平成 26 年度電気関係学会四国支部連合大会 講演論文集 （2014 徳島大学）
2014 SHIKOKU-SECTION JOINT CONVENTION RECORD OF THE INSTITUTES OF ELECTRICAL AND RELATED ENGINEERS (TOKUSHIMA)

1-14Synchronization Phenomena and Chaos Propagationin Coupled Chaotic Circuits of Ladder System
Shogo TAMADA

Kenta AGO
Yoko UWATE(Tokushima University)

Yoshifumi NISHIO

1. Introduction

3. Simulation results

In this study, we investigate synchronization phenomena and chaos propagation in coupled chaotic circuits,each circuit is coupled by the resistor in the ladder system.
The central circuit is set to generate chaotic attractor andthe other circuits are set to generate three-periodic attractors. Moreover, we measure the phase diﬀerence bycomputer simulations. We investigate the relationships
between the coupling strength and the number of coupling circuits.

In this study, we set the parameters of the system as
αc = 0.46, αp = 0.412, β = 3.0 and δ = 470.0. Figure 3
shows the observed attractors and the phase diﬀerencesin the case of N = 5. By increasing g, three-periodic attractors are aﬀected from chaotic attractor and all circuitsclose to the synchronous state.
X3X2

X1X4X5
τ= 0Z2Z1Z4X4X3X2X1Z3Z5X52. System model

τ=Z4 TFigure 1 shows the chaotic circuit. We propose a systemmodel in Fig. 2. In this system, each circuit is coupledby the resistor in the ladder system. The central circuit
is generated chaotic attractor and the other circuits aregenerated three-periodic attractors. We observe the chaos
propagation by changing the coupling strength.
121dFigure 1: Chaotic circuit.
C2PPPCcenter

Z2
Z2Z1Z3Z4Z4Z3Z5Z5τ= TZ2Z3Z4Z5Z1Figure 3: Attractors and phase diﬀerences (N = 5), αc =0.46, αp = 0.412, g = 0.01, β = 3.0 and δ = 470.0.

2Ci

C1Z1CN-1

CNP

P

Moreover, the synchronous state can be defined if theaverage of the phase diﬀerence between C1 and CN below 10◦ . Figure 4 shows the relation between the coupling strength and the number of coupling circuits when
all circuits are synchronized in chaotic state. As a result,the coupling strength increases by increasing the numberof coupling circuits when all circuits are synchronized inchaotic state.

Three-periodic attractor (αp )Chaotic attractor (αc )Figure 2: System model.
The normalized circuit equations of the system aregiven as follows:dxi

= αxi + zi



 dτ
dyi
(1)
= zi − f (yi )
 dτ


 dzi = −xi − βyi − g(2zi − zi−1 − zi+1 ),dτ
where i = 1, 2, · · ·, N , g represents the coupling strength,
α represents the chaos degree and N represents the number of the coupling circuits. f (yi ) in the formula can be
expressed as follows:()11δyi +− yi −.
(2)f (yi ) =2
δδThe number of circuit is used an odd number as N =5, 7, 9, · · · , 21. For the computer simulations, we calculate Eq. (1) using the fourth-order Runge-Kutta methodwith step size h = 0.01.

Figure 4: Relation between the coupling strength and thenumber of coupling circuits.

4. Conclusions
In this study, we have researched about synchronization
phenomena and chaos propagation in coupled chaotic circuits of the ladder system. By computer simulations, wehave observed that chaotic attractor of the central circuit
propagates to all circuits. Moreover, we define the synchronous state from a certain phase diﬀerences, we measure the coupling strength in the synchronous state. By
increasing the number of coupling circuits, the couplingstrength almost proportionally increases.

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