令和2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 (愛媛大学)
2020 SHIKOKU-SECTION JOINT CONVENTION RECORD OF THE INSTITUTES OF ELECTRICAL AND RELATED ENGINEERS (EHIME UNIV.)

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Synchronization Phenomena in Coupled
Two-Degrees-of-Freedom Chaotic Circuits by a Resistor
Naoto YONEMOTO

Katsuya NAKABAI Yoko UWATE( Tokushima University )Yoshifumi NISHIO

1. Introduction

3. Results

Asynchronous simultaneous oscillation is one of the typical non-linear phenomena and it is considered to exist in various high-dimensional systems in nature. Highdimensional chaotic circuits are actively studied, anda two-degrees-of-freedom chaotic circuit has been proposed [1]. We investigate synchronization phenomena when
two its chaotic circuits with asynchronous simultaneous oscillation are connected by a resistor.

In this study, three natural frequency cases are considered. The frequencies of x11 , x21 , x12 , x22 are corresponding to ω1 , ω2 , ω3 , ω4 , respectively. Figure 2 shows thecomputer simulation results.

2. System Model(a) ω1 = ω2 , ω3 = ω4 .

The circuit model is shown in Fig. 1. Two two-degreesof-freedom chaotic circuits are coupled by a resistor R.

(b) ω1 = ω2 , ω3 ̸= ω4 .

(c) ω1 ̸= ω2 , ω3 = ω4 .
Figure 2: Attractors and synchronization phenomena.

Figure 1: Circuit model.
The normalized circuit equations are described as follows:x11
˙ = −(y11 + z11 ) + α(x11 + x12 ) − δ(x11 + x12 −
x21 − x22 )



y11
˙ = x11z˙= β11 (x11 − f (z11 ))11x˙= −γ1 (y12 + z12 ) + αγ1 (x11 + x12 ) − γ1 δ(x11 +12x−x21 − x22 )12y˙=β12 x12
12

z12
˙ = β13 (x11 − f (z12 ))x21
˙ = −γ21 (y21 + z21 ) + αγ21 (x21 + x22 ) + γ21 δ(x11+x
12 − x21 − x22 )y˙= β21 x21

21

z
˙
=β22 (x21 − f (z21 ))
21



x˙
=−γ22 (y22 + z22 ) + αγ22 (x21 + x22 ) + γ22 δ(x1122
+x−x21 − x22 )12y˙=βx222223
(1)4. Conclusion
In this study, we confirmed that directly connecting two
previously separated circuits via a resistor induced synchronization between the two underlining subcircuits farthestfrom the newly created connection, while the two closest
subcircuits stayed asynchronous. As our future works, wewill investigate synchronization phenomena in the circuit
experiment.

z22
˙ = β24 (x21 − f (z22 )).

References

The characteristic equation for the diode is described asfollows:1(x + ε − |x − ε|).
f (x) = 2εIn the case Fig. 2(a), in-phase synchronization phenomena are confirmed between the upper subcircuits and between the lower subcircuits. The circuit parameters are chosen as α = 0.2, β11 = 6.0, β12 = 3.0, β13 = 18.0, β21 = 1.0,
β22 = 6.0, β23 = 3.0, β24 = 18.0, γ1 = 3.0, γ21 = 1.0,
γ22 = 3.0, δ= 0.1, and ε = 0.01. In the case Fig. 2(b), inphase synchronization phenomena are confirmed only between the upper subcircuits. The circuit parameters arechanged from case Fig. 2(a) to β12 = 2.0, β13 = 12.0,
γ1 = 2.0. In the case Fig. 2(c), although the subcircuits closest to the connection is asynchronous, it is confirmed thatthe lower subcircuits far from the connection are in-phasesynchronized. The circuit parameters are changed from caseFig. 2(a) to β21 = 2.0, β22 = 12.0, γ21 = 2.0.

(2)[1] Katunori SUZUKI, Yoshifumi NISHIO, and Shinsaku
MORI, “Twin Chaos - Simultaneous Oscillation ofChaos -”, The Institute of Electronics, Information andCommunication Engineers (IEICE), vol. j79-A, No. 3,
pp. 813-819, Mar. 1996.