IEEE Workshop on Nonlinear Circuit Networks
December 13-14, 2013

Syn hronization of Two Chaoti Cir uitswith Transmission Line Coupled by Resistors
Naoya Kondo, Yoko Uwate, Yoshifumi Nishio

Dept. Ele tri al and Ele troni Eng.,
Tokushima University
Email: { konnao, uwate, nishio} ee.tokushima-u.a .jp
Abstra t

m1 (v − E) + m0 E
m0 v
f (v) = m1 (v + E) − m0 E

In our past studies, we have investigated syn hro-nization phenomena ofoupled by thehaoti

ir uits with transmission lines

ross talk lo ated in the same dire tion or thereverse dire tion. In this study, we investigate two Chua

ir uits

with transmission lines lo ated in the reverse dire tion, when thelines areoupled by resistors. The original Chua

syn hronize easily, if the different

apa itors areresistors. However, the syn hronization

ir uits

(v > E)(|v| ≤ E) (v < −E)(2)

B. Cir uit Model

annot

onne ted byan be observed from ourir uit model.

I. I NTRODUCTION
In our previous study, the syn hronization of two haoti
ir uits with transmission line oupled by the ross talk wereinvestigated. We observed in-phase and anti-phase syn hronization phenomena from these ir uits in omputer simulations. In this study, we investigate the syn hronization of
two haoti ir uits with transmission lines oupled in reverseby resistors. The original Chua ir uits annot syn hronize
easily, if the different apa itors are onne ted by resistors. We
investigate the syn hronization, when the resistors' positionand size are hanged.
II. C IRCUIT M ODELFig. 2.

Two Chuair uits with transmission lines

oupled by resistors.

Figure 2 shows two Chua ir uits with transmission lines
oupled by resistors. We onsider the ase that the two Chuair uits are pla ed in the opposite dire tion as shown in Fig. 2.
We an derive the normalized ir uit equations of this ir uit
as follows.
ẋ0 = − (x0 − x1 ) − g (x0 )









ẋk = α y(k−1) − yk − γG xk − X(11−k) ẏ = β x − xA. Chua's Cir uit

In this study, we investigate two Chua ir uits in ludingtransmission lines oupled in reverse by resistors. Figure 1
shows the original Chua ir uit.

kk(k+1)Ẋ0 = − (X0 − X1 ) − g (X0 )



 Ẋk = α Y(k−1) − Yk − γG Xk − x(11−k)Ẏk = β Xk − X(k+1)(3)(k=1,2,· · · ,10)(a)Fig. 1.

(a) Chuair uit. (b) v − ia (x − 1) + bbx
g(x) = a (x + 1) − b(b)hara teristi s of nonlinear resistor f (v).

(4)

where variables and parameters are dened by the followingequations.

Cir uit equation of Fig. 1 is derived as follows.
dv0C0= −G0 (v0 − v1 ) − f (v0 )
dt



dv1= G0 (v0 − v1 ) − i1CS
dt





 LS di1 = v1dt(x > 1)(|x| ≤ 1) (x < −1)

α=(1)

C0
C0Gm
m0m1, β = 2 , γG =, a=, b=,
CS
G LS
G0
G0
G0

v0 = Ex0 , V0 = EX0 , vk = Exk , Vk = EXk ,ik = G0 Eyk , Ik = G0 EYk ,- 17 -dC0 dd=, “·”=.
dtG0 dτdτIII. S TABILITY ANALYSIS
We ondu t stability analysis. First, in order to investigate
the in-phase syn hronizations, the new variables orresponding
to the differen e between the two ir uits (x̂0 = x0 −X0 , x̂k =xk − Xk , ŷ0 = y0 − Y0 ) are introdu ed. The differential
equations of these variables (5) show the displa ement of thesolution from the syn hronous plane. Therefore, the stabilityof the in-phase syn hronization an be judged by the Ja obianmatrix's hara teristi roots at the origin (x̂0 = x̂k = ŷ0 = 0).
ˆ0 = −x̂0 + x̂1 + h(x̂0 ) ẋ
ˆẋ = α ŷ(k−1) − ŷk − γG x̂k + x̂(11−k) ˆkẏk = β x̂k − x̂(k+1)
−bx̂0 (|x̂0 | > 1)h(x̂0 ) =−ax̂0 (|x̂0 | ≤ 1)

(5)

(b)

( )(d)(6)

Next, in order to investigate the anti-phase syn hronizations,the new variables orresponding to the sum of the two ir uits(x̂0 = x0 + X0 , x̂k = xk + Xk , ŷ0 = y0 + Y0 ) are introdu ed.
The differential equations of these variables (7) show thedispla ement of the solution from the syn hronous plane.
Therefore, the stability of the anti-phase syn hronization anbe judged by the Ja obian matrix's hara teristi roots at theorigin (x̂0 = x̂k = ŷ0 = 0).
ˆ0 = −x̂0 + x̂1 + h(x̂0 ) ẋˆk = α ŷ(k−1) − ŷk − γG x̂k − x̂(11−k)ẋ ˆẏk = β x̂k − x̂(k+1)

(a)

(7)

(e)
Fig. 4.

