IEICE TRANS. FUNDAMENTALS, VOL.E84–A, NO.12 DECEMBER 2001

3099

PAPER

Synchronization and Its Analysis in Chaotic Systems
Coupled by Transmission Line
Junji KAWATA†a) , Yoshifumi NISHIO†† , and Akio USHIDA†† , Regular Members

SUMMARY In this study, synchronization phenomena inchaotic oscillators coupled by a transmission line are investigated.
In particular investigation using real circuits is done for the ﬁrst
time. It is conﬁrmed that the chaotic subsystems synchronize,
although signals propagating along the transmission line are affected by the time delay. Further the period-doubling bifurcation with varying the time delay and anti-phase synchronization
phenomena are observed in our circuit model. Also the voltage
distribution of transmission line is simulated in order to investigate whether the current ﬂowing through the transmission line
is constant or not. It is found that the subsystems synchronize
although the current through the transmission line keeps on varying.
key words: chaos synchronization, transmission line, characteristic impedance, time delay

1.

Introduction

Since synchronization of chaos has been reported by
Pecora and Carroll [1], it has received a great deal ofattention. It is extremely interesting that chaotic systems or signals synchronize although small error of initial values in a chaotic system is expanded as time goes.
For this background, a number of papers on chaos synchronization, for example [2]–[8] and so on, have been
published. Further communication schemes making useof chaos synchronization have been proposed in [9]–[13]and the eﬀects of non-ideal communication channels on
chaos synchronization have been investigated in [14]–[18].
However, almost studies on chaos synchronization
reported so far (such as [2]–[8]), have dealt with synchronization phenomena observed from chaotic oscillators coupled by lumped elements. In future, chaotic
systems will be integrated and their operation speed
will become higher for some engineering applications.
In such systems, the eﬀect of time delay can not be ignored. Thus we think that the investigation of systems
with time-delayed signal is very important.
As far as we know, studies on such systems havebeen hardly reported. In [19] two chaotic systems couManuscript received December 18, 2000.
Manuscript revised June 4, 2001.
†
The author is with the Department of Information Science and Systems Engineering, Tokushima Bunri University, Kagawa-ken, 769-2193 Japan.
††
The authors are with the Department of Electrical andElectronic Engineering, Tokushima University, Tokushimashi, 770-8506 Japan.
a) E-mail: kawata@is.bunri-u.ac.jp

pled by two delay lines are investigated and various
computer simulated results on synchronization are reported. In the system a state voltage passes through adelay line, an ampliﬁer and further a transformer, andreaches to another subsystem. So the system seems to
be complex and unnatural. Due to making use of ideal
delay line, simulation with varying the time delay have
only carried out. Further the results have not yet been
conﬁrmed by circuit experiments. It should go without saying that more detailed investigations must be
required for engineering applications of chaos.
So, in this paper, we investigate a chaotic system
coupled by transmission line. Remember that a transmission line introduces some time delay. In our models, two Chua’s circuits are coupled by a transmission
line instead of a lumped element. The circuit models,i.e. vC1 -coupled and vC2 -coupled systems, are shown in
Fig. 1.
First the models are investigated experimentally.
In our laboratory experiments, delay lines and coaxial
cables have been used as transmission line. The delayline is an element that transmits input signal waveforms
with ﬁdelity but delays time. It has been conﬁrmedthat the chaotic subsystems in both models synchronize even if the time delay exists. Moreover, for vC1 coupled system, interesting phenomena that the subsystems stop oscillations when the time delay becomeslarge, are observed.

Fig. 1 The circuit model. (a) vC1 -coupled system. (b)vC2 -coupled system.

IEICE TRANS. FUNDAMENTALS, VOL.E84–A, NO.12 DECEMBER 2001

3100

To make these phenomena more reliable, computer
simulations are further carried out. These models canbe simulated by using the method of characteristics[20], which has been proposed for transient simulation of transmission lines. It is found that the experimentally conﬁrmed phenomena can be also observed incomputer simulations.
In order to investigate synchronization phenomena in detail, the bifurcation diagrams with respect toboth the time delay and the characteristic impedance
are shown. The amazing phenomena including perioddoubling bifurcation and anti-phase synchronization
are observed from vC1 -coupled system. The time evolution of voltage distribution along the transmission line
is also given to examine whether the current throughthe transmission line is varying or not. It is found thatthe current through the transmission line is not constant and keeps on varying, which is diﬀerent from thecase of lumped element coupling.
2.

