International Symposium on Nonlinear Theory and its Applications
Xi’an, PRC, October 7–11, 2002

Information Transmission Using Signal Processing by Coupled Chaotic Circuits
Tomoya HAYASHI† , Yoshifumi NISHIO† , Martin HASLER‡ and Akio USHIDA††

Dept. of Electrical and Electronics Engineering,
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: hayatan@ee.tokushima-u.ac.jp‡

Dept. of Communication Systems,
Swiss Federal Institute of Technology Lausanne (EPFL)
EL-Ecublens, CH-1015 Lausanne, Switzerland

Abstract

2. Coupled chaotic circuits

At first, the coupled system with chaotic circuits investigated in the previous research is explained [4]. Figure 1
shows the circuit model. The circuit in Fig. 1(a) is athree-dimensional autonomous circuit generating chaotic
signal and was proposed by Inaba and Mori [5]. In thecircuit in Fig. 1(b), two Inaba’s circuits are coupled by
one coupling resistor R and an input signal is added tothe coupling resistor as a current I3 .
We assume the i − v characteristics of the diodes in
1. Introduction
the circuit by the two-segment piecewise linear function
Studies on engineering applications of chaos, such as as Eq. (1).
chaos communication systems and chaos cryptosystems,(1)
vd (ik ) = 0.5(rd ik + E − |rd ik − E|).
attract many researchers’ attentions. Because chaotic
signals possess infinite information, it could be possible By using the variables and the parameters in Eq. (2),to create novel engineering systems with great advan- the circuit equations are normalized as Eq. (3).
tages. However, at this moment, there are only a few
rr
CCsystems with some kinds of advantages over conventional
Ik =Exk , ik =Eyk ,engineering systems. Hence, basic researches on chaotic
L1L1pphenomena with future engineering applications in mind
L1(2)vk = Ezk , t = L1 Cτ, α =,are still quite important.
L2rr
r
On the other hand, the authors have researched aboutCC
C
β=r, γ=R, δ = rd.
many oscillatory circuits connected by a resistor [1]-[3].
L1
L1
L1
Using this connection of circuits, chaotic signals gener3
X
ated from chaotic circuits change the state of synchrodxk= β(xk + yk ) − zk − γxjnization to minimize the total of currents flowing in thedτ
j=1
coupled resistor. Moreover, we investigated phenomena
dykof the coupled chaotic circuits when a sinusoidal input= α {β(xk + yk ) − zk − fd (yk )}dτsignal is added [4]. The coupled system changes the state(3)of synchronization against the input signal with very limdzk
ited range of angular frequency. Therefore, this coupled= xk + ykdτsystem is useful for signal processing of input signals.
In this study, information transmission with coupled(k = 1, 2).
chaotic circuits can be considered. Chaotic signals in
In this study, we consider the case that a simple sithe coupled system synchronize depending on the amnusoidal signal is input. Namely, we use the following
plitude of input signals. We can transmit digital signals
sinusoidal function with the normalized amplitude Am
by amplitude shift keying. Computer simulations of digiand the normalized angular frequency ω as the input x3 .
tal signal transmission are carried out. The result of thisstudy suggests that it is possible to transmit information(4)x3 (τ ) = Am sin ωτ.
by the system with this coupled chaotic circuits.
In this study, information transmission with coupledchaotic circuits are considered. The coupled chaotic circuits are used as a receiver, and digital signals are transmitted by amplitude shift keying. The result of thisstudy suggests that it is possible to transmit information by the coupled chaotic circuits.

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I1

I1

i1

L1

I3
L1



I2
i2

i1
C

v1

L2

-r

vd

L1

C-r

v1

C

L2

v2-r

vd

L2vd

(b)

Figure 1: Chaotic circuit and the coupled chaotic circuits. α = 7.0, β = 0.14 and δ = 100.

