2006 RISP International Workshop
on Nonlinear Circuits and Signal Processing (NCSP'06)
Waikiki Beach Marriott, Honolulu,Hawaii, USA, March 3-5, 2006

Synchronization of Chaotic Circuits with Transmission Line
Yuki Nakaaji and Yoshifumi Nishio
Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
Phone: +81-88-656-7470, FAX: +81-88-656-7471
Email: nakaaji, nishio@ee.tokushima-u.ac.jp

Abstract
In this study, two Chua’s circuits with lossless transmission
lines placed in parallel are investigated. The effect of thecrosstalk is modeled by the connections via coupling capacitors or mutual inductors. By computer simulations, we confirm that two Chua’s circuits are synchronized in in-pahse oranti-phase by the crosstalk effect of the transmission lines.

circuits [4] [5], etc. However, almost researches consider thecouplings by lumped elements. We have investigated chaotic
phenomena observed from Chua’s circuit when the LC resonator is replaced by a transmission line [6]. Further, we havealso investigated synchronization phenomena of two Chua’scircuits coupled by a transmission line [7] [8].
In this study, we investigate synchronization of two Chua’scircuits with lossless transmission lines coupled by thecrosstalk of the transmission lines.

1. Introduction
Recently several researchers have investigated synchro- 2. Circuit Model
nization of chaotic circuits. Chua’s circuit is well known as
In this study we consider two Chua’s circuits with losslessa simple chaotic circuit. Chua’s circuit consists of one resistransmission
lines placed in parallel as shown in Figure ??.
tor, one capacitor, one nonlinear resistor called Chua’s diode,We
considerthatthe two circuits influence each other by theand an LC resonator. The v − i characteristics of the nonlinear resistor in Chua’s circuit are piecewise-linear as shown in
iRFigure ?? and are described by the following equation.
2R1iR = m2 v1 + (m0 − m1 ){|v1 + Bp1 | − |v1 − Bp1 |}21+ (m1 − m2 ){|v1 + Bp2 | − |v1 − Bp2 |} (1)
2+R

v1

C1C2

Bp1-v2-B p1

Bp2-B p1vR-

m0m2m1(a)R1+v10-Figure 2: Two Chua’s circuits with transmission lines.

i++-C 10iRL

v20

C 20i1R

where m0 , m1 and m2 are the slopes in the segments of thepeacewise-linear function, and Bp1 and Bp2 denote the breakpoints.
iR2effect of the cross talk of transmission lines.
In order to investigate the influence, we model the transmission lines placed in parallel by LC ladder circuits withfinite numbers of lumped elements.

(b)

Figure 1: (a) Chua’s circuit. (b) v − i characteristics of non- 2.1. Circuit Model 1 (Crosstalk via Capacitors)
linear resistor.
First, we model the transmission lines placed in parallel by
thecircuit shown in Fig. ??. In this model, the effect of theThere are many studies about coupled Chua’s circuits, forcrosstalkis given by the connections via coupling capacitors.
example, master-slave coupling of Chua’s circuits [1], mutual
coupling of Chua’s circuits [2], a ladder of Chua’s circuits, aring of Chua’s circuits [3], two-dimensional array of Chua’s
The circuit equations of this model can be derived as fol-- 353 -

i2R

R2

L21
i21

C1+i22

C1+v20C 20i1R-

C21R1-

v21
L11--

C12-

...

v23

C23L 12-+

C 2n-

...

v2n
L 1ni1n+v11C11Cni12+

v10C 10i2n

C2+

v22

C22+

v12v13

C13...

Bpj2,
Bpj1

C3k = C1k , (k=2, 3, ..., n), (l=1, 2, ..., n) and (j=1, 2).
2.2. Circuit Model 2 (Crosstalk via Mutual Inductors)

+

v1n

C 1n-

-

cj = Rj mj2 , dj =...

+i11+

L 2n

L 22-

Figure 3: Circuit Model 1.