Where Z0 is the characteristic impedance. ek is the
waveform generator to simulate the reﬂection and thesevalues are calculated by the following equations.
e1 (t) = 2v2 (t) − e2 (t − τ )e2 (t) = 2v1 (t) − e1 (t − τ )Also τ denotes the time delay of the transmission line.
Applying the characteristic model to our circuit
models, we can get the equivalent circuits shown in
Fig. 3 and the corresponding circuit equations.
vC1 -coupled system:
C1

C2

L

Circuit Model

Figure 1 shows the circuit models used in this paper. In
these models, the Chua’s circuit is used as each chaotic
subcircuit and two subcircuits are coupled by a transmission line. The Chua’s circuit is a extremely simple
system but it exhibits complex dynamics of bifurcationand chaos [21], [22].
In our simulations, the transmission line is modeled by the characteristic model [20], as shown in Fig. 2.

(1)

dvC1k
= G(vC2k − vC1k ) − g(vC1k )dt+ Y0 (ek (t − τ ) − vC1k ),dvC2k= G(vC1k − vC2k ) + iLk ,
dt

(2)

diLk
= −vC2kdt

vC2 -coupled system:
C1

C2

L

dvC1k
= G(vC2k − vC1k ) − g(vC1k ),dt
dvC2k= G(vC1k − vC2k ) + iLk
dt+ Y0 (ek (t − τ ) − vC2k ),

(3)

diLk
= −vC2kdt

Where G = 1/R, Y0 = 1/Z0 and g(·) is a piecewise
linear resistor deﬁned by
1g(v) = m1 v + (m0 − m1 )[|v + Bp | − |v − Bp |].
2(a)Fig. 2(b)(a) Transmission line and (b) its characteristic model.

Fig. 3After normalizing, we have the following dimensionless equations.

The equivalent circuits for (a) vC1 -coupled system and (b) vC2 -coupled system.

KAWATA et al.: SYNCHRONIZATION AND ITS ANALYSIS IN CHAOTIC SYSTEMS COUPLED BY TRANSMISSION LINE

3101

vC1 -coupled system:ẋk = α(yk − xk − f (xk ) + γ(wk − xk )),ẏk = xk − yk + zk ,żk = −βykin [3]. The measured values of synchronizing system are
as follows:(4)

vC2 -coupled system:ẋk = α(yk − xk − f (xk )),ẏk = xk − yk + zk + γ(wk − yk ),żk = −βyk

(5)

Where
1
f (x) = bx + (a − b)[|x + 1| − |x − 1|],2
and
α = C2 /C1 , β = C2 /LG2 , γ = Y0 /G,
t̂ = Gt/C2 , τ̂ = Gτ /C2 , “ ˙ ” = d/dt̂,
xk = vC1k /Bp , yk = vC2k /Bp , zk = iLk /GBp ,wk = ek (t̂ − τ̂ )/Bp , a = m0 /G, b = m1 /G.
3.

Synchronization Phenomena

3.1 Circuit Experiments
The resistor coupled Chua’s circuits have been investigated numerically and experimentally in [3]. In thisstudy, the resistor is replaced by an element with timedelay; i.e. transmission line. In our experiments, theChua’s circuits are built referring to [23]. The circuit
parameters are adjusted so that the circuits exhibit the
double scroll attractor, and so that they synchronize
when they are directly coupled by a resistor as reported

R = 1.435 kΩ, C1 = 9.50 nF, C2 = 103.8 nF,
L = 19.98 mH, rL = 40.25 Ω,
m0 = −0.75 mS, m1 = −0.41 mS,
Bp = 1.06 V.
These parameters will be ﬁxed throughout this paper.
As transmission line in our circuit models, a delay line or a coaxial cable is used. The delay line isan element that transmits input signal waveforms withﬁdelity but delays time only when it is matched withsuitable impedance at both ends. Namely how to use
the delay lines is the same as the usage of coaxial cables. Note that both the delay lines and the coaxial
cables are not matched in our all experiments. Table 1
shows the characteristics of the delay lines and coaxial
cables we used.
vC1 -coupled system:
The experimental results are illustrated in Fig. 4. As
found from the ﬁgures, though the signals from one
subsystem to another one through the transmission line
are aﬀected by the time delay, the chaotic subsystems
synchronize completely when the delay lines are used.
Table 1

The characteristics of real transmission lines.
Transmission LineZ0
τDelay Line #1
50 Ω
5 nsecFSL-5nDelay Line #2
500 Ω 500 nsec
FN-500B
Coaxial Cable #1
50 Ω
500 nsec
3D2V 100 m
Coaxial Cable #2
50 Ω
10 µsec
3D2V 2 km

Fig. 4 Synchronization in vC1 -coupled system (circuit experiments). (a) Delay Line
#1. (b) Delay Line #2. (c) Coaxial Cable #1. (d) Coaxial Cable #2.

