RESPONSE OF COUPLED CHAOTIC CIRCUITS TO SINUSOIDAL INPUT SIGNAL
Tomoya HAYASHI†, Yoshifumi NISHIO†, Martin HASLER‡ and Akio USHIDA††Tokushima University, Japan‡Swiss Federal Institute of Technology Lausanne, Switzerland
ABSTRACTIn this study, synchronization state of the coupled chaoticcircuits is investigated when a sinusoidal input signal is added
by both of computer simulations and circuit experiments. It
is confirmed that the synchronization state changes by thevalues of the amplitude and the angular frequency of theinput signal.
1. INTRODUCTION
Studies on engineering applications of chaos, such as chaos
communication systems and chaos cryptosystems, attract
many researchers’ attentions. Since, theoretically speaking, chaotic signals possess infinite information, it could bepossible to create novel engineering systems with great advantages. However, at this moment, there have been few
systems with some kinds of advantages over conventional
engineering systems. Hence, basic researches on chaotic
phenomena with future engineering applications in mind arestill quite important.
In this study, we investigate synchronization phenomena of two chaotic circuits coupled by a resistor when aninput signal is added. Because the circuit tends to minimize
the current through the coupling resistor in such a coupledsystem [1]-[3], synchronization phenomena change to compensate the effect of the input signal. More concretely, we
use a sinusoidal signal with the averaged amplitude and theaveraged frequency of the chaotic signals as a standard input signal. We investigate the synchronization phenomena
when the amplitude or the frequency of the sinusoidal input
signal is varied. The results suggest that coupled chaotic
circuits could be utilized to extract some kinds of featuresof unknown input signals.

3dIk
Ij
= rik + rIk − vk − R
dt
j=1
dik
L2
= rik + rIk − vk − vd (ik )dtdvk
C= ik + Ik
dt
(k = 1, 2)(1)

vd (ik ) = 0.5(rdik + E − |rd ik − E|)

(2)

L1

I1

I1

i1

I3
L1

L1

I2
i2

i1
C

v1

L2

-r

L1

C

v1-r

vd

C

L2

-r

vd

v2

L2
vd

R
(a)

(b)

Fig. 1. Circuit Model.

V1

I1

2. CIRCUIT MODEL

Fig. 2. An example of chaotic attractors. α = 7.0, β = 0.14
and δ = 100.

Figure 1 shows the circuit model. The circuit in Fig. 1(a)is a three-dimensional autonomous circuit generating chaos
shown in Fig. 2 and was proposed by Inaba and Mori [4].
In the circuit in Fig. 1(b), two Inaba’s circuits are coupledby one coupling resistor R and an input signal is added tothe coupling resistor as a current I 3 .
The circuit equations governing the circuit in Fig. 1(b)are given as Eq. (1). We assume the i − v characteristicsof the diodes in the circuit by the two-segment piecewise
linear function as Eq. (2).

By using the variables and the parameters in Eq. (3), thecircuit equations in Eq. (1) are normalized as Eq. (4).
CC
Ik =Exk , ik =Eyk ,L1
L1

L1vk = Ezk , t = L1 Cτ, α =,(3)L2CCC, γ=R, δ = rd
β=r
L1
L1
L10-7803-7448-7/02/$17.00 ©2002 IEEE

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3dxkxj= β(xk + yk ) − zk − γdτdyk= α {β(xk + yk ) − zk − fd (yk )}
dτx2

x1

x2
x3

x3

x1x1

(4)

x2x3

dzk
= xk + ykdτx1, X2
x1, X2, X3(k = 1, 2)

x1+X2+X3τ(a)

It is possible to consider various kinds of signals as input
signal x3 in future, for example, simple signals represented
by basic mathematical functions, chaotic signals obtained
from other chaotic circuits, audio/visual signals in real engineering systems and so on. In this study, as the first step
toward the total understanding of the response of the coupled circuits to various input signals, we consider the casethat a simple sinusoidal signal is input. Namely, we use
the following sinusoidal function with the normalized amplitude A m and the normalized angular frequency ω as theinput x 3 .
x3 (τ ) = Am sin ωτ

x1x2

Z2

Z1

j=1

Z1

x1

x2

Z2

x2

x1

x2
x3

x3

x1

x1
x2
x3
x1, X2
x1, X2, X3
x1+X2+X3
τ

(b)

(5)

