平成 21 年電気学会電子・情報・システム部門大会

TC4-5

CHAOS SYNCHRONIZATIONBY CROSSTALK OF TRANSMISSION LINES
Yoshifumi Nishio(Tokushima University)
AbstractiR

In this study, two Chua’s circuits with lossless transmission lines placed in parallel are investigated. By computer
simulations, it is conﬁrmed that the two Chua’s circuits canbe synchronized by the eﬀect of the crosstalk. Further, theexistence of the solution on the in-phase synchronization
can be explained theoretically.

EKeywords: chaos synchronization, Chua’s circuit, crosstalk, transmission line.

m0m1

1. Introduction
Chua’s circuit is one of the simplest autonomous chaoticcircuits. Chua’s circuit consists of a resistor, a capacitor, anonlinear resistor and an LC resonator as shown in Fig. 1.
The nonlinear resistor in Chua’s circuit has a piecewise linear v − i characteristics shown in Fig. 2.
Chua’s circuit whose LC resonator is replaced by a transmission line (Fig. 3) is referred to as the “time-delayed
Chua’s circuit.” 1−4 Because the time-delayed Chua’s circuit with C0 = 0 is governed by a one-dimensional difference equation, detailed analysis using the corresponding
one-dimensional map is possible 1,2 . However, in that system we can not observe the same chaotic phenomena as
those observed from the original Chua’s circuit, e.g., Double Scroll attractors. Hosny et al. have proposed an analytical method for solving nonlinear lumped circuits with alossless transmission line. As an example, chaotic behaviorof the time-delayed Chua’s circuit is analyzed for the caseof C0 6= 0 3 . Kawata and the author also investigated thesame circuit with more general transmission line 4 .

Fig. 2: v − i characteristics of nonlinear resistor.

GiRC0Fig. 3: Time-delayed Chua’s circuitwith transmission lines could be synchronized by the eﬀectof the crosstalk. 5 In this study, the synchronization of two
Chua’s circuits linked by the crosstalk is analyzed in moredetail, especially, paying the attentions to the synchronization plane.

G2. Chua’s Circuits with Transmission Lines
Placed with Same Direction

iiR
C0v0-Ev0CFirst, two Chua’s circuits with transmission lines are considered when they are placed in parallel as shown in Fig. 4.
Crosstalk of the transmission lines can be expressed by coupling capacitors CM and mutual inductors LM after replacing the transmission lines by lumped ladder LC circuits asshown in Fig. 5.
The circuit equations governing the circuit model in Fig. 5
can be described by the following equations.

v
L−(x10 − x11 ) − f (x10 )−(x20 − x21 ) − f (x20 )α{(1 + γC )(y1(k−1) − y1k ) + γC (y2(k−1) − y2k )}α{(1 + γC )(y2(k−1) − y2k ) + γC (y1(k−1) − y1k )}β{x1k − x1(k+1) − γL (x2k − x2(k+1) )}β{x2k − x2(k+1) − γL (x1k − x1(k+1) )}(1)
where k = 1, · · · , n, y10 = x10 − x11 , y20 = x20 − x21 , andx1(n+1) = x2(n+1) = 0. Nonlinear function corresponding

Fig. 1: Chua’s circuit.

ẋ10ẋ20ẋ1kẋ2kẏ1kẏ2k

In this study, two Chua’s circuits with lossless transmission
lines placed in parallel are considered. Transmission lines
placed in parallel cause the crosstalk phenomena. Crosstalk
phenomena appear in long powerlines or very high speed
VLSI and its eﬀect is normally not preferable. Nakaaji andthe author have already conﬁrmed that two Chua’s circuits

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平成 21 年電気学会電子・情報・システム部門大会

GiR
C0GiR
C0Fig. 4: Chua’s circuits with transmission lines placedwith same direction.

Ls i21

Gi2R

Lsv22

v21
C0

Csv20

Lsi22

Cs

LMCMCMG

Ls i11i23(a) α = 0.5, β = 17.0, αγC = γL = 0.5.

v23LMCsCsLMCM
Lsv2n

LsCM
Lsi12i2ni13
LMi1RC0Csv10

v11v12

Cs

Cs

v13

Cs

v1n

Ls
i1n

Fig. 5: Discretized circuit model of Chua’s circuitswith transmission lines placed with same direction.
to the v − i characteristics of the nonlinear resistor can be
written as
f (x) = bx − (a − b)|x − 1| − |x + 1|.
2(b) α = 0.4, β = 19.0, αγC = γL = 0.4.

Fig. 6: Computer simulation results.
a = −1.2, b = −0.75.

(2)

n = 10,

Variables and parameters are deﬁned by the following equations.
v10 = Ex10 ,v20 = Ex20 ,v2k = Ex2k ,i1k = GEy1k ,a=m0,
Gb=m1,
G

v1k = Ex1k ,i2k = GEy2k ,C0,
CS + 2CMCMLM
γC =, γL =,CS
LS

α=C0 LS,β= 2 2G (LS − L2M )C0dt=
τ, “ · ” =,
G
dτ

(3)
GiR

(k = 1, · · · , n).

