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Adv. Studies Theor. Phys., Vol. 6, 2012, no. 10, 497 - 509

Numerical Simulation of Unidirectional Chaotic

Synchronization of Non-Autonomous Circuit and

its Application for Secure Communication

Mustafa Mamat1, Zabidin Salleh

1, Mada Sanjaya WS

1,4,

Noor Maizura Mohamad Noor2

and Mohd Fadhli Ahmad3

1Department of Mathematics2Department of Computer Sciences

3Department of Maritime Technology

Universiti Malaysia Terengganu, Kuala Terengganu, Malaysia4Department of Physics, Universitas Islam Negeri Sunan Gunung Djati

Bandung, Indonesia

Abstract. The nonlinear chaotic non-autonomous fourth order system is

algebraically simple but can generate complex chaotic attractors. In this paper,

non-autonomous fourth order chaotic oscillator circuits were designed and

simulated. Also chaotic non-autonomous attractor is addressed suitable for chaotic

masking communication circuits using Matlab and MultiSIM programs. We

have demonstrated in simulations that chaos can be synchronized and applied to

signal masking communications. We suggest that this phenomenon of chaos

synchronism may serve as the basis for little known chaotic non-autonomous

attractor to achieve signal masking communication applications. Simulation

results are used to visualize and illustrate the effectiveness of non-autonomous

chaotic system in signal masking. All simulations results performed on

non-autonomous chaotic system are verify the applicable of secure

communication.

Keywords: Chaotic synchronization, unidirectional coupling, double bell

attractor, non-autonomous chaotic circuit, secure communication

Introduction

There are two types of chaotic systems, autonomous and non-autonomous.

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498 M. Mamat et al

Although there are many known autonomous chaotic oscillators very few

non-autonomous have been introduced in the literature. Non-autonomous chaotic

circuits form a class of systems which produce chaos while being driven by an

external time varying source. The amplitude and frequency of the sinusoidal

signal both contribute to the chaotic dynamics of the system.

Chaos behavior can occur everywhere, even in very simple and low-dimensional

nonlinear systems. The well known Poincare-Bendixon theorem [1,2], requires

an autonomous continuous time state space model to be at least three-dimensional

in order to have bounded chaotic solutions. On the other hand, for

non-autonomous systems, chaos can appear in two-dimensional models. There are

many examples, such as Lorenz [3], Rssler [4] systems that have been widely

studied. Electronic circuits that consist of two nonlinear elements can be used toverify theoretical predictions. As an example, nonlinear Duffing forced oscillators

have been experimentally studied [5]. Chaotic and chaotic synchronization in

ecological and biological neural networks have been numerically studied [6-9].

Another popular example is the nonlinear chua's circuit, built and experimentally

examined [10,11]. Chaos and chaotic systems have many fields of applications.

One of the popular practical application is secure communication.

Synchronization of chaotic systems and chaos based secure communications have

become an area of active research in recent years[12-16]. Different approaches are

proposed and being pursued.

Chaotic signals depend very sensitively on initial conditions, have unpredictablefeatures and noise like wideband spread spectrum. So, it can be used in various

communication applications because of their features of masking and immunizing

information against noise. The fundamental of chaos communication are the

synchronization of two chaotic systems under suitable conditions if one of the

systems is driven by the other. Since Pecora and Carrol [17,18] have demonstrated

that chaotic systems can be synchronized, the research in synchronization of

couple chaotic circuits is carried out intensively and some interesting applications

such as communications with chaos have come out of that research.

This paper focuses on design of non-autonomous chaotic oscillator and signal

masking circuits. The brief is organized as follows. In Section II, mathematicalmodels of the non-autonomous chaotic oscillator system are studied. In Section III,

numerical simulation and MultiSIM circuit design and their simulations of the

non-autonomous chaotic oscillator system are obtained. In Section IV, the

unidirectional coupling method is applied to synchronize non-autonomous chaotic

oscillator system. In Section V, chaotic masking communication circuits and their

simulations of the non-autonomous chaotic oscillator system are realized also

Matlab and MultiSIM. Section VI contains conclusions.

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Numerical simulation of unidirectional chaotic synchronization 499

Figure 1: 4th order non-autonomous chaotic circuit models[19].