IEICE TRANS. FUNDAMENTALS, VOL.E84–A, NO.12 DECEMBER 2001

3102

Fig. 5 Synchronization in vC2 -coupled system (circuit experiments). (a) Delay Line
#1. (b) Delay Line #2. (c) Coaxial Cable #1. (d) Coaxial Cable #2.

However the coaxial cables are used, then the subsystems stopped oscillation. From Figs. 4(a) and (c),it seems that oscillation stops if only τ changes from
5 nsec to 500 nsec and Z0 remains 50 Ω. Further, from
Figs. 4(b) and (c), the subsystems seem to synchronize
again, when Z0 is varied from 50 Ω to 500 Ω even if τ isstill 500 nsec.
vC2 -coupled system:
The circuit experiments for vC2 -coupled system is alsodone. The experimental results are illustrated in Fig. 5.
The subsystems synchronize for all cases. For the casethat 2 km long of coaxial cable was used, surprisingly
enough, two Chua’s circuits synchronized. Unfortunately we do not have any transmission lines with larger
delay, thus we could not conﬁrm whether what kinds ofphenomena in real system will happen for larger delay.
3.2 Computer Simulations
In this subsection, we conﬁrm whether chaos synchronization observed in real circuits are achieved numerically. In our simulations, Eqs. (4) and (5) are solved by
the fourth order Runge-Kutta method. The parametervalues and the initial conditions of chaotic subsystemsare
α = 10, a = −1.2, b = −0.75, Bp = 1,
x1 = 1.0, y1 = 0.02, z1 = 0.0,x2 = −0.7, y2 = 0.4, z2 = 0.08.
Taking a suitable value of β the chaotic subsystem exhibits the well-known double scroll attractor.
Moreover, the initial conditions of the waveformgenerators in the equivalent circuits are set to zero, i.e.,ei (t) = 0, (t ∈ [−τ, 0]).

Fig. 6 Synchronization in vC1 -coupled system (computer simulations). (a) γ = 10 and τ̂ = 0.001. (b) γ = 10 and τ̂ = 0.01.
(c) γ = 1 and τ̂ = 0.01.

But we have conﬁrmed that the same phenomena canbe observed for the random initial conditions.
vC1 -coupled system:
By varying the values of γ and τ̂ , we simulate the vC1 coupled system for β = 15. Figure 6 shows the results.
It can be observed that the chaotic subsystems synchronize completely for γ = 10 and τ̂ = 0.001 (i.e.
Z0 = 143.5 Ω and τ = 149 nsec). When only the timedelay increases as τ̂ = 0.01 (i.e. τ = 1490 nsec), eachchaotic subsystems stop oscillation. But the value of γ
is also varied as γ = 1 (i.e. Z0 = 1435 Ω), the subsys-

KAWATA et al.: SYNCHRONIZATION AND ITS ANALYSIS IN CHAOTIC SYSTEMS COUPLED BY TRANSMISSION LINE

3103

Fig. 7 Synchronization in vC2 -coupled system (computer simulations). (a) γ = 30 and τ̂ = 0.05. (b) γ = 30 and τ̂ = 0.2. (c)γ = 15 and τ̂ = 0.2.

tems become to synchronize again even for τ̂ = 0.01.
By taking the relationship between Z0 , τ and γ, τ̂ into
account, these results are the same as the case of experiments.
vC2 -coupled system:
In the similar way to vC1 -coupled system, numerical investigations are carried out. The results for β = 18 areshown in Fig. 7. The chaotic subsystems in vC2 -coupledsystem also synchronize perfectly for γ = 30 and τ̂ =0.01 (i.e. Z0 = 47.8 Ω and τ = 1490 nsec). If only thetime delay increases to τ̂ = 0.2 (i.e. τ = 29.8 µsec),
however, the orbits of chaotic subsystems diverged asshown in Fig. 7(b). It seems that the orbits of subsystems diverge with increasing the time delay. This
could not be observed in circuit experiments, because
the time delay probably is small, even for 2 km coaxial
cable. Further we found that similar to vC1 -coupledsystem, the subsystems get to synchronize by varying
the value of γ to 15 (i.e. 95.7 Ω).
4.

