ISCAS 2000 - IEEE International Symposium on Circuits and Systems, May 28-31, 2000, Geneva, Switzerland

ON SYNCHRONIZATION PHENOMENA IN CHAOTIC SYSTEMS COUPLED BY
TRANSMISSION LINE
Junji Kawatay, Yoshifumi Nishioz and Akio Ushidaz
yTokushima Bunri University, Kagawa, JAPANzTokushima University, Tokushima, JAPANABSTRACT
In this study, synchronization phenomena in chaotic
oscillators coupled by a transmission line are investigated. In particular investigation using real circuits isdone for the rst time. We report the very interestingresults.
1. INTRODUCTION
Since synchronization of chaos has been reported by
Pecora and Carroll [1], it has received a great deal ofattention. A large number of recent papers have been
published in this area. However, almost studies onchaos synchronization reported so far, have dealt withsynchronization phenomena observed from chaotic oscillators coupled by lumped elements as in [2].
A few studies on systems coupled by elements withtime delay have been reported. In [3] t wo chaotic systems coupled by t wo delay lines are investigated. It has
been shown via numerical experiments that t wo chaotic
circuits synchronized when the time delay exists.
We have also reported synchronization phenomena
in a chaotic system coupled by transmission line [4]. It
is con rmed by simulations that the subsystems synchronize by adjusting the characteristic impedance ofthe line when the time delay increases. However theexperimental studies on such systems have not beenreported so far.
In this study, following our previous study, we investigate t wo chaotic circuits coupled by a transmission line.In particular the systems are assessed by laboratory experiments as well as computer simulations.
These circuits are simulated by using the method ofcharacteristics [5]. In our experiments, delay lines andcoaxial cables are used as transmission line. The delayline is a element that transmits input signal waveforms
delity but delays time.
We show by experiments that vC 2 -coupled synchronization is achieved, which coincide our previous resultsin [4]. It is also con rmed by simulations and experiments that vC 1 -coupled subsystems synchronize only

for small delays. A very interesting phenomena, that
oscillation of the Chua's circuit stops when coaxial cables are used, is observed from vC 1 -coupled system.
2. CIRCUIT MODEL
Fig. 1 shows vC 1 -coupled system that is one of the circuit models used in this paper. vC 2 -coupled system isomitted because of limitations of space. In these models the Chua's circuit is used as each chaotic subcircuitand t wo subcircuits are coupled by a transmission line.
The Chua's circuit is a extremely simple system but
it exhibits the complex dynamics of bifurcation andchaos.
In our simulations, the transmission line is just modeled using the method of characteristics, where thetransmission line is replaced by the characteristic model[5] as shown in Fig. 2. After normalizing, we obtainthe following circuit equations.
vC 1 -coupled system:x_ k = (yk ; xk ; f (xk ) + (wk ; xk ));y_k = xk ; yk + zk ;(1)
z_k = ; ykvC 2 -coupled system:x_ k = (yk ; xk ; f (xk ));y_k = xk ; yk + zk + (wk ; yk );(2)
z_k = ; yk
Wheref (x) = bx + 12 (a ; b)[jx + 1j ; jx ; 1j](3)
and= C2 =C1 ; = C2 =LG2 ; = Y0 =G;^t = Gt=C2 ; ^ = G=C2 ; \ _ " = d=dt^;
xk = vC 1k =Bp ; yk = vC 2k =Bp ; zk = iLk =GBp ;wk = ek (t^ ; ^)=Bp ; a = m0 =G; b = m1 =G:

Y0 and  are the characteristic admittance and timedelay of the transmission line respectively, and ek isthe waveform generator to simulate the re ection in
the characteristic model.

0-7803-5482-6/99/$10.00 ©2000 IEEE

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3. NUMERICAL AND EXPERIMENTAL RESULTS
We have already investigated the vC 2 -coupled system
by simulations [4]. We rst make experiment to conrm the results. The Chua's circuits are built, basedon [6]. The circuit parameters are adjusted so thatthe circuits exhibit the double scroll attractor, and sothat they synchronize when they are directly coupled
by a resistor as reported in [2]. The measured valuesof synchronizing system are as follows:
R = 1:435[k ]; C1 = 9:50[nF]; C2 = 103:8[nF];
L = 19:98[mH]; rL = 40:25[ ];m0 = ;0:75[mS]; m1 = ;0:41[mS]; Bp = 1:06[V]Remaining the circuit parameters, the resistor is replaced by a delay line or coaxial cable. The delay line
is a element that transmits input signal waveforms delity 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 the delay lines and the coaxial cables
are not matched in our experiments. Table 1 shows thecharacteristics of the delay lines and coaxial cables weused1 ,Table 1: The characteristics of transmission lines.
Transmission Line Z0
Delay Line #150[ ] 5[nsec]
FSL05-050A
Delay Line #2 500[ ] 500[nsec]
FN-500B
Coaxial Cable #1 50[ ] 500[nsec]
3D2V 100 m
Coaxial Cable #2 50[ ] 10[sec]
3D2V 2 km
where Z0 denotes the characteristic impedance. The
experimental results are illustrated in Fig. 3. The subsystems synchronize for all cases. For the case that 2km long of coaxial cable was used, surprisingly enough,t wo Chua's circuits synchronized. These results havegood agreement with the simulation results [4]. Unfortunately we do not have any transmission lines withlarger delay, thus we could not con rm whether whatkinds of phenomena will happen for larger delay.
Next we investigate the vC 1 -coupled system by experiments. With changing the circuit con guration,
experimental investigations were done. Fig. 4 shows
the results. For the case that the delay lines are used,
the subsystems synchronized as shown in Fig. 4(a),(b).
1 The delay lines are made by Showa Densen Corporation.
The 3D2V cables are made in accordance with JIS (JapaneseIndustrial Standard). With binding the coaxial cables rounddrum, all experiments were done.

