fig. 2.

3. Synchronization between the Chen-Lee and Lorenz systems

In this section, we design control law for achievingsynchronization between the Chen-Lee and Lorenz systems.

Definition 1. It is said that synchronization occursbetween master system (1) and slave system (2) such that

limt→∞ ∥yi(t)− xi(t)∥ = 0, (i = 1, 2, 3).

Now, for our synchronization scheme, let us defineerror signals between the Chen-Lee chaotic system andLorenz chaotic system in the sense of Definition 1 as

e1(t) = y1(t)− x1(t)

e2(t) = y2(t)− x2(t)

e3(t) = y3(t)− x3(t). (5)

The time derivative of error signal (5) is

e1(t) = y1(t)− x1(t)

e2(t) = y2(t)− x2(t)

e3(t) = y3(t)− x3(t). (6)

By substituting (3) and (4) into (6), we have the followingerror dynamics

e1 = a1y2 − a1y1 + x2x3 − ax1 + u1

= −a1e1 − (a1 + a)x1 + a1y2 + x2x3 + u1

e2 = b1y1 − y1y3 − y2 − x1x3 + bx2 + u2

= −be2 + (b− 1)y2 + b1y1 − y1y3 − x1x3 + u2

e3 = y1y2 − c1y3 −1

3x1x2 + cx3 + u3

= −c1e3 + (c− c1)y3 + y1y2 −1

3x1x2 + u3. (7)

Here, our goal is to achieve synchronization betweenthe Chen-Lee and Lorenz systems. For this end, thefollowing theorem shows that chaotic systems (3) and (4)can be synchronized effectively by the following designedcontroller.

Theorem 1. Chaotic Chen-Lee system (3) and Lorenzsystem (4) can be synchronized asymptotically for anydifferent initial conditions with the following controller:

u1 = −x2(t)x3(t)− a1y2(t) + (a1 + a)x1(t)

u2 = y1(t)y3(t) + x1(t)x3(t)− b1y1(t)− (b− 1)y2(t)

u3 = −y1(t)y2(t) +1

3x1(t)x2(t)− (c− c1)y3(t). (8)

Proof. Let us take the following Lyapunov function candi-date

V =1

2(e21 + e22 + e23). (9)

By differentiating Eq. (9), we get

V = e1e1 + e2e2 + e3e3. (10)

By applying our controller (8) and error dynamics (7) to Eq.(10), we obtain

V = e1(−a1e1 − (a1 + a)x1 + a1y2 + x2x3 + u1

)e2(−be2 + (b− 1)y2 + b1y1 − y1y3 − x1x3 + u2

)e3(−c1e3 + (c− c1)y3 + y1y2 −

1

3x1x2 + u3

)= −a1e1 − be2 − c1e3

= −

e1e2e3

T 10 0 00 10 00 0 8

3

e1e2e3

≡ −eTPe < 0, (11)

which guarantees the stability of error systems in the senseof Lyapunov theory. This implies that the error signals satisfylimt→∞ ∥ei(t)∥ = 0 (i = 1, 2, 3). This completes the proof.

4. Numerical simulationIn order to demonstrate the validity of proposed ideas, nu-

merical simulation via Matlab software is presented. Fourth-order Runge-Kutta method with sampling time 0.001[sec] isused to solve the system of differential equations (3) and(4).The system parameters are used by a = 5, b = 10, c =3.8, a1 = 10, b1 = 28, c1 = 8/3 in numerical simulation.The initial conditions for master and slave system are givenby x(0) = (−5,−7,−10)T and y(0) = (0,−1,−1)T ,respectively. Fig. 3 shows that error signals go to zeroasymptotically. It means synchronization occurs betweenstate of xi(t) and state of yi(t), (i = 1, 2, 3).

−30−20

−100

1020

30

−30

−20

−10

0

10

20

30−18

−16

−14

−12

−10

−8

−6

−4

−2

x1

x2

x 3

Fig. 1: Chaotic motion of Chen-Lee system

−20

−10

0

10

20

−30

−20

−10

0

10

20

30−10

0

10

20

30

40

50

y1

y2

y 3

Fig. 2: Chaotic motion of Lorenz system

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−2

0

2

4

6

e 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−2

0

2

4

6

e 2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−5

0

5

10

15

e 3

time

Fig. 3: Error signals of numerical example

5. Circuit design and analysisIn this section, we present circuit design and analysis

for proposed synchronization scheme. As previously stated,chaotic systems have some errors between theoretical systemparameters and practical system parameters. So we willconduct some process for elimination of these errors.

5.1 Chen-Lee circuitFor the circuit design of mathematical dynamic model (3),

we use transformed Chen-Lee chaotic system because ofsome problems. Based on electronic circuit of Eq.(3), therange of state variables is over the limit of power supply. So,the reasonable transformation is to multiply 10 by nonlinearterm.