Detailed Analysis

4.1 Bifurcation Diagram
The results in previous section let us ﬁnd the following
two issues.
• When the characteristic impedance is ﬁxed, oscillations of the subsystems stop or orbits of the subsystems diverge as the time delay increases.
• By varying the value of characteristic impedance,
the subsystems become to synchronize even if timedelay increases.
Consequently it is natural to think that the systems

Fig. 8 One parameter bifurcation diagrams (computer simulations). (a) vC1 -coupled system (β = 15 and γ = 10). (b)vC1 -coupled system (β = 15 and τ̂ = 0.001). (c) vC2 -coupledsystem (β = 18 and γ = 30). (d) vC2 -coupled system (β = 18
and τ̂ = 0.05).

may give rise to bifurcation. To conﬁrm this expectation, we plot bifurcation diagrams with respect to thetime delay and the characteristic impedance.
Figure 8 shows the bifurcation diagrams. Where
the horizontal axes correspond to the normalized time

IEICE TRANS. FUNDAMENTALS, VOL.E84–A, NO.12 DECEMBER 2001

3104

Fig. 9 Anti-phase synchronization for vC1 -coupled system (computer simulations). β =15, γ = 10 and τ̂ = 0.4.

delay τ̂ for Figs. 8(a) and (c), and correspond to γ whichis related to the characteristic impedance, for Figs. 8(b)and (d). Vertical axis is the voltage vC1 . Surprisingly
enough, for vC1 -coupled system, rich bifurcation phenomena including the period-doubling route to chaosand several periodic windows are observed with varyingτ̂ or γ. Note that the two subsystems are synchronizing completely for the parameter values in these ﬁgures(except the range of divergence). The bifurcation diagrams for diﬀerent values of γ or τ̂ are omitted, because
almost similar diagrams were obtained.
In [19] it has been also reported that their circuit
model exhibits the period-doubling bifurcation undercertain conditions. However, their subsystems do notsynchronize completely during the bifurcation route.
Further the time delay increases, anti-phase synchronization is observed as shown in Fig. 9. This isvery interesting phenomena, because anti-phase synchronization can not be observed from resistor coupledsystems. More detailed analysis of the synchronization
is our future research.
On the other hand, in vC2 -coupled system, chaos
synchronization can be observed for relatively wide
range of time delay, since periodic orbits do not emerge
even if the time delay becomes fairly large. By increasing the time delay further, the trajectories eventually
diverge. Note that in vC2 -coupled system the equilibrium points are always unstable unlike vC1 -coupled system. Moreover, many types of periodic window takeplace according as values of τ̂ .
Next, we make detailed laboratory experiments forvarious diﬀerent length of time delay in order to conﬁrm
whether the bifurcation phenomena observed from oursimulations are also generated in real system.
Experiments were carried out by cascading someof delay lines listed in Table 2, for example we connect 12 of Delay Line AT-50 to realize τ = 600 nsec
for Fig. 10(a). Note that any transmission lines with asame Z0 can be cascaded to get larger time delay. The
results for vC1 -coupled system are shown in Fig. 10. It
is found that the period-doubling bifurcation phenomena can be also observed in real system. Unfortunately,
we have not conﬁrmed anti-phase synchronization experimentally, which is due to the shortage of time de-Table 2The delay lines with Z0 = 100 Ω.

Transmission Line
Delay Line AT-01
Delay Line AT-05
Delay Line AT-10
Delay Line AT-50

τ
1 nsec5 nsec10 nsec
50 nseclay† .
4.2 Voltage Distribution
At ﬁrst thought, we expected that the current throughthe transmission line may be zero or constant, because
in systems coupled by a lumped element the current
ﬂowing the coupling element is zero or constant. Thus
we investigate the current distribution of the transmission line.
Let the current and voltage of transmission line
be i(x, t) and v(x, t), respectively. Since i(x, t) =v(x, t)/Z0 , we can investigate the voltage distribution
alternatively. In our simulations, the transmission line
is divided to 50 segments of transmission lines, and theneach of them are replaced with the characteristic models.
The results with the subsystems synchronizing areshown in Figs. 11 and 12. In turn from left, the attractor in the left subcircuit, the attractor in the right one,the state of synchronization, and the voltage distribution are given. In the ﬁgure of voltage distribution, theaxis toward the right direction is the distance from theleft end of transmission line, the upward axis is the voltage vC1 or vC2 , and axis toward the depth corresponds
to time.
As you can see from this ﬁgure, the voltage distribution is not constant for both systems. So we canconclude that though the current ﬂowing through thetransmission line is varying, the chaotic subsystems
synchronize. This is so wonder, because the current
through the coupling element in synchronizing systems
coupled with lumped element is constant.