However both Chua's circuits stopped oscillating when
the coaxial cables are used (see Fig. 4(c),(d)). Theselet us nd the following t wo issues.
 For the same characteristic impedance, oscillations of the system stop when time delay increases.
 By varying the characteristic impedance, the subsystems synchronize even if time delay increases.
Second item can be also applied to the vC 2 -coupledsystem [4].
To make these phenomena more reliable, we further simulate the vC 1 -coupled system with the use of
the method of characteristics [5]. In our numerical experiments, 4th order Runge-Kutta method is used, andwe set the stepsize as 0.0001 and the parameters andinitial conditions of t wo subcircuits as follows:= 10; = 15; a = ;1:2; b = ;0:75; Bp = 1;
x1 = 1:0; y1 = ;0:2; z1 = 0:3;
x2 = 0:7; y2 = 0:4; z2 = 0:08:
Simulation results are shown in Fig. 5. The phenomena in circuit experiments could be also observed incomputer simulations.
4. 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.
Hereafter we intend to analyze the systems qualitatively. The systems may have more exciting phenomena according to the characteristic impedance and timedelay.
5. REFERENCES[1] L. M. Pecora and T. L. Carroll, \Synchronization inchaotic systems," Phys. Rev. Lett., vol. 64, no. 4, pp.
821-824, 1990.
[2] L. O. Chua et al., \Chaos synchronization in Chua's
circuit," J. Circuits, Syst., Comput., vol. 3, no. 1, pp.
93-108, 1993.
[3] R. Koike et al., \Synchronization of t wo c haotic circuits coupled by delay line," Proc. of ECCTD'97, vol.
3, pp. 1280{1285, 1997.
[4] J. Kawata, Y. Nishio and A. Ushida, \Chaos sync hronization by transmission line coupling," Proc. ofNOLTA'96, pp. 109{112, 1996.
[5] F. H. Branin, Jr., \Transient analysis of lossless transmission line," Proc. IEEE, vol. 55, pp. 2012-2013, Nov.
1967.
[6] M. P. Kennedy, \Robust OP amp realization of Chua'scircuit," Frequenz, vol. 46, pp. 66-80, 1992.

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Transmission LineRiL1+L

vC 21C2C1;

g(vC 11 )g(vC 12 )++vC 11vC 12;;R

iL2+C1vC 22C2L;

Figure 1: The circuit model. vC 1 -coupled system.
Characteristic Model

RiL1+L

vC 21+C2C1;

vC 11;

g(vC 11 )g(vC 12 )Z0e1+ +; ;

Z0e2

R+

vC 12C1;C2+iL2vC 22L;

Figure 2: The equivalent circuit of Fig. 1.

(a)

(b)

(c)
(d)
Figure 3: Synchronization in vC 2 -coupled system. vC 11 vs. vC 12 . (a) Delay Line #1, (b) Delay Line #2, (c)Coaxial Cable #1, (d) Coaxial Cable #2.

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(a)

(b)

(c)
(d)
Figure 4: Experimentally observed phenomena in vC 1 -coupled system. vC 21 vs. vC 22 . (a) Delay Line #1, (b) DelayLine #2, (c) Coaxial Cable #1, (d) Coaxial Cable #2.
2.0

2.0

y2

y20.0;2.0;2.0
2.0

0.00.0(a)y1

2.0;2.0;2.0
2.0

y20.0(b)y12.0

y20.0;2.0;2.00.0

0.0(c)y1

2.0;2.0;2.00.0(d)

y1

2.0

Figure 5: Numerically observed phenomena in vC 1 -coupled system. (a) = 10; ^ = 0:001, (b) = 1; ^ = 0:01,(c) = 10; ^ = 0:01, (d) = 10; ^ = 0:1.

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