Mathematical Models of Non-Autonomous Chaotic Circuit

The low frequency response of a 4th order non autonomous, nonlinear electronic

circuit has been studied. The electronic circuit consists of two active elements,

one linear negative conductance and one nonlinear resistor exhibiting a

symmetrical piecewise linearv-i characteristic of N-type. The circuit contains also

two capacitances C1 and C2, two inductances L1 and L2 and a sinusoidal input

source Vs(t) [19]. Applying Kirchoffs law, the non-autonomous circuit is

described by four differential equations:

+=

=

+=

=

)(2222

2

11121

1

1222

2

11 1

tVrivdt

diL

rivvdt

diL

iivgdt

dvC

iidt

dvC

SLCL

LCCL

LLCnC

NLC

(1)

where ftvtV mS 2sin)( = is the input sinusoidal signal with amplitude vm andfrequency f, while R2

denotes the internal resistance of the source with the

function g(vC1) is defined

[ ]PCPCCCN BvBvmmvmvgi ++== 1101101 )(2

1)( (2)

where1m and 0m are the slopes in the inner and outer regions, respectively, and

PB denote the breakpoints.

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500 M. Mamat et al

Figure 2: Nonlinear resistor function iN[19].

Numerical Simulation and Circuit Implementation

We present numerical simulation to illustrate the dynamical behavior of

non-autonomous chaotic circuit from system (1). For numerical simulation of

chaotic systems defined by a set of differential equations such as non-autonomous

chaotic circuit, different integration techniques can be used in simulation tools. In

the Matlab numerical simulations, ODE45 solver yielding a fourth-order

Runge-Kutta integration solution has been used.

According to these numerical simulations, the circuits chaotic dynamics and

double-bell attractors are shown in Figure 3. For showing the dynamics of the

system (1), the parameter set given as fixed parameters, see Table 1. We have

studied systems response in low frequencies range. Particularly, the numerical

simulation and experimental simulation of phase portraits vC2 vs. vC1 for thefrequency f= 50 Hz and the amplitude vm= 2 volt of input sinusoidal signal VS (t)

are shown in this section.

Table 1: Circuit parameters.

Element Description Value Tolerance

L1 Inductor 100mH %10L2 Inductor 300mH %10C1 Capacitor 33nF %5C2 Capacitor 75nF %5R1 Resistor 1k %5R2 Resistor 1 %5R3 Resistor 2k %5R4 Resistor 2k %5R5 Resistor 2k %5R6 Resistor 1k %5R7

Resistor

15.5k

%5

R8 Resistor 4.1k %5R9 Resistor 297k %5U1,2 TL082CD

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Numerical simulation of unidirectional chaotic synchronization 501

gn Negative

Resistor

-0.475

ms

m0 Gradient 5 ms

m1Gradient

-0.35

ms

BP Breakpoint

voltage

1.2

volt

Now we shall prove that the strange attractor shown in Figure 3. is actually

chaotic in nature [20]. For this we will first calculate all the Lyapunov exponentsassociated with the strange attractor shown in Figure 3(a). The spectrum of

Lyapunov exponent is shown in Figure 3(b). One can see that the largest

Lyapunov exponent thus calculated is positive, showing that the strange attractor

is chaotic in nature.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-6

-4

-2

0

2

4

6

Vc1

Vc2

Phase portrait non-autonomous chaotic circuit

0 5 10 15 20 25 30 35 40-50

-40

-30

-20

-10

0

10

20Dynamics of Lyapunov exponents

Time

Lyapunov

exponents

(a) Phase portrait vC1 vs vC2

(b) Spectrum Lyapunov Exponent

0 0.02 0.04 0.06 0.08 0.1-4

-3

-2

-1

0

1

2

3

4

Times

Vc1

Time series non-autonomous chaotic circuit

0 0.02 0.04 0.06 0.08 0.1

-10

-5

0

5

10

Times

Vc2

Time series non-autonomous chaotic circuit

(c) Time series vC1

(d) Time series vC2

Figure 3: Numerical simulation results forf= 50 Hz and vm = 2 volt.

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502 M. Mamat et al

V1

2000mV

50 Hz

0Deg

L2300mH

U1

OPAMP_3T_VIRTUAL

R3

2k

R1

1k

C133nF

R7

15.5k

R6

1k

U2

OPAMP_3T_VIRTUAL

R21

R4

2k

R5

2k

R9297

R8

4.1k

L1

100mH

C275nF

Figure 4: Implementation of non-autonomous chaotic circuit.

(a) Phase portrait vC1 vs vC2

(b) Time series vC1

(c) Time series vC2

Figure 5: MultiSIM simulation results forf= 50 Hz and vm = 2 volt.

