NOLTA 2006
11-14 september
Bologna - ITALY

Synchronization of Chaotic Circuits Linked by Cross Talk
Yuki Nakaaji and Yoshifumi Nishio
Department of Electrical and Electronic Engineering, Tokushima University
Email: {nakaaji, nishio}@ee.tokushima-u.ac.jp
sion lines could be synchronized by the effect of the crosstalk [7]. In this study, we investigate the synchronizationof the two Chua’s circuits linked by the cross talk in detail,especially, paying our attentions to the breakdown of chaos
synchronization.

Abstract—In this study, we investigate synchronization oftwo Chua’s circuits with a lossless transmission lines. Two circuits with transmission lines placed in parallel cause the crosstalk phenomena. The effect of the cross talk is modeled by theconnections via coupling capacitors or mutual inductors.
By computer simulations, we investigate the synchronizationof the two chaotic circuits linked by the cross talk. Especially,in this study, we pay our attentions to the breakdown of chaos
synchronization.

2. Basic Circuit Model [7]In our previous research, we reported that two Chua’scircuits with lossless transmission lines could be synchronized. The circuit model is shown in Fig. 2.

1. Introduction

i2R

Recently, several researchers have investigated synchronization of chaotic circuits. Chua’s circuit is one of thesimplest autonomous chaotic circuits. Chua’s circuit consists of a resistor, a capacitor, a nonlinear resistor and anLC resonator. The nonlinear resistor in Chua’s circuit hasa piecewise linear v − i characteristics as shown in Fig. 1
and it is described by the following equation;iR = m2 v1 + 0.5(m0 − m1 )[|v1 + Bp1 | − |v1 − Bp1 |]+ 0.5(m1 − m2 )[|v1 + Bp2 | − |v1 − Bp2 |],+-B p1++v1

C2-

Figure 2: Two Chua’s circuits with transmission lines.

(1)

We modeled the transmission lines by LC ladder circuitswith finite numbers of lumped elements. Further, we modeled the cross talk effect by coupling capacitors or mutualinductors. The circuit equations could be derived as follows;
dvj0
dtdv
Cj1 dtj1
dv
Cjk dtjk
diLjl dtjl

Cj0

Bp2-B p1

v2vR
m0-

m2
m1(a)

v10-

iC1

R1+L

Bp1-

C 10iRRv20

C 20
i1R

where m0 , m1 and m2 are the slopes in the segments ofthe piecewise linear function, and Bp1 and Bp2 denote thebreakpoints. There are many studies about coupled Chua’s
iR

R2(b)

==
=
=

vj1 −vj0−iRj (vj0 )Rjd(vj1 −v(j+1)1 )vj1 −vj0
ij1 − R+C1
dt
jd(vjk −v(j+1)k )ijk −ij(k−1) +Ck
dtd(ijk −i(j+1)k )Mvj(l+1) −vjl + 2 kdt
R C10(2)where v31 = v11 , vj(n+1) = 0, i31 = i11 , (k=2, 3, ..., n),(l=1, 2, ..., n) and (j=1, 2). From Eqs. (1) and (2), we could
obtain the normalized circuit equations as follows;

Figure 1: (a) Chua’s circuit. (b) v − i characteristics ofnonlinear resistor.
circuits, for example, master-slave coupling of Chua’s circuits [1], mutual coupling of Chua’s circuits [2], a ladder of Chua’s circuits, a ring of Chua’s circuits [3], twodimensional array of Chua’s circuits [4], etc. However,almost researches consider the couplings by lumped elements. We have investigated chaotic phenomena observed
from Chua’s circuit when the LC resonator is replaced bya transmission line [5]. Further, we have also investigatedsynchronization phenomena of two Chua’s circuits coupled
by a transmission line [6].
We consider two Chua’s circuits with lossless transmission lines placed in parallel. Transmission lines placedin parallel cause the cross talk phenomena. Cross talkphenomena appear in long powerlines or very high speed
VLSI and its effect is normally not preferable. We have already confirmed that the two Chua’s circuit with transmis-x˙10x˙11= x11 − x10 − f (x10 )= α11 (y11 − x11 + x10 )−β11 (y21 − x21 + x20 )x1k
˙= α1k (y1k − y1(k−1) ) − β1k (y2k − y2(k−1) )x˙20 = ζ(x21 − x20 − f (x20 ))x˙21 = α21 (y21 − ζ(x21 − x10 ))−β21 (y11 − x11 + x10 )x2k
˙= α2k (y2k − y2(k−1) ) − βjk (y1k − y1(k−1) )y˙jl = γjl (xj(l+1) − xjl ) + βl (x(j+1)(l+1) − xjl(3)where xj(n+1) = 0,f (xj0 ) = cj xj0 + 0.5(aj − bj )(|xj0 + 1| − |xj0 − 1|)+ 0.5(bj − cj )(|xj0 + dj | − |xj0 − dj |),(4)43t = R1 τ, “ ˙ 00 =dBp1
, vjk = Bp1 xjk , ijl =yjl ,dτR1