Z1

x2

x1

3. RESPONSE TO SINUSOIDAL INPUT

x1x2

Z2x2x3x3

x1

x1

Figure 3 shows the computer simulated results of Eq. (4)using the fourth-order Runge-Kutta method. At first, in order to understand the typical response of the coupled system, we fix the angular frequency of the input signal asω = 1.037 which is close to the value of the averaged
angular frequency of the chaotic signals obtained from thechaotic circuits. Further, the amplitude of the input signal
Am is chosen using the averaged amplitude of the chaotic
signals as a standard value.
Figure 3(a) shows the result for the case of A m = 0,
namely no input signal is added. The two chaotic signals
almost synchronize at anti-phase. Figure 3(b) shows the result for the case of Am = 1.17, namely the sinusoidal signal
with the averaged amplitude and the averaged angular frequency of the chaotic signal is input. In this case, the twochaotic signal x1 and x2 and the input signal x 3 almost synchronize at three-phase. Figure 3(c) shows the result forthe case of Am = 2.34, namely the sinusoidal signal with
about doubled amplitude of the chaotic signal is input. Evenin this case, the shapes of the chaotic attractors observedin the two chaotic circuits are not influenced by the input
signal. And the two chaotic signals almost synchronize at
in-phase and they almost synchronize to the input signal atanti-phase.
Next, we show the circuit experimental results in Fig. 4.
We can observe similar results to the computer simulatedresults.
4. DETAILED INVESTIGATIONIn the previous section, we fixed the angular frequency of
the input signal as the averaged angular frequency of thechaotic signal and investigated the response of the coupledx2x3x1, X2x1, X2, X3x1+X2+X3
τ(c)Fig. 3. Computer simulated results. α = 7.0, β = 0.14,
γ = 0.03, δ = 100 and ω = 1.037. (a) Am = 0. (b) Am =1.17. (c) Am = 2.34.
I [0.2 mA / div]

I [0.2 mA / div]

I1

I1

I2

I2

I3

I3
time [50 µs / div]

time [50 µs / div]

(a)

(b)

I [0.2 mA / div]

I1
I2
I3∗
time [50 µs / div](c)Fig. 4.
Circuit experimental results.
L 1 =100mH,
L2 =10mH, C=33nF and R=320Ω. (a) Am = 0 (no inputsignal). (b) Am = 1.17 (averaged amplitude of the chaoticsignals). (c) Am = 2.34 (doubled amplitude of the chaoticsignals). *In (c) vertical scale of I3 is doubled.

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[deg]

system only for the cases of the three typical values of theamplitude of the input signal. In this section these values
are varied continuously.
In order to investigate the synchronization in detail, wedefine the phase difference of the two chaotic signals. By
using the time when the chaotic signals take extrema as
shown in Fig. 5, the phase difference θ[deg] is defined asfollows.
τ20 − τ10× 360(6)θ=τ11 − τ10240180θ

900-9001

Am

22.5

22.5(a)
[deg]

210
180

Average ofx1

θ

900x2

-30

0

1Am(b)

τ1 0 τ 2 0

τ1 1Fig. 6. Phase difference θ as varying Am . α = 7.0, β =0.14, γ = 0.03, δ = 100 and ω = 1.037. (a) θ. (b) Averageof θ.

τFig. 5. Definition of the phase difference θ.
4.1. Variation of Amplitude
Figure 6(a) shows the results for the case that the angularfrequency of the input is fixed as ω = 1.037 and the amplitude Am is varied continuously. In the figure, 1500 data ofθ are plotted after the system settles to the steady state foreach Am . As observed in the previous section, θ is almost
around 180 ◦ (anti-phase) for Am = 0, θ is almost around120◦ (three-phase) for Am = 1.17 and θ is almost around 0 ◦(in-phase) for Am = 2.34. Further, between these typicalvalues of the amplitude, the phase difference continuouslyvaries.
Figure 6(b) shows the average of θ in Fig. 6(a). Because
the fluctuation caused by chaotic feature is averaged, we cansee the variation of θ more clearly.
The results in Fig. 6 suggest that it is possible to distinguish the amplitude of the input sinusoidal signal by thephase difference between the two chaotic signals.
4.2. Variation of Angular Frequency
Next, we fix the amplitude of the input signal as A m = 1.17
and vary the angular frequency ω. Figure 7(a) shows 1500
data of θ and Fig. 7(b) shows their average. In this casethe coupled system responds to the input clearly only forvery limited values of angular frequency. Namely, as wecan see from Fig. 7(b), the two chaotic signals almost synchronize at anti-phase for wide parameter region and they
synchronize to the input signal at three-phase only aroundω = 1.037.
Figures 7(c) and (d) are the magnifications of the partof Figs. 7(a) and (b), respectively. From the figures we cansee that the phase difference between the two chaotic signals varies from about 90 ◦ to about 120 ◦ when ω is varied
around 1.037.
The results in Fig. 7 suggest that the coupled system
could be useful as a filter to distinguish the frequency of theinput sinusoidal signal.

