平成 27 年度電気関係学会四国支部連合大会 講演論文集 (2015 高知工科大学)
2015 SHIKOKU-SECTION JOINT CONVENTION RECORD OF THE INSTITUTES OF ELECTRICAL AND RELATED ENGINEERS (KOCHI)

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Chaos Propagation in Coupled Chaotic Circuits
Takahiro CHIKAZAWA

Shogo TAMADA Yoko UWATE( Tokushima University)

1. Introduction

Yoshifumi NISHIO

and γ is the coupling strength. f (yi ) is described as follows:

In this study, we investigate chaos propagation in coupledchaotic circuits when one circuit is set to generate chaotic
attractor and the other circuits are set to generate threeperiodic attractors. Moreover, we observe how to propagate
chaos by increasing the coupling strength.

f (yi ) =12

11− yi −δδyi +.

(2)

The coupling strength which connected with 6th is γc , andthe others are γp . We define αc to generate the chaotic
attractor, and αp is defined to generate the three-periodic
attractors.

2. System model
The chaotic circuit is shown in Fig. 1. This circuit consists of a negative resistor, two inductors, a capacitor anddual-directional diodes. We propose a system model inFig. 2. In this system, 6th circuit generates chaotic attractor and the other circuits generate three-periodic attractors.
In order to investigate the chaos propagation, we use twoparameters of coupling strength. Each circuit is coupled by
the resistor.

In this study, we set the parameters of the system as
αc = 0.460, αp = 0.412, β = 3.0 and δ = 470.0. Figure 3 shows the initial state when all circuits are not connected. Figure 4 shows the observed attractors by increasing γp when we fix the coupling strength as γc = 0.0007.
The 5th circuit is only propagated the chaotic attractor of
6th chaotic circuit (see. Fig. 4(a)). The chaotic attractorof 5th circuit propagates to both side of the neighbor circuits (see. Fig. 4(b)). The states of all circuits become thechaotic attractors (see. Fig. (c)). As a result, three-periodic
attractors are affected from chaotic attractor by increasing
γp .

i2

i1
L1

L2

Civ

C-r

3. Simulation results

vdz1

z2z3z4z5z6(a)γp=0.0000

x1

Figure 1: Chaotic circuit.

x2

1st2ndx3

2ndx4

3rdx5

4thx6

5th

6th

Figure 3: Chaos propagation (γc = 0.0000).

Pγpγp
3rd

1stPPC 6thγpγcP

Three-periodic attractors (αp )CChaotic attractor (αc )

z1

z2

z3

z4

z5z6(a)γp=0.0007

x1

z1

γpx2

z2x3

z3x4

z4x5z5x6z6(b)γp=0.0010

P5th

γpx1

P

z1

4thx2

z2x3

z3x4

z4x5z5x6z6(c)γp=0.0015

x1

Figure 2: System model.

1stThe normalized circuit equations of the system are given
as follows:x2
2ndx3
3rdx4
4thx5
5thx6
6th

Figure 4: Chaos propagation (γc = 0.0007).

dxi
= αxi + zi

dτ

dyi


= zi − f (yi )4. Conclusions

dτNX

dzi


γij (zi − zj )=−x−βy−
ii

dτ


i,j=1



In this study, we have investigated chaos propagation incoupled chaotic circuits as our proposed system. By thecomputer simulations, We have observed that the chaotic
attractor is propagated to the other circuits. The threeperiodic attractors are affected from the chaotic attractors
when the coupling strength increase.
As our future works, we develop the network model into
cubic and more complex. Considering the other network ofchaotic circuits is important subjects for us.

(1)(i, j = 1, 2, · · ·, N ).

In Eq. (1), N is the number of coupled chaotic circuits

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