αjk =Cjk C10 (C(j+1)k − Ck ),2
2−Ck2 − (Cjk+ 1)Ck + Cjk
C(j+1)kβjk =Cjk C10 Ck,
22−Ck2 − (Cjk+ 1)Ck + Cjk
C(j+1)kβl =R12 C10 Ml,(L2l − Ml )(L1l − Ml ) − Ml2

γjl =R12 Cj0
C10, ζ=, aj = Rj m0 ,Ljl
C20(a) α1k =16.000000005, α2k =16.0, and β=4.0.

(b) α1k =16.000000010, α2k =16.0, and β=4.0.
4Bp2,
bj = Rj m1 , cj = Rj m2 , d =Bp1"result1.dat"
32

C3k = C1k , (k=2, 3, ..., n), (l=1, 2, ..., n) and (j=1, 2).
Some examples of chaos synchronization obtained bycomputer simulations are shown in Figs. 3 and 4. We confirmed that the circuits were synchronized in anti-phase in
the case of mutual inductors and in in-phase in the case ofcoupling capacitors.

x10 + x20

10-1-2-3-4times(c) Time waveform of x10 + x20 . Parameters are the same as (b).

Figure 5: Breakdown of synchronization. Cross talk effectis modeled by mutual inductors.
4. Cross Talk from Opposite Direction
Now, we consider the two Chua’s circuits with losslesstransmission lines placed in parallel but from the opposite
direction as shown in Fig. 7.
Similar to the circuit model in Fig. 2, we model the lossless transmission lines by LC ladder circuits with finite
numbers of lumped elements as shown in Fig. 8.
Further, we consider that the effect of the cross talk ismodeled by mutual inductors as shown in Fig. 9.
We carry out computer simulation for this models by using the 4th order Runge-Kutta method. In this simulation,
we consider only the case that the two Chua’s circuits areidentical. So, we can rewrite the parameters as follows;

Figure 3: Anti-phase synchronization via mutual inductors(γ=1.0 and β=4.0).

Figure 4: In-Phase synchronization via coupling capacitors(γ=1.0 and β=0.092).

α1 = α2 = α, γ1 = γ2 = γ, a1 = a2 = a,b1 = b2 = b,c1 = c2 = c, d1 = d2 = d.

(5)

In the following simulations, we fix the parameters as follows;a = −1.2,
b = −0.75, c = 10,
d = 8,(6)
α = 18,γ = 1,
n = 10.

3. Breakdown of Synchronization
In order to understand the synchronization phenomena
via cross talk effects of the transmission lines, we investigate the breakdown of the synchronization when the circuit
parameters are gradually changed.
We fix the coupling parameter β and increase the error
between the two Chua’s circuits. Figure 5 shows the computer simulated results when the cross talk effect is modeled by mutual inductors. As the error between the Chua’s
circuits α1k − α2k increases, the solution escapes from thesynchronized plane more often. Figure 5(c) shows the time
waveform of x10 + x20 . We can see from this figure thatthe solution behaves under the on-off intermittency.
Figure 6 shows the computer simulated results when thecross talk effect is modeled by coupling capacitors. Wecan observe the similar phenomena to the case of mutualinductors. However, the breakdown seems to be less sensitive to the error of the parameters than the case of mutualinductors.