  

!#"  



R(a)  τ  
      




2.1. Basic response to sinusoidal input signal
Figure 2 shows the computer simulated results ofEq. (3) using the fourth-order Runge-Kutta method. At
first, in order to understand the typical response of thecoupled system, we fix the angular frequency of the input
signal as ω = 1.037 which is close to the value of the averaged angular frequency of the chaotic signals obtained
from the chaotic circuits. Further, the amplitude of theinput signal Am is chosen using the averaged amplitudeof the chaotic signals as a standard value.
Figure 2(a) shows the result for the case of Am = 0,
namely no input signal is added. The two chaotic signals
almost synchronize at anti-phase. Figure 2(b) shows theresult 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 thiscase, the two chaotic signal x1 and x2 and the input signal x3 almost synchronize at three-phase. Figure 2(c)shows the result for the case of Am = 2.34, namely thesinusoidal signal with about doubled amplitude of thechaotic signal is input. Even in this case, the shapesof the chaotic attractors observed in the two chaotic circuits are not influenced by the input signal. And the twochaotic signals almost synchronize at in-phase and theyalmost synchronize to the input signal at anti-phase.
We show the circuit experimental results in Fig. 3. We
can observe similar results to the computer simulatedresults.

τ

τ

Figure 2: 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.
average of θ in Fig. 5(a). Because the fluctuation causedby chaotic feature is averaged, we can see the variationof θ more clearly. The results in Fig. 5 suggest that
it is possible to distinguish the amplitude of the input
sinusoidal signal by the phase difference between the twochaotic signals.
Next, we fix the amplitude of the input signal as
Am = 1.17 and vary the angular frequency ω. Figure 6(a) shows 1500 data of θ and Fig. 6(b) shows theiraverage. In this case the coupled system responds tothe input clearly only for very limited values of angularfrequency. Namely, as we can see from Fig. 6(b), thetwo chaotic signals almost synchronize at anti-phase forwide parameter region and they synchronize to the input signal at three-phase only around ω = 1.037. The
results in Fig. 6 suggest that the coupled system could
be useful as a filter to distinguish the frequency of theinput sinusoidal signal.
3. Signal detecting by coupled chaotic circuits

As the results shown previously, the coupled chaoticcircuits respond to only a certain limited signal. FigIn order to investigate the synchronization in detail, ure 7 shows how the coupled system can detect a sinuwe define the phase difference of the two chaotic signals. soidal signal from the multiplexed signal consisting of
By using the time when the chaotic signals take extrema two sinusoidal signals with different angular frequencies.
as shown in Fig. 4, the phase difference θ[deg] is defined In this simulations, two sinusoidal signals are added asas follows.
Eq. (6), and the coupled chaotic circuits detect the firstτ20 − τ10
θ=× 360.
(5) sinusoidal signal.
τ11 − τ102.2. Response to sinusoidal input signal

Figure 5(a) shows the results for the case that the(6)x3 (τ ) = Am sin ωτ + Am sin(ω + ∆ω)τ.
angular frequency of the input is fixed as ω = 1.037
and the amplitude Am is varied continuously. In the If the phase difference θ of the two chaotic signals isfigure, 1500 data of θ are plotted after the system settles around 120 degree, it means that the coupled system canto the steady state for each Am . Figure 5(b) shows the detect the first sinusoidal signal from the multiplexed sig--822-

[deg][deg]

I [0.2 mA / div]

210

240

180

180

I1



I2θθ

90

e of90I30time [50 µs / div]0(a)I [0.2 mA / div]

-900I [0.2 mA / div]

1Am

2-30

2.50(a)I1

I1I2I2I3I3∗

1Am

22.5(b)Figure 5: Phase difference θ as varying Am . α = 7.0,
β = 0.14, γ = 0.03, δ = 100 and ω = 1.037. (a) θ.
(b) Average of θ.

time [50 µs / div]

time [50 µs / div]

(c)

(b)

Figure 3: Circuit experimental results. L1 =100mH,
L2 =10mH, C=33nF and R=320Ω. (a) Am = 0 (no input signal). (b) Am = 1.17 (averaged amplitude of thechaotic signals). (c) Am = 2.34 (doubled amplitude of
the chaotic signals). *In (c) vertical scale of I3 is doubled.