4.3. Variation of Amplitude and Angular Frequency
At last, Fig. 8 shows the results obtained by varying bothof the amplitude A m and the angular frequency ω. Figure8(a) and (b) shows the results for the cases of γ = 0.03
and γ = 0.06, respectively. We can see that the parameterregion where the coupled system respond to the input signal
is changed by the value of the coupling strength.

5. RESPONSE TO RECTANGULAR INPUT
Additionally, we confirm the response of the circuits to rectangular input. Figure 9 shows the computer simulated results when the rectangular signal with the same frequency
and amplitude as the sinusoidal signal in Fig. 3, is added tothe circuits as I3 . The two chaotic signals almost synchronize similarly. We show the circuit experimental results inFig. 10.

6. CONCLUSIONSIn this study, we have investigated the synchronization stateof the coupled chaotic circuits when an sinusoidal input signal was added by both of computer simulations and circuit
experiments. We confirmed that the synchronization statechanged by the values of the amplitude and the angular frequency of the input signal.
The results of this study suggest that the coupled chaotic
circuits could be useful to distinguish the sinusoidal input
signals. Our future researches are to investigate the responseof the coupled system to more complicated input signals andto utilize the system to extract some kinds of important features of unknown input signals.

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[deg]360(a)0θ

Average of θ

180

180
1.1

2.5
2.01.05

ω
003

ω

1.5
1.0

1.0

60.5

Am

0.95 0(a)

(b)

[deg]

360

0
Average of θ
180
1.1

Average of

θ

180

2.5
2.0

1.05

ω

1.5
1.0

1.0
0.5

Am

0.95 0003

ω

6Fig. 8. Phase difference θ as varying ω and Am . α = 7.0,
β = 0.14 and δ = 100. (a) γ = 0.03. (b) γ = 0.06.

(b)
[deg]

360

θx1x2Z2

Z1x2

180x2x1x3x3x1x10
1.02x21.035

ω

1.05x3(c)

x1, X2[deg]

360

x1, X2, X3
x1+X2+X3
τ(a)Average ofθ

180

Z1x2

x10
1.02

1.035

ω

x1x2

Z2

x1x2x3x3x1

1.05x2(d)x3Fig. 7. Phase difference θ as varying ω. α = 7.0, β = 0.14,
γ = 0.03, δ = 100 and Am = 1.17. (a) θ. (b) Average of θ.
(c) Magnification of (a). (d) Magnification of (b).

x1, X2
x1, X2, X3
x1+X2+X3
τ(b)

7. REFERENCES[1] Y. Nishio, K. Suzuki, S. Mori, and A. Ushida, “Synchronization in mutually coupled chaotic circuits,”
Proc. of ECCTD’93, vol.1, pp.637-642, Aug. 1993.
[2] Y. Nishio and A. Ushida, ”Spatio-Temporal Chaos inSimple Coupled Chaotic Circuits,” IEEE Trans. Circuits
and Syst. I, vol. 42, no. 10, pp. 678-686, Oct. 1995.
[3] Y. Nishio and A. Ushida, “Quasi-synchronization phenomena in chaotic circuits coupled by one resistor,”
IEEE Trans. Circuits and Syst. I, vol. 43, no. 6, pp. 491496, June 1996.
[4] N. Inaba and S. Mori, ”Chaotic Phenomena in Circuitswith a Linear Negative Resistance and an Ideal Diode,”
Proc. of MWSCAS’88, pp. 211-214, Aug. 1988.

Fig. 9. Computer simulated results. α = 7.0, β = 0.14,
γ = 0.03, and δ = 100. (a) Am = 1.17. (b) Am = 2.34.
I [0.2 mA / div]

I [0.2 mA / div]

I1

I1

I2

I2

I3

I3∗
time [50 µs / div](a)time [50 µs / div](b)

Fig. 10.
Circuit experimental results.
L 1 =100mH,
L2 =10mH, C=33nF and R=320Ω. (a) Am = 1.17 (averaged amplitude of the chaotic signals). (b) A m = 2.34
(doubled amplitude of the chaotic signals). *In (c) verticalscale of I3 is doubled.

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