4.1. Cross Talk via Mutual Inductor M
Figure 10 shows the synchronization of the two circuits.
It is interesting to observe that the two circuits can be synchronized even if they are placed from the opposite direction. The parameter m corresponds to the overlap of thetwo transmission lines. Namely, m = 10(= n) means that
all of inductors are coupled to one of the other inductors viamutual inductors. While m = 5 means that only the halfof the transmission lines is influenced each other. Further,m=1 means that only the last inductors in the two transmission lines are coupled each other.
4.2. Cross Talk via Mutual Inductor −M
Next, we consider the case that the cross talk effect ismodeled by negative mutual inductors. Figure 11 shows44

iR (v20)

R2iR (v10)

+

R1

v20

C 20-

C 10+

v10-

Figure 7: Two Chua’s circuits with transmission lines placed from the opposite direction.
L2nR1

C 10L11C 11C 2nL13L12C 12

L2n-1

C 13

C 2n-1

L2n-2
C 2n-2C 14

L23L22

C 23
L1nC 22...

C 1n-2L1n-2C 24
L1n-1

C 1n-1

C 1n

L21R2C 21

C 20

...
Figure 8: Discrete model by LC ladder circuits with finite number of lumped elements.

(a) α1k =16.001, α2k =16.0 and β=0.1.

(a) m = 10 and β = 6.5.

(b) α1k =16.003, α2k =16.0 and β=0.1.

(b) m = 5 and β = 9.

5"result10.date" every 10432x10 - x20

10-1(c) m = 1 and β = 14.

-2-3-4

Figure 10: Simulation results of cross talk via mutual inductor M .

-5times(c) Time wave form of x10 − x20 . Parameters are the same as (b).

ness of the synchronization against the coupling strength
is changed according to the position of the coupling of thetransmission line

Figure 6: Breakdown of synchronization. Cross talk effectis modeled by coupling capacitors.
the simulation results for the case that M is negative. In
this case, we confirm that the two circuits synchronize inin-phase.

5. Conclusions
In this study, we have considered the two Chua’s circuitswith lossless transmission lines placed in parallel. We havemodeled the effect of the cross talk by mutual inductors.
By computer simulations, we have investigated various interesting phenomena related with chaos synchronization.
In our future work, we investigate Chua’s circuit withlossy transmission line, because real transmission lines
should have loss actually. Furthermore, we investigate
crosstalk phenomena between conductor boards placed inparallel and apply the result to chaotic circuits.

4.3. Effect of Coupling Strength
Finally, we carry out computer simulation as varying β,namely, the coupling strength. Figure 12 shows the simulation result when m = 3. From Fig. 12, we can say thatthe synchronization is robust against the coupling strength.
While Fig. 13 shows the simulation result when m = 5.
This results show that the synchronization is sensitive tothe change of the coupling strength.
At the moment, we can not explain why the robust45

i 2n

i 2n-1i 2n-2i 23

i 22

i 21...
M1+v 2n+v 10-+

i 11+v 11v 12--M2+

v 2n-1-i 12M3Mm+v 2n-2-

i 13++v 14--

i 1n-2i 1n-1+v 1n-2...

+v 22-+v 13+v 23v 1n-1--+v 21--+v 20-i 1n+v 1n-

Figure 9: Model of cross talk effect by mutual inductors.

(a) m = 10 and β=−6.5.

(a) m = 3 and β = 6.3.

(b) m = 5 and β = −9.

(b) m = 3 and β = 7.0.

(c) m = 1 and β = −14.

(c) m = 3 and β = 8.0.

Figure 11: Simulation results of cross talk via mutual inductor −M .

Figure 12: Effect of coupling strength when m = 3.

References[1] J.A.K. Suykens, P.F. Curran and L.O. Chua, “Master-Slave
Synchronization Using Dynamic Output Feedback,” International Journal of Bifurcation and Chaos, vol. 7, no. 3, pp.
671-679, 1997.
[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.
[4] A. Perez-Munuzuri, V. Perez-Munuzuri, V. Perez-Villar andL.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.
[5] J. Kawata, Y. Nishio and A. Ushida, “Analysis of Chua’s Circuit with Transmission Line,” IEEE Transactions on Circuitsand Systems I, vol. 44, no. 6, pp. 556-558, 1997.
[6] J. Kawata, Y. Nishio and A. Ushida, “Chaos Synchronization by Transmission Line Coupling,” Proc. of NOLTA’96,
pp. 109-112, 1996.
[7] Y. Nakaaji and Y. Nishio, “Synchronization of Chaotic Circuits with Transmission Line,” Proc. of NCSP’06, pp. 353356, 2006.

(a) m = 5 and β = 8.7.

(b) m = 5 and β = 9.5.

(c) m = 5 and β = 9.9.

Figure 13: Effect of coupling strength when m = 5.

46