[deg]

[deg]

360360

θ

θ

180003

ω

(a)

6e of

180003ω

6(b)Figure 6: 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.

x1x2

τ 10 τ 1 1

τ 20

τ

the case that four pairs of the coupled chaotic circuits
are not connected, namely each pair works as an detecFigure 4: Definition of the phase difference θ.
tor dependently. The results are shown in Fig. 10(b). In
this case, the coupled chaotic circuits cannot decode anynal. This result suggests that more than 10% difference information from the multiplexed signal.
of angular frequency of input signals is needed.
Figure 8 shows how the coupled system can detect asinusoidal signal when N sinusoidal signals are multi- 5. Conclusions
plexed as an input signal;
In this research, we have investigated that the coupledN
chaoticcircuits could detect sinusoidal signals. FurtherX
x3 (τ ) =Am sin{ω + (i − 1)∆ω}τ.
(7) more, we showed the example of information transmisi=1
sion using signal processing by the coupled chaotic circuits. It suggests that multiplexed information can be
The coupled chaotic circuits detect the first sinusoidaltransmitted by using this coupled system.
signal from the multiplexed signal for N ≤ 10.
It is necessary to consider the tolerance over the noiseof a system in detail. Moreover, the response speed of
4. Information transmission
this system is slow and is not practical. It is necessary
In this section, an example of information transmis- to consider how to improve.
sion using the signal processing by the coupled chaoticcircuits is shown.
References
In this simulations, 4 ASK signals are multiplexed andinput to the coupled chaotic circuits. At first, four pairs [1] Yoshifumi Nishio and Akio Ushida, “Spatioof coupled chaotic circuits are connected as shown in
Temporal Chaos in Simple Coupled Chaotic CirFig. 9. These coupled chaotic circuits are tuned to oscilcuits,” IEEE Transactions on Circuits and Systems
late with the averaged angular frequencies ω +(i−1)∆ω.
I, vol. 42, no. 10, pp. 678-686, Oct. 1995.
The simulated results are shown in Fig. 10(a), We cansee that each pair of coupled chaotic circuits can de- [2] Yoshifumi Nishio, Martin Hasler and Akio Ushida,“Markov Chain Modeling of Chaotic Wandering incodes the corresponding ASK signal. Next, we consider-823-θ     

∆ω

Figure 7: Phase difference θ as varying ∆ω. α = 7.0,β = 0.14, γ = 0.03, δ = 100, Am = 1.17, ω1 = 1.035 andω2 = 1.035 + ∆ω.

Figure 9: Circuits for information transmission.

x1(ASK sinusoidal signal 1)

θ




x2
(ASK sinusoidal signal 2)

x3(ASK sinusoidal signal 3)x4(ASK sinusoidal signal 4)





total of 4 signals

180| θ1 | [deg]Figure 8: Phase difference θ when N sinusoidal signals
are multiplexed as an input signal. ∆ω = 0.2, α = 7.0,
β = 0.14, γ = 0.03, δ = 100 and Am = 1.17.

0180| θ2 | [deg]0180| θ3 | [deg]Simple Coupled Chaotic Circuits,” Proc. of ECCTD’01, vol. 2, pp. 105-108, Aug. 2001.

0180| θ4 | [deg]0[3] Yoshifumi Nishio and Akio Ushida, “Chaotic Wandering and Its Analysys in Simple Coupled ChaoticCircuits,” IEICE Transactions on Fundamentals, vol.
E85-A no. 1, pp. 248-255, Jan. 2002.

τ



τ

x1
(ASK sinusoidal signal 1)
x2(ASK sinusoidal signal 2)
x3(ASK sinusoidal signal 3)[4] Tomoya Hayashi, Yoshifumi Nishio, Martin Hasler
and Akio Ushida, “Response of Coupled Chaotic Circuits to Sinusoidal Input Signal,” Proc. of ISCAS’02,vol. 3, pp. 461-464, May 2002.

x4(ASK sinusoidal signal 4)
total of 4 signals

180| θ1 | [deg][5] Naohiko Inaba and Shinsaku Mori, “Chaotic Phenomena in Circuits with a Linear Negative Resistance and an Ideal Diode,” Proc. of MWSCAS’88,
pp. 211-214, Aug. 1988.

0180| θ2 | [deg]0180| θ3 | [deg]0180| θ4 | [deg]0Figure 10: Information Transmission by multiplexing 4ASK signals. (a) Coupled chaotic circuits are connected,(b) not connected. α = 7.0, β = 0.14, γ = 0.03, δ = 100,
Am = 1.17 or 2.34 and ∆ω = 0.05.

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