−3 −2 −1 0 1 2 3−4

−2

0

2

4

x1

x 2

−3 −2 −1 0 1 2 3−2

−1.5

−1

−0.5

0

x1

x 3

−4 −2 0 2 4−2

−1.5

−1

−0.5

0

x2

Chen−Lee system of Eq.(12)

x 3

−40 −20 0 20 40−40

−20

0

20

40

x1

x 2

−40 −20 0 20 40−20

−15

−10

−5

0

x1

x 3

−40 −20 0 20 40−20

−15

−10

−5

0

x2

Chen−Lee system of Eq.(3)

x 3

Fig. 4: Comparing with original Chen-Lee and modifiedChen-Lee systems

Consider the following transformed Chen-Lee equations

x1(t) = ax1(t)− 10x2(t)x3(t)

x2(t) = −bx2(t) + 10x1(t)x3(t)

x3(t) = −cx3(t) +10

3x1(t)x2(t), (12)

where a = 5, b = 10, c = 3.8.This system can be more easily operated with analog circuitbecause all the state variables gave similar dynamic rangeand circuit voltages remain well within the range of typicalpower supply limits. In order to present effect of previousprocess, Fig. 4 is given which shows phase to phase portraitof original Chen-Lee system (3) and modified Chen-Leesystem (12) of x1 − x2, x1 − x3, x2 − x3 respectively. In

Fig. 4, we can note that the state value of modified Chen-Lee system (12) is similar the state value of original Chen-Lee system (3) divided by 10 but inherent chaotic behavioris not changed. It means we can use the transformed Chen-Lee system (12) for our synchronization scheme because thisprocess keep the range of state variables less than the limitof electronic device and transformed equations behave samechaotic motions. The analog circuit of transformed Chen-LeeEq.(12) is shown in Fig. 5.

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B B

C C

D D

E E

F F

G G

H H

I I

A2Y

XR2

R1

R4

A3Y

XR6

R5

C3

A4Y

XR12

R14

R11 R13

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U1A

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U1B

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R32

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C1

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R7

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R85

C2

VCC

VCC

VCC

VCC

VEE

VEE VEE

VEE

VEE

VEE

VCC

VCC

11

0

R9

R10

VCC

VCC

VEE

VEE

14

0

10

12

16R15

R16

18

0

0

0

15

19

17

20

13

Fig. 5: The circuit of Chen-Lee system

The electrical equations of the circuit are given by

x1 =1

R4C1

(R3

R1x1 −

R3R10

R2R9x2x3

)x2 =

1

R8C2

(−R7R10

R5R9x2 +

R7

R6x1x3

)x3 =

1

R14C3

(−R13R16

R11R15x3 +

R13

R12x1x2

), (13)

where we can note that Eq.(13) is equivalent to Eq.(12)after some calculation and applying the required electricalparameters such as: R2, R5, R6 = 10kΩ; R1 = 20kΩ;R4, R8, R14 = 1MΩ; R3, R7, R9, R10, R11, R13, R15 =100kΩ; R12 = 30kΩ; R16 = 380kΩ; Ci = 1µF, (i =1, 2, 3). The operational amplifiers are considered to be ideal,the time step is 0.001 [s] and the initial condition of mastercircuit is x(0) = (0.02, 0.02, 0.02) [V]. Fig. 6 displays phaseto phase portrait of master system of x1 − x2, x1 − x3,x2−x3, respectively, in left side and time to state x1, x2, x3,respectively, in right side.

Fig. 6: Chaotic phase of Chen-Lee system

5.2 Lorenz circuit

As the same reason, we transformed Lorenz chaoticsystem(4) into following equations

y1(t) = a1(y2(t)− y1(t))

y2(t) = b1y1(t)− 20y1(t)y3(t)− y2(t)

y3(t) = 20y1(t)y2(t)− c1y3(t), (14)

where a1 = 10, b1 = 28, c1 = 8/3.

−20 −10 0 10 20−40

−20

0

20

40

y1

y 2

−20 −10 0 10 200

20

40

60

y1

y 3

−40 −20 0 20 400

20

40

60

y2

Lorenz system of Eq.(4)

y 3

−1 −0.5 0 0.5 1−2

−1

0

1

2

y1

y 2

−1 −0.5 0 0.5 10

1

2

3

y1

y 3

−1.5 −1 −0.5 0 0.5 1 1.50

1

2

3

y2

Lorenz system of Eq.(14)

y 3

Fig. 7: Comparing with original Lorenz and modified Lorenzsystems

As comparing with Eq.(4), the transformed equation ischanged nonlinear terms which are multiplied by 20. Fig. 7displays phase to phase portrait of original Lorenz system(4) and modified Lorenz system (14) of x1 − x2, x1 − x3,x2 − x3 respectively. Like the preceding, we can note thatthe state value of modified Lorenz system (14) is similar thestate value of original Lorenz system (4) divided by 20. Butwe can also know inherent chaotic behavior is not changed.The analog circuit of transformed Lorenz equation (14) isshown in Fig. 8.