C0G

Figure 6 shows the computer calculated results of Eq. (1).
We can see that the two circuits can be synchronized completely as Fig. 6(a), if the coupling is relatively large. As
decreasing the coupling parameters, the synchronization becomes incomplete, although the solution tends to stay close
to the synchronization plane as Fig. 6(b).

iRC03. Chua’s Circuits with Transmission Lines
Placed with Opposite Direction

Fig. 7: Chua’s circuits with transmission lines placedwith opposite direction.

Next, the same two Chua’s circuits with transmission lines
are considered, but they are placed in parallel from theopposite direction as shown in Fig. 7.

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平成 21 年電気学会電子・情報・システム部門大会

With the same notations of the variables and the parameters as Eq. (3), the circuit equations are derived as follows.

The following dependent variables are deﬁned.
pxk
pyk−(x10 − x11 ) − f (x10 )−(x20 − x21 ) − f (x20 )α{(1 + γC )(y1(k−1) − y1k )+ γC (y2(n−k) − y2(n−k+1) )}ẋ2k = α{(1 + γC )(y2(k−1) − y2k )+ γC (y1(n−k) − y1(n−k+1) )}ẏ1j = β{x1k − x1(k+1) + γL (x2(n−k) − x2(n−k+1) )}ẏ2j = β{x2k − x2(k+1) + γL (x1(n−k) − x1(n−k+1) )}2ẏ1n = β(1 − γL)x1n2ẏ2n = β(1 − γL)x2n(4)where j = 1, · · · , n − 1, k = 1, · · · , n, y10 = x10 − x11 , andy20 = x20 − x21 .
Figure 8 shows the computer calculated results of Eq. (4).
Similar to the case of the same direction, the two circuits
can be synchronized, even though the transmission lines areplaced from the opposite direction.
ẋ10ẋ20ẋ1k=====x1k − x2k
y1k − y2k(5)where k = 0, · · · , n. The diﬀerence system can be derived
from Eq.(1) and is described using the variables as follows.
ṗx0ṗxkṗyk===−(px0 − px1 ) − {f (x10 ) − f (x20 )}α(py(k−1) − pyk )β(1 + γL )(pxk − px(k+1) )(6)

where k = 1, · · · , n and px(n+1) = 0. Since the term f (x10 )−f (x20 ) becomes apx0 or bpx0 when the solution stays in thesame linear region, the solution pxk = pyk = 0 for all kapparently satisﬁes Eq. (6). This means that the solutionof Eq. (1) exists on the in-phase synchronization space.
Next, let us consider the case of the opposite direction. Using the same method, the diﬀerence system of Eq.(4) canbe derived as follows.
−(px0 − px1 ) − {f (x10 ) − f (x20 )}α{(1 + γC )(py(k−1) − pyk )− γC (py(n−k) − py(n−k+1) )}ṗyj = β{pxj − px(j+1) − γL (px(n−j) − px(n−j+1) )}2ṗyn = β(1 + γL)pxn(7)
where j = 1, · · · , n − 1 and k = 1, · · · , n. Same as theprevious case, the solution pxk = pyk = 0 for all k satisfy
Eq. (7) and the solution of Eq. (4) also exists on the in-phasesynchronization space.
ṗx0ṗxk==5. Conclusions
In this study, two Chua’s circuits with lossless transmission lines placed in parallel have been investigated. By computer simulations, it was conﬁrmed that two Chua’s circuits
could be synchronized by the eﬀect of the crosstalk. Further, the existence of the solution on the in-phase synchronization could be explained theoretically.
Investigating the stability of the synchronization plane isone of important future problems.

(a) α = 0.5, β = 17.0, αγC = γL = 0.5.

References[1] A.N. Sharkovsky, “Chaos from a Time-Delayed Chua’sCircuit,” lEEE Transactions on Circuits and Systems I,
vol. 40, no. 10, pp. 781-783, Oct. 1993.
[2] Yu.L. Maistrenko, V.L. Maistrenko and S.I. Vikul,“Bifurcations of Attracting Cycles from Time-Delayed
Chua’s Circuit,” International Journal of Bifurcation andChaos, vol. 5, no. 3, pp. 653-671, 1995.
[3] E.A. Hosny and M.I. Sobhy, “Analysis of Chaotic Behavior in Lumped-Distributed Circuits Applied to theTime-Delayed Chua’s Circuit,” lEEE Transactions onCircuits and Systems I, vol. 41, no. 12, pp. 915-918, Dec.
1994.

(b) α = 0.4, β = 19.0, αγC = γL = 0.47.

Fig. 8: Computer simulation results.
a = −1.2, b = −0.75.

n = 10,[4] Junji Kawata, Yoshifumi Nishio and Akio Ushida,“Analysis of Chua’s Circuit with Transmission Line,”
IEEE Transactions on Circuits and Systems I, vol. 44,
no. 6, pp. 556-558, Jun. 1997.

4. Synchronization Plane[5] Yuki Nakaaji and Yoshifumi Nishio, “Synchronizationof Chaotic Circuits Linked by Cross Talk,” Proc. ofNOLTA’06, pp. 43-46, Sep. 2006.

In order to analyze the synchronization theoretically, thesynchronization plane can be considered.

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