The complete implementation of the non-autonomous chaotic circuit design using

MultiSIM software is shown in Figure 4. The function of nonlinear resistor as

see in Figure 2, are implemented with the analog operational amplifier. By

comparing Figure 3, and Figure 5, it can be concluded that a good qualitative

agreement between the numerical integration of (1) using Matlab, and the circuits

simulation using MultiSIM.

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Numerical simulation of unidirectional chaotic synchronization 503

Unidirectional Chaotic Synchronization

Synchronization between chaotic systems has received considerable attention and

led to communication applications. With coupling and synchronizing identicalchaotic systems methods, a message signal sent by a transmitter system can be

reproduced at a receiver under the influence of a single chaotic signal through

synchronization. This work presents the study of numerical simulation of chaos

synchronization for non-autonomous chaotic circuit.

Synchronization of chaotic motions among coupled dynamical systems is an

important generalization from the phenomenon of the synchronization of linear

system, which is useful and indispensable in communications. The idea of the

methods is to reproduce all the signals at the receiver under the influence of a

single chaotic signal from the driver. Therefore, chaos synchronization provides

potential applications to communications and signal processing. However, tobuild secure communications system, some other important factors, need to be

considered [21].

Numerical Simulation

We use the same parameters as in Table 1, all parameters are identical except for

their control value vm, in which the transmitter system vm1 is 2.001 V and the

receiver system vm2 is 2.000 V. The following master-slave (unidirectional

coupling) configuration, as described in [21], is used:

( )

( )

+=

=

+=

+=

+=

=

+=

=

tfvrivdt

diL

rivvdt

diL

iivgdt

dvC

vvgiidt

dvC

tfvrivdt

diL

rivvdt

diL

iivgdt

dvC

iidt

dvC

mLCL

LCCL

LLCnC

CCCNLC

mLCL

LCCL

LLCnC

NLC

2222222222

22

1212122212

12

122222222

22

12111212

12

1122221

21

1111112111

11

112121121

21

11111

11

2sin

)2sin(

2

(3)

withCC Rg /1= the coupling strength andRcare variable resistors, see Figure 7.

The asymptotic synchronized situation is defined as

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504 M. Mamat et al

0)()(lim 1211 = tvtv CCt (4)

First synchronization between identical systems is considered. We consider

coupling throughCC Rg /1= . It can be seen in Figure 6 that synchronization occurs if

Rc does not exceed 1 K.

.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Vc11

Vc12

Unidirectional Chaotic Synchronization

0 0.02 0.04 0.06 0.08 0.1

-3

-2

-1

0

1

2

3

Time

e

=V

c11-Vc12

Unidirectional Chaotic Sy nchronization

(a)RC= 10 k

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Vc11

Vc12

Unidirectional Chaotic Synchronization

0 0.02 0.04 0.06 0.08 0.1

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Time

e

=V

c11-Vc12

Unidirectional Chaotic Synchronization

(b)RC= 5 k

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Vc11

Vc12

Unidirectional Chaotic Synchronization

0 0.02 0.04 0.06 0.08 0.1

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Time

e

=

Vc11-Vc12

Unidirectional Chaotic Sy nchronization

(c)RC= 1 k

Figure 6: Unidirectional chaotic synchronization phase portrait and error

vC11-vC12 numerical results.

Synchronization numerically appears for a coupling strength 1CR k as

shown in Figure 6. For different initial condition, if the resistance coupling

strength 1>CR k, the synchronization cannot occur as shown in Figure 6(a),

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Numerical simulation of unidirectional chaotic synchronization 505

and Figure 6(b); Figure 6(b) shown that it takes longer time to achieve the

synchronization. The synchronization occurs when 1C

R k with errors

01211 = CC vve imply the complete synchronization for this resistance

coupling strength as shown in Figure 6(c).

Analog Circuit Simulation

Figure 7 shows the circuit schematic for implementing the unidirectional

synchronization of non-autonomous chaotic circuit system. We use TL082CD

op-amps, appropriate valued resistors, inductor and capacitors for MultiSIM

simulations. Figure 8 also shows MultiSIM simulation results of this circuit.

V1

2001mV

50 Hz

0Deg

L2300mH

U1

OPAMP_3T_VIRTUAL

R3

2k

R1

1k

R21

R4

2k

R5

2k

L1100mH

C275nF

C175nF

R7

15.5k

R6

1k

U2

OPAMP_3T_VIRTUAL

R9297

R8

4.1k

V2

2000mV

50 Hz

0Deg

L3300mH

U3

OPAMP_3T_VIRTUAL

R10

2k

R11

1k

R121

R13

2k

R14

2k

L4

100mH

C375nF

C475nF

R15

15.5k

R16

1k

U4

OPAMP_3T_VIRTUAL

R17297

R18

4.1k

R191

Unidirectional Chaotic Sy nchronization

U5

OPAMP_3T_VIRTUAL

Figure 7: Unidirectional chaotic synchronization non-autonomous circuit.

