BIFURCATION AND COMPLEX PHENOMENA IN CHAOTIC SYSTEMSCOUPLED BY TRANSMISSION LINE
Junji Kawata† , Yoshifumi Nishio‡,∗ and Akio Ushida‡† Tokushima Bunri University, Kagawa, JAPAN‡ Tokushima University, Tokushima, JAPAN∗ Swiss Federal Institute of Technology, Lausanne, Switzerland
ABSTRACTIn this study, bifurcation and complex phenomena in synchronizing chaotic oscillators coupled by a lossless transmission line are investigated. The period-doubling bifurcation with varying the time delay is observed in our circuit
model. Further the voltage distribution of transmission lineis simulated in order to investigate whether the current flowing through the transmission line is constant or not. It isfound that the subsystems synchronize although the current
through the transmission line keeps on varying.

which is different from the case of lumped element coupling.
2. CIRCUIT MODEL

1. INTRODUCTION
Since synchronization of chaos has been reported by Pecora
et al. [1], it has received a great deal of attention. A large
number of recent papers have been published in this area.
However, almost studies on chaos synchronization reported
so far, have dealt with synchronization phenomena observed
from chaotic oscillators coupled by lumped elements as in[2]. We think that the investigation of systems with timedelayed signal is very important, because there exists thetime delay of signals in the natural fields and real systems
utilizing chaos will be affected by the time delay.
A few studies on systems coupled by elements with timedelay have been reported. In [3] two chaotic systems coupled by two delay lines are investigated. It has been shownvia numerical experiments that two chaotic circuits synchronized when the time delay exists.
We have also reported synchronization phenomena in achaotic system coupled by transmission line [4], [5]. It is
confirmed by simulations and experiments that the subsystems synchronize by adjusting the characteristic impedanceof the line when the time delay increases.
Further, in this paper, we present some new results. The
bifurcation diagrams with respect to the time delay are shown.
The period-doubling bifurcation is observed from v C1 -coupledsystem. The voltage distribution of the transmission line isalso given in order to investigate whether the current throughthe line is varying or not. It is found that the current flowing through the coupling transmission line keeps on varying,

Fig. 1 shows the circuit models used in this paper. In thesemodels, the Chua’s circuit is used as each chaotic subcircuitand two subcircuits are coupled by a transmission line. The
Chua’s circuit is a extremely simple system but it exhibits
the complex dynamics of bifurcation and chaos.
In our simulations, the transmission line is modeled using the method of characteristics, where the transmission
line is replaced by the characteristic model [6]. After normalizing, we obtain the following circuit equations.
vC1 -coupled system:ẋkẏkżkα(yk − xk − f(xk ) + γ(wk − xk )),xk − yk + zk ,−βyk(1)vC2 -coupled system:ẋk=α(yk − xk − f(xk )),ẏkżk==xk − yk + zk + γ(wk − yk ),−βyk(2)Where k = 1, 2,
1f(x) = bx + (a − b)[|x + 1| − |x − 1|],
2(3)

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.
Y0 and τ are the characteristic admittance and time delay of
the transmission line respectively, and e k is the waveformgenerator of the characteristic model to simulate the reflection.

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0-7803-6685-9/01/$10.00©2001 IEEE===3. NUMERICAL RESULTS
The following has been confirmed numerically and experimentally in our previous studies [4], [5].
• The subsystems synchronize when the time delay exists.
• When only the time delay increases, oscillations of
the subsystems stop for v C1 -coupled system and thetrajectories of subsystems diverge for v C2 -coupled system.
• By adjusting the characteristic impedance, the subsystems in both models become to synchronize when
the time delay increases.
3.1. Bifurcation Diagram
As mentioned above, since the chaotic subsystems stop oscillation or their orbits diverge according to the time delay,
it is natural to think that the systems may give rise to bifurcation. Thus we plot bifurcation diagrams with respect tothe time delay. The circuit parameters are fixed as follows:
α = 10, a = −1.2, b = −0.75, B p = 1,
x1 = 1.0, y1 = 0.02, z1 = 0.0,x2 = −0.7, y2 = 0.4, z2 = 0.08.
Figs. 3 and 4 show the bifurcation diagrams. Where thehorizontal and vertical axes correspond to the time delayand voltage vC1 , respectively. Note that the two subsystems are synchronized completely for the parameter values
in these figures (except the range of divergence). Surprisingly enough, for v C1 -coupled system, rich bifurcation phenomena including the period-doubling route to chaos andseveral periodic windows are observed with decreasing thedime delay even if the two subsystems are synchronizedcompletely. In [3] it has been also reported that their circuit model exhibits the period-doubling bifurcation undercertain conditions. However, their subsystems did not synchronize completely during the bifurcation route.
On the other hand, in v C2 -coupled system, chaos synchronization can be observed for relatively wide range oftime delay, since periodic orbits do not emerge even if thetime delay becomes fairly large. By increasing the time delay further, the trajectories eventually diverge. Moreover,many types of periodic window take place according as values of τ̂ .
3.2. Voltage Distribution
At first thought, we expected that the current through thetransmission line may be zero or constant, because in systems coupled by a linear resistor the current flowing the coupling resistor is zero or constant. Thus we investigate thecurrent distribution of the transmission line.

