平成25年度電気関係学会四国支部連合大会 講演論文集 （2013 徳島大学）
2013 SHIKOKU-SECTION JOINT CONVENTION RECORD OF THE INSTITUTES OF ELECTRICAL AND RELATED ENGINEERS (TOKUSHIMA)

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Synchronization of Two Chaotic Circuitswith LC Ladders Coupled by Resisters
Naoya Kondo, Yoko Uwate and Yoshifumi Nishio(Tokushima University)

1. Introduction
In this study, we investigate two simple chaotic circuitswith LC ladders coupled by resistors. By computer simulations, it is confirmed that the two chaotic circuits can besynchronized. Further, the existence of the solution on thein-phase synchronization can be explained theoretically.

2. Circuit Model
Figure 2: Characteristic roots.

Figure 1: Circuit model (Gm : coupling resistor).
The chaotic circuit used in this study is a modified
Chua’s circuit [1]. We consider the case that the two modified Chua’s circuits are placed in the opposite direction
as shown in Fig. 1. We can derive the normalized state
equations of this circuit as follows.

ẋ =−(x0 −x1 )−g(x0 ) 0
ẋk =α{(y(k−1) −yk )+γG (xk −X(11−k) )}ẏk =β (xk −x(k+1) )(c)(d)(e)(f)

Figure 3: Computer simulation results. (a) γG = 0.12.
(b) γG = 0.13. (c) γG = 0.15. (d) γG = 0.20. (e) γG= 0.31. (f) γG = 0.32.

Ẏk =β (Xk −X(k+1) )g(x) = bv − (a − b) {|x − 1| − |x + 1|}/2
where variables and parameters are defined by the followingequations.

For Figs. 3(b), (c), (d) and (e), the synchronization
turned into a complete in-phase synchronization as indicated in Fig. 1. However, for Fig. 3(a), the in-phase synchronization plane does not seem to be stable on the contrary to the result in Fig. 1. For the gammaG value inFig. 3(f), the attractor becomes point attractor, we could
not confirm the stability of the synchronization planes. Additionally, as far as we carried out computer simulations,
we could not confirm anti-phase synchronization of chaos
although the existence of the stable region is confirmed theoretically.

v0 =Ex0 , V0 =EX0 , vk =Exk ,Vk =EXk , ik =G0 Eyk , Ik =G0 EYk ,α=C0 /CS , β=C0 /(GLS ), γG =Gm /G0 , a=m0 /G0 , b=m1 /G0 ,t=G0 /C0 τ, “·”=d/dτ.

3. Simulation results
We fix the parameters α = 0.5, β = 0.15, a = −1.25,
b = −0.75, and investigate the influence of the coupling
parameters γG on synchronization. In order to check theexistence of synchronization states, we define the following
dependent variables and substitute into Eq. (1).
In−phase: x̂0 =x0 −X0 , x̂k =xk −Xk , ŷk =yk −Yk )

(b)

(1)

Ẋ0 =−(X0 −X1 )−g(X0 ) Ẋk =α{(Y(k−1) −Yk )+γG (Xk −x(11−k) )}(a)

4. Conclusions
In this study, we investigated the synchronization of twosimple chaotic circuits with LC ladders coupled by resistors.
By carrying out computer calculations, we observed chaossynchronizations.

(2)Anti−phase: x̂0 =x0 +X0 , x̂k =xk +Xk , ŷk =yk +Yk )Next, the characteristic roots of the synchronization
planes are investigated and these values are summarized
in Fig. 2. Figure 3 shows some examples of computer simulation results. The left figures are the attractors and theright figures are the synchronization states between the twocircuits.

References[1] Yuki Nakaaji and Yoshifumi Nishio, “Synchronizationof Chaotic Circuits with Transmission Line,” Proc. ofNCSP’06, pp. 353-356, Mar. 2006.

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