(a)RC= 10 k

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506 M. Mamat et al

(b)RC= 5 k

(c)RC= 1 k

Figure 8: Unidirectional chaotic synchronization phase portrait and time series

results.

Synchronization with MultiSIM simulation appears for a coupling strength

1CR k as shown in Figure 8. For different initial condition, if the resistance

coupling strength 1>CR k, the synchronization cannot occur as shown in

Figure 8(a) and Figure 8(b); the synchronization occurs when 1CR k with

errors 01211 = CC vve imply the complete synchronization for this resistance

coupling strength as shown in Figure 8(c).

Application for Secure Communication Systems

Due to the fact that output signal can recover input signal, it indicates that it is

possible to create secure communication for a chaotic system. The presence of the

chaotic signal between the transmitter and receiver has proposed the use of chaos

in secure communication systems. The design of these systems depends as we

explained earlier on the self synchronization property of the chaotic

non-autonomous attractor. Transmitter and receiver systems are identical except

for their control value vm, in which the transmitter system is 2.001 V and the

receiver system is 2.000 V as shown in Figure 9.

It is necessary to make sure the parameters of transmitter and receiver are

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Numerical simulation of unidirectional chaotic synchronization 507

identical for implementing the chaotic masking communication [12-16]. In this

masking scheme, the square wave signals of amplitude 2V and frequency 0.5 kHz

is added to the synchronizing driving chaotic signal in order to regenerate a clean

driving signal at the receiver. Thus, the message has been perfectly recovered by

using the signal masking approach through synchronization in the chaotic

non-autonomous attractor. Computer simulation results have shown that the

performance of chaotic non-autonomous attractor in chaotic masking and message

recovery.

The square wave signal is added to the generated chaotic x signal, and the S(t) =x

+ i(t) is feed into the receiver. The chaotic x signal is regenerated allowing a

single subtraction to retrieve the transmitted signal, [x+i(t)]-xr = i(t), Ifx = xr.

Figure 9 shows the circuit schematic for implementing the chaoticnon-autonomous attractors Chaotic Masking Communication. Figure 10 shows

MultiSIM simulation results of this Chaotic Masking Circuit.

V1

2001mV

50 Hz

0Deg

L2300mH

U1

OPAMP_3T_VIRTUAL

R3

2k

R1

1k

R21

R4

2k

R5

2k

L1

100mH

C275nF

C175nF

R7

15.5k

R6

1k

U2

OPAMP_3T_VIRTUAL

R9297

R8

4.1k

V2

2000mV

50 Hz

0Deg

L3300mH

U3

OPAMP_3T_VIRTUAL

R10

2k

R11

1k

R121

R13

2k

R14

2k

L4

100mH

C375nF

C475nF

R15

15.5k

R16

1k

U4

OPAMP_3T_VIRTUAL

R17297

R18

4.1k

U6 U7

U8R20

1kR21

1k

R221k

R231k

R24

1kR251k

R26

1k

U9U10

R27

1k

R28

1k

R29

1k

R301k

V6

-1 V 2 V

0.5msec 2msec

i(t) S(t)

i'(t)

Adder

Substractor

Buffer

R19

1

U5

OPAMP_3T_VIRTUAL Figure 9: Non-autonomous chaotic attractor masking communication circuit.

(a)

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508 M. Mamat et al

(b)

(c)

Figure 10: MultiSIM outputs of non-autonomous chaotic attractor Masking

Communication Circuit: (a) Information signal i(t); (b) chaotic maskingtransmitted signal S(t); (c) retrieved signal i(t).

Conclusions

This paper focuses on the chaotic oscillator circuit and the identical

synchronization of the Fourth order non-autonomous chaotic attractor and its

applications in signal masking communications. In this paper, non-autonomous

chaotic circuit system is studied in detail, the system has rich chaotic dynamics

behaviors. We have demonstrated in simulations that chaos can be synchronized

and applied to secure communications. We suggest that this phenomenon of chaos

synchronism may serve as the basis for little known non-autonomous chaotic

attractor to achieve secure communication. Chaos synchronization and chaos

masking were realized using MultiSIM programs.