†
The maximum time delay in our experiments is 1100
nsec (20 of AT-50 and 10 of AT-10 are used).

KAWATA et al.: SYNCHRONIZATION AND ITS ANALYSIS IN CHAOTIC SYSTEMS COUPLED BY TRANSMISSION LINE

3105

Fig. 10 Phase portraits of period-doubling bifurcation for vC1 -coupled system (circuit
experiments). (a) ﬁxed point for τ = 600 nsec. (b) period-1 attractor for τ = 250 nsec.
(c) period-2 attractor for τ = 200 nsec. (d) period-4 attractor for τ = 165 nsec. (e) spiralattractor for τ = 150 nsec. (f) double scroll attractor for τ = 50 nsec.

Fig. 11 Voltage distribution for vC1 -coupled system (computer simulations). β = 15.
(a) γ = 10 and τ̂ = 0.001. (b) γ = 1 and τ̂ = 0.01.

IEICE TRANS. FUNDAMENTALS, VOL.E84–A, NO.12 DECEMBER 2001

3106

Fig. 12 Voltage distribution for vC2 -coupled system (computer simulations). β = 18.
(a) γ = 30 and τ̂ = 0.01. (b) γ = 15 and τ̂ = 0.1.

5.

Conclusion

In this study, we have reported very interesting phenomena in a chaotic system coupled by a transmission
line. Chaos synchronization is also achieved in such
system as well as systems coupled with lumped elements. It is found that in our models bifurcation andcomplex phenomena occur with variation of the timedelay, which can not be observed in resistor coupledsystems. Also the current through the coupling transmission line is not constant, which is diﬀerent from thecase of lumped element coupling. The systems may
have more exciting phenomena according to the characteristic impedance and time delay.
Our goal is to make clear synchronization mechanisms and various nonlinear phenomena in our models,
but unfortunately we have not yet reached a level which
we can consider about them. Thus, for the purpose, we
primary have to investigate our systems more fully andthen begin qualitative analysis. Moreover we will thinkover some applications using our models.
References[1] L.M. Pecora and T.L. Carroll, “Synchronization in chaotic
systems,” Phys. Rev. Lett., vol.64, no.4, pp.821–824, 1990.
[2] J.C. Alexander, J.A. Yorke, and Z. You, “Riddled basins,”
Int. J. Bifurc. Chaos, vol.2, pp.795–813, 1992.
[3] L.O. Chua, M. Itoh, L. Kocarev, and K. Eckert, “Chaos
synchronization in Chua’s circuit,” J. Circuits, Syst., Comput., vol.3, no.1, pp.93–108, 1993.
[4] E. Ott and J.C. Sommerer, “Blowout bifurcations: The occurrence of riddled basins,” Phys. Lett. A, vol.188, pp.39–
47, 1994.
[5] P. Ashwin, J. Buescu, and I. Stewart, “Bubbling of attractors and synchronization of chaotic oscillators,” Phys. Lett.