Let the current and voltage of transmission line be i(x, t̂)and v(x, t̂), respectively. Since i(x, t̂) = v(x, t̂)/Z0 , we caninvestigate the voltage distribution. In our simulations, thetransmission line is divided to 50 segments of transmission
lines, and then each of them are replaced with the characteristic models.
The results with the subsystems synchronizing are shownin Figs. 5 and 6. 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 thefigure of voltage distribution, the axis toward the right direction is the distance from the left end of transmission line,
the upward axis is the voltage v C1 or vC2 , and axis toward
the depth corresponds to normalized time t̂.
As you can see from this figure, the voltage distribution
is not constant for both systems. So we can conclude thatin our systems the current is flowing through the transmission line, which is different to the case of lumped element
coupling.
4. CONCLUSION
In this study, we have reported very interesting phenomena
in a chaotic system coupled by a lossless transmission line.
Chaos synchronization is also achieved in such system aswell as systems coupled with lumped elements. It is foundthat in our models bifurcation phenomena occurs with variation of the time delay. Also the current flowing throughthe coupling transmission line keeps on varying, which isdifferent from the case of lumped element coupling.
Hereafter we intend to investigate the systems more fully(e.g. by 2 parameter bifurcation diagrams and so on) andthen analyze the systems qualitatively. The systems may
have more exciting phenomena according to the characteristic impedance and time delay.
5. REFERENCES[1] L. M. Pecora and T. L. Carroll, “Synchronization in chaoticsystems,” 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 two chaotic circuits coupled by delay line,” Proc. of ECCTD’97, vol. 3, pp. 1280–1285, 1997.
[4] J. Kawata, Y. Nishio and A. Ushida, “Chaos synchronizationby transmission line coupling,” Proc. of NOLTA’96, pp. 109–112, 1996.
[5] J. Kawata, Y. Nishio and A. Ushida, “On synchronization
phenomena in chaotic system coupled by transmission line”,
Proc. IEEE ISCAS’2000, vol. 3, pp. 479-482, 2000.
[6] F. H. Branin, Jr., “Transient analysis of lossless transmissionline,” Proc. IEEE, vol. 55, pp. 2012-2013, Nov. 1967.
[7] M. P. Kennedy, “Robust OP amp realization of Chua’s circuit,” Frequenz, vol. 46, pp. 66-80, 1992.

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Transmission LineRi L1vC21LC2C1+vC11vC21−−−iL2+++Rg (vC21)

g (vC11)C1

vC22

C2L−(a)Transmission LineRg (vC11)+

vC11

C1i L1iL2++

vC21

C2C2

vC21

C1−−−−g (vC21)+

vC22

L LR(b)Figure 1: The circuit model. (a) v C1 -coupled system. (b) vC2 -coupled system.
Characteristic ModelRi L1++

vC21

L

C2

C1g (vC11)
Z0

vC11

Z0+

C1

C2+

iL2

vC22L

e2−−+

vC21+

e1−

Rg (vC21)−−−(a)R

g (vC11)
+

vC11

C1

Characteristic Modeli L1C2+iL2

vC21

LZ0Z0++

e1−−R
iL2+L

vC22g (vC21)+C2

C1vC21e2−−−−(b)Figure 2: The equivalent circuits for (a) v C1 -coupled system and (b) vC2 -coupled system.
4.0
x1

0.0

−4.0
0.0

0.002

0.004

τ^0.006

0.008

0.01

Figure 3: One parameter bifurcation diagram for v C1 -coupled system (β = 15, γ = 10).

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4.0x1

0.0
Divergence
−4.0
0.0

0.04

0.08

τ^0.120.16

0.2

Figure 4: One parameter bifurcation diagram for v C2 -coupled system (β = 18, γ = 30).
2.0

2.0

1.0

y1

y2

y20.0

0.0

0.0

−2.0
−4.0

−2.0
4.0 −4.0−1.0
4.0 −1.0

4.0

v (x, ^t )

^t

0.0
0.0

x1

0.0

x2

0.0

y1

x

1.0

(a)

2.0

2.0

1.0

y1

y2

y2

0.0

0.0

0.0

4.0

v (x, ^t )

^t

0.0
−2.0
−4.0

0.0

x1

−2.0
4.0 −4.0

0.0

x2

−1.0
4.0 −1.0

0.0

y1

x

1.0

(b)
Figure 5: Voltage distribution for v C1 -coupled system (β = 15). (a) γ = 10, τ̂ = 0.001 (b) γ = 1, τ̂ = 0.01.
2.0

2.0

4.0

y1

y2

x2

0.0

0.0

0.0

−2.0
−4.0

−2.0
4.0 −4.0−4.0
4.0 −4.0

2.0v (x,τ^)

τ^

0.0
0.0

x1

0.0

x2

0.0

x1

x

4.0

(a)

2.0

2.0

4.0

y1

y2

x2

0.0

0.0

0.0

2.0v (x, ^t )

^t

0.0
−2.0
−4.0

0.0

x1

−2.0
4.0 −4.0

0.0

x2

−4.0
4.0 −4.0

0.0

x1

4.0

x(b)

Figure 6: Voltage distribution for v C2 -coupled system (β = 18). (a) γ = 30, τ̂ = 0.01 (b) γ = 15, τ̂ = 0.1.

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