令和3年度 電気・電子・情報関係学会 四国支部連合大会 講演論文集
2021 Shikoku-Section Joint Convention Record of the Institutes of Electrical and Related Engineers

1-20
Chaos propagation in Ladder Chaotic Circuits by Switching Coupling Strength
Naoto YONEMOTO Yoko UWATE
Yoshifumi NISHIO( Tokushima University)
1. Introduction

3. Results

Recently, chaos propagation has attracted much attention. In this study, chaos propagation is researched byswitching the coupling strength based on the diﬀerence ofthe voltages of the neighboring circuits.

The chaos propagation is changed by switching the coupling strength with the diﬀerence of xn and xn+1 as athreshold. The first method is to switch all the couplings
simultaneously, and the second method is to switch eachcoupling at a certain intervals. How to switch the couplingstrength is shown in Fig. 3.

2. System Model
The chaotic circuit is shown in Fig. 1. The system modelis shown in Fig. 2. In this system, the circuit in the one endof the system generates chaotic attractor and the other circuits generate three-periodic attractors. Ten chaotic circuits
next to each other are coupled together with resistors.

i1i2CL2v

L1-r

Cn

Figure 3: The algorithm for controlling chaotic propagation by coupling strength.(the threshold widths are all 0.2)

vdThe horizontal axis is the value of xn representing thecurrent flowing in L1 , and the vertical axis shows the valueof zn representing the voltage applied to C.

Figure 1: Circuit model.

1stChaos

2nd

3rd

4th

5th

6th

7th

8th

9th

10th

(a)
(b)

Chaotic attractor (αc)(c)

Three-periodic attractor (αp)(d)

Figure 2: System model.

Figure 4: Four patterns of chaos propagation.

The normalized circuit equations are described as follows:x˙n = αxn − zny˙n = zn − f (yn )z˙k = −xn − βyn −γn,n−1 (zn − zn−1 )− γn,n+1 (zn − zn+1 ) (n = 1, 2, · · ·, N )

(1)

where
γ1,0 =γN,N +1 = 0
f (yn ) = 2δ (|yn + 1δ | − |yn − 1δ |).

When all coupling strength is switched at the same time,the maximum number of circuits to propagate is 4 or 5.
However, when the coupling strength is switched for eachcoupling, propagation can be classified into four patterns asshown in the Fig. 4. When the first threshold width is 0 ≤|xn - xn+1 | ≤ 0.1, the chaos propagation can be suppressedas shown in the Fig. 4(a). When the voltage diﬀerence isless than 0.6 and the respective threshold widths are 0.1,0.2, and 0.3 regardless of the order, the propagation stateis the same as in Fig. 4(b). Using the value of the threshold
width in Fig.3, the propagation can be seen in Fig. 4(c).
As shown in Fig. 4(d), when the first or second threshold
width is 0.4, the propagation spreads out greatly.

4. Conclusion

For the computer simulations, we calculate Eq. (1) using
the fourth-order Runge-Kutta method with step size h =0.005. We set the parameters of this circuit model as follows; αc =0.460, αp = 0.413, β = 3.0 and δ = 470.0.

In this study, chaos propagation by switching the coupling strength was considered. It was found that there may
be a regularity of propagation depending on the thresholdof the coupling strength. As our future works, it will beconfirmed whether the observed regularity holds when thebifurcation parameter of chaos, the time to generate chaos,and the time to continue coupling are changed.

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