A, vol.193, pp.126–139, 1994.
[6] J.F. Haegy, T.L. Carroll, and L.M. Pecora, “Experimental
and numerical evidence for riddled basins in coupled chaotic
systems,” Phys. Rev. Lett., vol.73, pp.3528–3531, 1994.
[7] M. Wada, Y. Nishio, and A. Ushida, “Blowout bifurcation
and riddled basin in simple coupled chaotic circuits,” Proc.
ECCTD’97, vol.3, pp.1269–1273, 1997.
[8] M. Hasler and Y.L. Maistrenko, “An introduction to thesynchronization of chaotic systems: Coupled skew tent
maps,” IEEE Trans. Circuits & Syst. I, vol.44, no.10,
pp.856–866, Oct. 1997.
[9] L. Kocarev, K.S. Halle, K. Eckert, and L.O. Chua, “Experimental demonstration of secure communication via chaotic
synchronization,” Int. J. Bifurc. Chaos, vol.2, no.3, pp.709–
713, Sept. 1992.
[10] K.S. Halle, C.W. Wu, M. Itoh, and L.O. Chua, “Spread
spectrum communication through modulation of chaos,”Int. J. Bifurc. Chaos, vol.2, pp.469–477, 1992.
[11] H. Dedieu, M.P. Kennedy, and M. Hasler, “Chaos shift keying: Modulation and demodulation of a chaotic character
using self-synchronizing Chua’s circuits,” IEEE Trans. Circuits & Syst. II, vol.40, no.10, pp.634–642, Oct. 1993.
[12] C.W. Wu and L.O. Chua, “A simple way to synchronize
chaotic systems with applications to communications,” Int.
J. Bifurc. Chaos, vol.3, pp.1619–1627, 1993.
[13] M. Itoh, H. Murakami, and L.O. Chua, “Communication
systems via chaotic modulations,” IEICE Trans. Fundamentals, vol.E77-A, no.6, pp.1000–1005, June 1994.
[14] T.L. Carroll and L.M. Pecora, “The eﬀect of ﬁltering on
communication using synchronized chaotic circuits,” Proc.
IEEE ISCAS’96, vol.3, pp.174–177, May 1996.
[15] R. Caponetto, L. Fortuna, M. Lavorgna, G. Manganaro,
and L. Occhipinti, “Experimental study on chaos synchronization with non-ideal transmission channel,” Proc. IEEE
GLOBECOM’96, pp.2083–2087, 1996.
[16] P. Celka, “Synchronization of chaotic optical dynamical systems through 700 m of single mode ﬁber,” IEEE Trans. Circuits & Syst. I, vol.43, no.10, pp.869–872, Oct. 1996.
[17] L.O. Chua, T. Yang, G. Zhong, and C.W. Wu, “Synchronization of Chua’s circuits with time-varying channels andparameters,” IEEE Trans. Circuits & Syst. I, vol.43, no.10,

KAWATA et al.: SYNCHRONIZATION AND ITS ANALYSIS IN CHAOTIC SYSTEMS COUPLED BY TRANSMISSION LINE3107

pp.862–868, Oct. 1996.
[18] J. Kawata, Y. Nishio, H. Dedieu, and A. Ushida, “On theinﬂuence of transmission line on communication system using chaos synchronization,” IEICE Trans. Fundamentals,
vol.E81-A, no.8, pp.1713–1724, Aug. 1998.
[19] R. Koike, H. Sekiya, N. Miyabayashi, S. Moro, and S. Mori,“Synchronization of two chaotic circuits coupled by delayline,” Proc. ECCTD’97, vol.3, pp.1280–1285, 1997.
[20] F.H. Branin, Jr., “Transient analysis of lossless transmission line,” Proc. IEEE, vol.55, pp.2012–2013, Nov. 1967.
[21] L.O. Chua, M. Komuro, and T. Matsumoto, “The double
scroll family,” IEEE Trans. Circuits & Syst. I, vol.33, no.11,
pp.1072–1117, Nov. 1986.
[22] L.O. Chua, C.W. Wu, A. Huang, and G. Zhong, “A universal circuit for studying and generating chaos,” IEEE Trans.
Circuits & Syst. I, vol.40, no.10, pp.732–761, Oct. 1993.
[23] M.P. Kennedy, “Robust OP amp realization of Chua’s circuit,” Frequenz, vol.46, pp.66–80, 1992.

Junji Kawata
was born in Kagawa,Japan, on 1967. He received the B.E. and
M.E. degrees form Tokushima University,
Tokushima, Japan, in 1990, 1992, respectively. He is currently working towards
the Ph.D. degree. His research interests
are in circuit simulation and chaotic communication. Mr. Kawata is a member of
the IEEE.

Yoshifumi Nishio
received the B.E.,
M.E., and Ph.D. degrees in electrical engineering from Keio University, Yokohama
Japan, in 1988, 1990, and 1993, respectively. In 1993, he joined the Department of Electrical and Electronic Engineering at Tokushima University, Tokushima Japan, where he is currently an Associate Professor. His research interests
are in chaos and synchronization phenomena in nonlinear circuits. Dr. Nishio is amember of the IEEE.

Akio Ushidareceived the B.E.
and M.E. degrees in electrical engineering from Tokushima University in 1961
and 1966, respectively, and the Ph.D. degree in electrical engineering from University of Osaka Prefecture in 1974. He wasan associate professor from 1973 to 1980
at Tokushima University. Since 1980 hehas been a Professor in the Department
of Electrical Engineering at the university.
From 1974 to 1975 he spent one year as
a visiting scholar at the Department of Electrical Engineeringand Computer Sciences at the University of California, Berkeley. His current research interests include numerical methods
and computer-aided analysis of nonlinear system. Dr. Ushida isa member of the IEEE.