The electrical equations of the circuit are given by

y1 =1

R5C1

(R4

R1y2 −

R3

R2 +R3

(1 +

R4

R1

)y1

)y2 =

1

R13C2

(R12

R11y1 −

R12R9

R10R8y2 −

R12R9

R10R7y1y3

)y3 =

1

R18C3

(R17

R16y1y2 −

R17R14

R15R6y3

), (15)

where we can note that Eq.(15) is equivalent to Eq.(14)after rescaling time by a factor of 100. And the requiredelectrical parameters are as following: R1, R2, R11 = 10kΩ;R3, R4, R8, R9, R15, R17 = 100kΩ; R5, R13, R18 = 1MΩ;R6 = 300kΩ; R7, R16 = 5kΩ; R10, R12 = 280kΩ; R14 =800kΩ; Ci = 1µF, (i = 1, 2, 3). The operational amplifiersare considered to be ideal, the time step is 0.001 [s] and theinitial condition of master circuit is x(0) = (0.01, 0.01, 0.01)[V]. Fig. 9 displays phase to phase of master system ofy1 − y2, y1 − y3, y2 − y3, respectively, in left side and timeto state y1, y2, y3, respectively, in right side.

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3

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A A

B B

C C

D D

E E

F F

G G

H H

U1A

3

2

4

8

1

U1B

5

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U2A

3

2

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1

U2B

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U3A

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U3B

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U4B

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R1

R2

R3

R4

VEE

VEE

VCC

VCC

0

23

22R5

24

C1

VEE

VEE

VCC

VCC

25

0

A1Y

XR7

R8 R9

R10

R11

R12

27

28

0

29

VCC

VCC

VEE

VEE

30

0

R1331

C2

VCC

VCC

VEE

VEE

32

A2Y

X

R6

R14

R15

R16

R17

R1834

38

35

0

36

37

C3

39

0

0

0

2

1

4

Fig. 8: The circuit of Lorenz system

Fig. 9: Chaotic phase of Lorenz system

5.3 Synchronization circuitAs transforming Eqs. (3) and (4) to Eqs.(12) and (14),

respectively, the control inputs of Theorem 1. should be alsochanged as follows:

u1 = −10x2x3 − a1y2 + (a1 + a)x1

u2 = 20y1y3 + 10x1x3 − b1y1 − (b− 1)y2

u3 = −20y1y2 +10

3x1x2 − (c− c1)y3. (16)

Fig. 10: Simulation results without control

To show the effect of control input, first of all, we runthe circuit without control inputs. Fig. 10 displays phase tophase and time to phase portraits of master and slave systemsfor this case. One can see that the errors do not approach tozero as expected since the control inputs are not applied.

Finally, the circuit of the whole synchronizing systemis given in Fig. 11. The circuit consists of three parts:master systems, slave systems, and controllers. Then, Fig.12 displays that synchronization between Chen-Lee chaoticsystem and Lorenz chaotic system is achieved by controlinputs as expected.

6. ConclusionIn this paper, we have investigated the synchronization

problem for the Chen-Lee and Lorenz chaotic systems. Ourproposed control scheme is verified by numerical simula-tion of the system. It should be noted that we includedcircuit analysis for the synchronization between the differentchaotic systems for the first time.

AcknowledgementsThis research was supported by Basic Science Research

Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science andTechnology (2010-0009373).

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VEE VEE

VEEVEE

VCC VCC

VCCVCC

R24

C4

A1Y

XR26

R25 R27

R28

R29

R30

R34

C5

A5Y

X

R37

R38

R39

R35

R36

R43

C6

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0

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31

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VEE

VCC

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VEE

VCC

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VCC

VEE21

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VCC

VEE U9A

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VCC

VCC

VCC

VCC

VEE

VEE

VEE

VEE

R23

R33

R32

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A10Y

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A13Y

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A14Y

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VCC

VCC

VEE

VEE

R44

R45

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R54

R61

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VEE

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VCC

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R50

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40

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20

65

8

37

36

56

59

63

Master system

Slave system

Controller

Fig. 11: The circuit for controlled systems

Fig. 12: Simulation results with control

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[2] U.E. Vincent, Synchronization of Rikitake chaoticattractor using active control. Physics Letters A 343,133-138, 2005.

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