2006 IEEE Workshop on Nonlinear Circuit Networks (NCN'06)
Tokushima University, Tokushima, Japan, Dec. 9, 2006

Research on Synchronization of Chua’s Circuit withTransmission Lines
Yuki Nakaaji

Yoshifumi Nishio

Dept. Electrical and Electronic Eng.,
Tokushima University
Email: nakaaji@ee.tokushima-u.ac.jp

Dept. Electrical and Electronic Eng.,
Tokushima University
nishio@ee.tokushima-u.ac.jp

Abstract— In this study, we investigate synchronization of two
Chua’s circuits with a lossless transmission lines. Two circuitswith transmission lines placed in parallel cause the cross talkphenomena.
By computer simulations, we investigate the synchronizationof the two chaotic circuits linked by the cross talk. .

β = 6.5.

I. I NTRODUCTION
Recently, several researchers have investigated synchronization of chaotic circuits. Chua’s circuit is one of the simplest
autonomous chaotic circuits. Chua’s circuit consists of aresistor, a capacitor, a nonlinear resistor and an LC resonator.
By replacing parallel LC resonator in original Chua’s circuit,we can consider it is “time-delayed Chua’s circuit”.[1] We
consider two “time-delayed Chua’s circuits” placed in parallel.
Transmission lines placed in parallel cause the cross talkphenomena.
II. C ROSS TALK FROM O PPOSITE D IRECTION
In our previous research, we reported that two Chua’s circuits with lossless transmission lines could be synchronized.[2]We model the lossless transmission lines by LC ladder circuitswith finite numbers of lumped elements as shown in Fig. 1.
We carry out computer simulation for this models by using the4th order Runge-Kutta method. In this simulation, we consider
only the case that the two Chua’s circuits are identical.
L2nR1C 10L11C 11C 2nL13L12C 12L2n-1

C 13

C 2n-1L2n-2...

C 2n-2

C 14

C 1n-2

L1n-2

C 24
L1n-1

C 1n-1

C 1n

L23

L22

C 23
L1n

C 22

L21

Fig. 2.

Simulation results of cross talk via mutual inductor M .

β=−6.5.
Fig. 3.

Simulation results of cross talk via mutual inductor −M .

It is interesting to observe that the two circuits can be synchronized even if they are placed from the opposite direction.
III. R ESEARCH
We define equations as follows;
pkqlp˙0
C 20p˙1p˙kq˙1q˙2...

Fig. 1. Discrete model by LC ladder circuits with finite number of lumpedelements.

..
.

We could obtain the normalized circuit equations as follows;q˙lxj0
˙xj1
˙xjk
˙yj1
˙yj2
˙

===
==..
.
y˙jl

=(1)

γjl (xj(l+1) −xjl ) − β(x(j+1)2 − xj1 )wherex3k = x1k , xj(n+1) = 0, i31 = i11 , (k=2, 3, ..., n), (l=1,2, ..., n) and (j=1, 2),f (xj0 )=cj xj0 +0.5(aj −bj )(|xj0 +1|−|xj0 −1|)+0.5(bj −cj )(|xj0 +dj |−|xj0 −dj |),

(3)

=
=
=
=
=
=

C0
C1 p1 −p0 −f (p0 )α(q1 −p1 +p0 )α(qk −qk−1 )γ(q2 −q1 )−β(qn )γ(q3 −q2 )−β(qn−1 −qn )(4)γ(ql+1 −ql )−β(x1 −x2 )This equation shows existence of the solution on the synchronous plane because pk = ql = 0 satisfy Eq. (4).

Cj0
Cj1 xj1 −xj0 −f (xj0 )

αj1 (yj1 −xj1 +xj0 )
αjk (yjk −yj(k−1) )γj1 (xj2 −xj1 )−β(−x(j+1)n )γj2 (xj3 −xj2 )−β(x(j+1)n −x(j+1)(n−1) )x1k −x2ky1l −y2l

we can obtain as follows equation

R2C 21==(2)

IV. C ONCLUSIONS
By computer simulations, we have investigated various
interesting phenomena related with chaos synchronization. We
confirmed that the existence of the solution on the synchronousplane. In our future work, we investigate the stability of thesynchronous plane.
R EFERENCES[1] J. Kawata, Y. Nishio and A. Ushida, “Analysis of Chua’s Circuit withTransmission Line,” IEEE Transactions on Circuits and Systems I, vol. 44,
no. 6, pp. 556-558, 1997.
[2] Y. Nakaaji and Y. Nishio,“Synchronization of Chaotic Circuit Linked
by Cross Talk,” Proceedings of International Symposium on Nonlinear
Theory and its Applications, pp. 43-46, Sep. 2006-5-