Next, we model the transmission lines placed in parallel bythe circuit shown in Figure ??. In this model, the effect of
the crosstalk is given by the connections via mutual inductors.
The normalized circuit equations of this model can be derived
i2R

R2

L21

lows:

i21

...

i22
M1+L 2n

L 22i2nM2+Mn+++...
vvvv
v
CCCC
C
vj1 − vj0
LiRL
L=− iRj (vj0 )...
Rjiiid(vj1 − v(j+1)1 )vj1 − vj0
= ij1 −+ C1vvv
v
v
...
CCCCC
Rjdtd(vjk − v(j+1)k )= ijk − ij(k−1) + Ck
dt
= vj(l+1) − vjl
Figure 4: Circuit Model 2.
(2)where v31 = v11 , vj(n+1) = 0, (k=2, 3, ..., n), (l=1, 2, ..., n) as follows:and (j=1, 2).
x˙10 = x11 − x10 − f (x10 )From (1) and (2), we can obtain the normalized circuitx˙11 = α11 (y11 − (x11 − x10 ))equations as follows:x1k
˙= α1k (y1k − y1(k−1) )x
˙= ζ(x21 − x20 − f (x20 ))20
x˙10 = x11 − x10 − f (x10 )x˙= α21 (y21 − ζ(x21 − x10 ))21x˙11 = α11 (y11 − x11 + x10 ) − β11 (y21 − x21 + x20 )x˙= α2k (y2k − y2(k−1) )2kx1k
˙= α1k (y1k − y1(k−1) ) − β1k (y2k − y2(k−1) )y˙= γjl (xj(l+1) − xjl ) + βl (x(j+1)(l+1) − xjl )jl
x˙= ζ(x − x − f (x ))

dvj0
Cj0
dt
dvj1
Cj1
dt
dvjk
Cjk
dt
dijl
Ljl
dt

20

20

20

1R-

22

21

21

1

2211--

11+-21

20-+1212--1313-+1n-1n-

20x˙21x2k˙y˙jl= α21 (y21 − ζ(x21 − x10 ) − β21 (y11 − x11 + x10 )where xj(n+1) = 0, x3(l+1) = x1(l+1) , and= α2k (y2k − y2(k−1) ) − β2k (y1k − y1(k−1) )= γjl (xj(l+1) − xjl )C10
αjk =,(3)Cjkwhere xj(n+1) = 0,
R12 C10 Ml
βl =,1(L2l − Ml )(L1l − Ml ) − Ml2
f (xj0 ) = cj xj0 + (aj − bj )(|xj0 + 1| − |xj0 − 1|)
2R12 C10 (L(j+1)l − Ml )1+ (bj − cj )(|xj0 + dj | − |xj0 − dj |), (4)
,
γjl =
2
(L2l − Ml )(L1l − Ml ) − Ml2t = R1 τ, “ ˙ 00 =Bp1
d, vjk = Bp1 xjk , ijl =yjl ,dτ
R1

αjk =

Cjk C10 (C(j+1)k − Ck ),2 + 1)C + C 2 C−Ck2 − (Cjkkjk (j+1)kβjk =Cjk C10 Ck,
2 + 1)C + C 2 C−Ck2 − (Cjkkjk (j+1)kγjl =R12 Cj0,
Ljl

1n1n+

1111122n2n12+10102323(5)where L3l = L1l , (k=2, 3, ..., n), (l=1, 2, ..., n) and (j=1, 2).
3. Simulation Results
We carry out computer simulation for the above-mentioned
two models of the circuit by using the 4th order Runge-Kutta
method. In the simulations, we consider only the case thatthe two Chua’s circuits are identical. Hence, we rewrite theparameters as
α1 = α2 = α,a1 = a2 = a,c1 = c2 = c,C10
ζ=, aj = Rj mj0 , bj = Rj mj1 ,C20- 354 -β1 = β2 = β,b1 = b2 = b,d1 = d2 = d.

γ1 = γ2 = γ,(6)

In the following simulations, we fix the parameters as follows:a = −1.2, b = −0.75, c = 10, d = 8, α = 16.

(7)

3.1. Results of Circuit Model 1(a)Figure ?? shows the simulation results of the Circuit Model
1. In this simulation, we vary β, namely, the couplingstrength.
We can confirm that the two circuits synchronize in inphase even when each circuit produce the double scroll attractor. However, for a relatively small coupling parameter value,
the synchronization breaks down as shown in Fig. ??(f). Figure ?? shows the one-parameter bifurcation diagram when γ
is chosen as a control parameter. Namely, in this figure, thecoupling strength is fixed and nonlinear strength of each circuit is changed. The horizontality axis is γ and vertical axis
is x10 when the solution hits the hyperplane X11 =0. From
Fig. ??, we can see that the bifurcation is similar to that of
the original Chua’s circuit.

(b)

(c)

3.2. Results of Circuit Model 2(d)Figure ?? shows the simulation results of the Circuit Model
2. In this simulation, we vary β, namely, the couplingstrength.
Interestingly, in this case, we can confirm that the two circuits synchronize in anti-phase. Figure ?? shows the oneparameter bifurcation diagram when γ is chosen as a control
parameter. From Figure ??, we can say that the bifurcation issimilar to the case of the capacitor coupling.