Acknowledgement. The authors gratefully acknowledge the financial support

from the Ministry of Higher Education, Malaysia under the FRGS Vot 59173.

References

[1] Alligood, K.T., Sauer, T.D., & Yorke, J.A. 1996. Chaos: An Introduction toDynamical Systems. Springer-Verlag, New York.

[2] Hilborn, H.C. 1994. Chaos and Nonlinear Dynamics. Oxford UniversityPress, New York.

[3] Lorenz, E.N. 1963. Deterministic non-periodic flow. Journal of theAtmospheric Sciences, vol. 20, pp. 130141.

[4] Rossler, O.E. 1976. An equation for continuous chaos. Phys. Lett. A 57,397-398.

[5] Volos, C.K., Kyprianidis, I.M., & Stouboulos, I.N. 2007.Synchronization of two Mutually Coupled Duffing type Circuits,

International Journal of Circuit, Systems and Signal processing, 1(3),

274-281.

7/31/2019 mamatASTP9-12-2012

13/13

Numerical simulation of unidirectional chaotic synchronization 509

[6] Mada, S.W.S., Mamat, M., Salleh, Z., & Mohd, I. 2011. BidirectionalChaotic Synchronization of Hindmarsh-Rose Neuron Model. Applied

Mathematical Sciences, 5(54), 2685 2695.

[7] Mada, S.W.S., Mamat, M., Salleh, Z., Mohd, I., & Muhammad Noor, N.M.2011. Numerical Simulation Dynamical Model of Three Species Food

Chain with Holling Type-II Functional Response. Malaysian Journal of

Mathematical Sciences 5(1),1-12.

[8] Mamat, M., Mada, S.W.S., Salleh, Z., & Ahmad, M.F. 2011. NumericalSimulation Dynamical Model of Three Species Food Chain with

Lotka-Volterra Linear Functional Response, Journal of Sustainability

Science and Management, 6(1), 44-50.

[9] Salleh, Z., Mada, S.W.S., Mamat, M., & Noor, N.M.M., 2011. TheDynamics of a Three Species Food Chain Interaction Model withMichaelis-Menten Type Functional Response, Journal of Sustainability

Science and Management, 6(2), 215-223.

[10] Matsumoto, T. 1984. A chaotic attractor from Chua's circuit,IEEE Trans.Circuits Syst., CAS 31(12),1055-1058.

[11] Kennedy, M.P. 1992. Robust Op Amp Implementation of Chuas Circuit.Frequenz, 46(3-4), 66-80.

[12] Cuomo, K.M., & Oppenheim, A.V. 1993. Circuit implementation ofsynchronized chaos with applications to communications, Physical Review

Letters, 71(1), 6568.

[13] Feng, J.C., & Tse, C.K. 2007. Reconstruction of Chaotic Signals withApplications to Chaos-Based Communications. Tsinghua University Pressand World Scientific Publishing Co. Pte. Ltd.,

[14] Lee, T.H., & Park, J.H. 2010. Generalized functional projectivesynchronization of Chen-Lee chaotic systems and its circuit implementation.

International Journal of the Physical Sciences, 5(7), 1183-1190.

[15] Pehlivan, I., Uyaroglu, Y., & Yogun, M. 2010. Chaotic oscillator design andrealizations of the Rucklidge attractor and its synchronization and masking

simulations. Scientific Research and Essays, 5(16), 2210-2219.

[16] Mada, S.W.S., Maulana, D.S., Mamat, M., & Salleh, Z. 2011. NonlinearDynamics of Chaotic Attractor of Chua Circuit and Its Application for

Secure Communication.J.Oto.Ktrl.Inst(J.Auto.Ctrl.Inst), 3 (1),1-16.

[17] Pecora, L., & Carroll, T. 1990. Synchronization in Chaotic Systems.Physical Review Letters, 64, 821-823.

[18] Pecora, L., & Carroll, T. 1991. Driving systems With Chaotic Signals.Physical Review Letters, 44, 2374-2383.

[19] Papadopoulou, M.S., Kyprianidis, I.M., & Stouboulos, I.N. 2008. ComplexChaotic Dynamics of the Double-Bell Attractor, WSEAS Transactions on

Circuit and Systems, 7, 13-21.

[20] Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A. 1985. DeterminingLyapunov Exponents From a Time Series. Physica D, 16, 285-317.

[21] Kapitaniak, T. 2000. Chaos for engineers. 2nd ed. Springer-Verlag, Berlin.