(f)Attra tors of their uit in Fig. 2. (a) γG = 0.12. (b) γG = 0.13. ( )γG = 0.15. (d) γG = 0.20. (e) γG = 0.31. (f) γG = 0.32.

an not onrm the stability of the syn hronization planes.
Additionally, as far as we arry out omputer simulations, we
an not onrm anti-phase syn hronization of haos although
the existen e of the stable region is onrmed theoreti ally.
B. Conne ted Outside Capa itors in Transmission Line

IV. S IMULATION RESULTS
We x the parameters α = 0.5, β = 0.15, a = −1.25, b =−0.75, and investigate the inuen e of the oupling parameters
γG on syn hronization.
A. Conne ted All Capa itors in Transmission LineFig. 5.

Two Chuaresistors at the outside(a)Fig. 3.

Computer simulation results of their uits with transmission line

oupled in reverse byapa itors.

(b)
ir uit in Fig. 2. (a) Stability

analyses result. (b) Computer simulations result.

Figure 3 shows some examples of omputer simulation
results of Fig. 2. The left gures are the attra tors of theChua ir uit with transmission line and the right gures are thesyn hronization states between the two ir uits. For Figs. 4(b),( ), (d) and (e), the syn hronization turn into a omplete inphase syn hronization as indi ated in Fig. 2. However, forFig. 4(a), the in-phase syn hronization plane does not seemto be stable on the ontrary to the result in Fig. 2. For theγG value in Fig. 4(f), the attra tor be omes point attra tor, we(b)(a)Fig. 6.

Computer simulation results of their uit in Fig. 5. (a) Stability

analyses result. (b) Computer simulations result.

Figure 5 shows two Chua ir uits with transmission line
oupled in reverse by resistors only at some outside apa itors.
Figures 6 and 7 show some examples of omputer simulation
results. The in-phase syn hronization is not seen, when thedouble s roll attra tors appear.

- 18 -

(a)Fig. 7.

( )Attra tors of the(b)

(a)

(b)

(d)

( )

(d)

ir uit in Fig. 5. (a) γG = 0.20. (b) γG = 0.21. ( )γG = 0.33. (d) γG = 0.34.

C. Conne ted Inside Capa itors in Transmission Line

Fig. 10.

(e)Attra tors of the(f)ir uit in Fig. 8. (a) γG = 0.18. (b) γG = 0.19.

( ) γG = 0.75. (d) γG = 0.76. (e) γG = 1.39. (f) γG = 1.40.

development of the analysis te hnique of signal ir uits withtransmission lines and the elu idation of the syn hronizationme hanism of haoti ir uits in lude the distributed onstantparameters.
Fig. 8.

Two Chuaresistors at the inside

ir uits with transmission line

R EFERENCES

oupled in reverse byapa itors.

[1℄ Yuki Nakaaji and Yoshifumi Nishio, Syn hronization of ChaotiCir-uits with Transmission Line, Pro . of NCSP'06, pp. 353-356, Mar.
2006.
[2℄ Yuki Nakaaji and Yoshifumi Nishio, Syn hronization of Chaoti

Cir-uits Linked by Cross Talk, Pro . of NOLTA'06, pp. 43-46, Sep.2006.

(a)Fig. 9.

Computer simulation results of the(b)ir uit in Fig. 8. (a) Stability

analyses result. (b) Computer simulations result.

Figure 8 shows two reverse Chua ir uits with transmission
line oupled by resistors only at some inside apa itors.
Figures 9 and 10 show some examples of omputer simulation results. The number of oupling resistors is de reased,
however the syn hronization range by γG of the oupling
parameters be omes large. The syn hronization of doubles roll attra tors an be seen in a wide range of the oupling
parameter γG .
V. C ONCLUSIONSIn this study, we investigated two Chua ir uits with transmission lines oupled in reverse by resistors. We onrmedthat these ir uits were syn hronized in in-phase by omputer simulations. Moreover, even small number of oupling
resistors ould realize the syn hronization in a wide rangeof oupling parameter γG . From these results, we expe t the- 19 -