(e)

4. Conclusions(f)x0In this study, we have investigated two Chua’s circuits withFigure 5: Synchronization phenomena of Circuit Model 1
losslesstransmission lines placed in parallel. We have mod(γ=1.0). (a) β=0.150, (b) β=0.128, (c) β=0.115, (d) β=0.092,
eledtheeffectof the crosstalk by the connections via coupling(e) β=0.0905, (f) β=0.0900.
capacitors or mutual inductors. By computer simulations, wehave confirmed that two Chua’s circuits are synchronized in3
in-pahse or anti-phase by the crosstalk effect of the transmisBifurcation 1
2.5
sion lines.
2In our future work, we investigate Chua’s circuit with lossy
transmission line, because real transmission lines should have
1.5
loss actually. Furthermore, we investigate crosstalk phenom1
ena between conductor boards placed in parallel and apply
0.5
the result to chaotic circuits.
0-0.5

References

-1
-1.5
-2
0.8

0.85

0.9

0.95

11.05

1.1

1.15

1.2

1.25

beta

Figure 6: Bifurcation diagram of Circuit Model 1 (β=0.1).

[1] J.A.K. Suykens, P.F. Curran and L.O. Chua, “MasterSlave Synchronization Using Dynamic Output Feedback,” International Journal of Bifurcation and Chaos,vol. 7, no. 3, pp. 671-679. 1997.

- 355 -[2] G.O. Zhong, C.W. Wu and L.O. Chua, “Torus-Doubling
Bifurcations in Four Mutually Coupled Chua’s Circuits,” IEEE Transactions on Circuits and Systems I,
vol. 45, no. 2, pp. 186-193. 1998.
[3] I.P. Marino, V. Perez-Munuzuri and M.A. Matias,“Desynchronization Transitions in Rings of Coupled
Chaotic Oscillators,” International Journal of Bifurcation and Chaos, vol. 8, no. 8, pp. 1733-1738. 1998.

(a)[4] M.J. Ogorzalek, A. Dabrowski and W. Dabrowski, “Hyperchaos, Clustering and Cooperative Phenomena in
CNN Arrays Composedof Chaotic Circuits”, Proceedings of IEEE Int. Workshop on Cellular Neural Networks and Their Applications (CNNA’94), pp. 315-320,1994.

(b)[5] A. Perez-Munuzuri, V. Perez-Munuzuri, V. Perez-Villarand L.O. Chua, “Spiral waves on a 2-D array of nonlinear circuits,” IEEE Transactions on Circuits and Systems I, vol. 40, no. 11, pp. 872-877. 1993.

(c)[6] J. Kawata, Y. Nishio and A. Ushida, “Analysis of Chua’s
Circuit with Transmission Line,” IEEE Transactions onCircuits and Systems I, vol. 44, no. 6, pp. 556-558, June
1997.

(d)

[7] J. Kawata, Y. Nishio and A. Ushida, “Chaos Synchronization by Transmission Line Coupling,” Proc. of 1996
International Symposium on Nonlinear Theory and its
Applications (NOLTA’96), pp. 109-112, 1996.

(e)[8] J. Kawata, Y. Nishio and A. Ushida, “On Synchronization Phenomena in Chaotic Systems Coupled by Transmission Line,” ISCAS 2000 IEEE International Symposium on Circuits and Systems, May 28-31, 2000.

(f)

Figure 7: Synchronization phenomena of Circuit Model
2 (γ=1.0). (a) β=4.9, (b) β=4.7, (c) β=4.3, (d) β=4.0,(e) β=3.8, (f) β=3.5.
3[9] M. Itoh, T. Yang and L.O. Chua, “Experimental Studyof Impulsive Synchronization of Chaotic and Hyperchaoti Circuits,” International Journal of Bifurcation
and Chaos, vol. 9, no. 7, pp. 1393-1424. 1999.
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Bifurcation 2

2.5
21.5

[11] F.Y. Chang, “Waveform Relaxation Analysis of RLCG
Transmission Lines,” IEEE Transactions on Circuits
and Systems I, vol. 37, no. 11, pp. 1394-1415, Nov.
1990.

x0

1
0.5
0-0.5
-1-1.5
-2
0.8

0.9

11.1
beta

1.2

1.3

1.4

Figure 8: Bifurcation diagram of Circuit Model 2 (β=3.5).

- 356 -