平成 29 年度電気関係学会四国支部連合大会 講演論文集 (2017 愛媛大学)
2017 SHIKOKU-SECTION JOINT CONVENTION RECORD OF THE INSTITUTES OF ELECTRICAL AND RELATED ENGINEERS (EHIME)

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Synchronization Phenomena in a Ringof van der Pol Oscillators Coupledby Time-Varying Resistor
Toshiya MATSUNASHI

Daiki NARIAI Yoko UWATE(Tokushima University)

Yoshifumi NISHIO

1. Introduction

3. Simulation results

Synchronization is one of the fundamental phenomena innature and it is observed over the various fields.
In this study, we propose coupled oscillatory systems suchas a ring of van der Pol oscillators coupled by Time-Varying
Resistor (TVR). We investigate synchronization phenomena observed in the proposed circuit system by changing
the frequency of TVR.

The simulation results of the system model are shown
from Fig. 3 to Fig. 6. The value of the parameters are setto ε = 0.1, γ1 = ± 0.01, γ2 = ± 0.01. A frequency of TVR1and TVR2 sets with f1 and f2 . Where, fn = 2πωn . The
figure on the left shows phase diﬀerence when the initialcondition set with in-phase. The figure on the right showsphase diﬀerence when the initial condition set with 3-phase.
In case of f1 = f2 = 0.09, we can observe in-phase synchronization phenomena and 3-phase synchronization phenomena. We can observe both synchronization phenomena due
to change by initial value. By chaging f1 and f2 , we observe
only 3-phase synchronization phenomena, it is observed regardless of initial value. In case of f1 = 0.09, f2 = 0.1, we
can observe both synchronization phenomena. In case of f1= 0.03, f2 = 0.219, circuit2 and circuit3 exhibit synchronization phenomena regardless of initial value. We observe
synchronization phenomena by changing the frequency.

2. System modelFigure 1 shows a system model constituted van der Poloscillators (VDP). We use a ring of van der Pol oscillators,
three VDP are connected by Time-Varying Resistor (TVR1and TVR2). We realize the TVR by switching a positive
and a negative resistor periodically as shown in Fig. 2. We
investigate synchronization phenomena, how changing by
the frequency of TVR1 and TVR2.
circuit1(a) in-phase
R1TVTV
R1

VDP
VDPgCLVDPVDPTVR2

circuit2(b) 3-phase

Figure 3: Phase diﬀerence (f1 = f2 = 0.09).

circuit3(a) System model.

(b) van der Pol oscillator.

Figure 1: Circuit model.
Figure 4: Phase diﬀerence (f1 = f2 = 0.1).
γ1,γ22π0.01

Figure 5: Phase diﬀerence (f1 = 0.09, f2 = 0.1).
0ωτ−0.01

Figure 6: Phase diﬀerence (f1 = 0.03, f2 = 0.219).

Figure 2: Characteristics of the TVR.
4. Conclusions
We have proposed a system model using a ring of threevan der Pol oscillators coupled by TVR. We can observevarious synchronization phenomena by varying frequencyof TVR. When two frequencies of TVR (f1 , f2 ) equal,
sometimes we can observe synchronization phenomena, andsometimes 3-phase synchronization phenomena. However
when we changed frequency f1 and f2 , three oscillators ofa ring possible to see behavior diﬀerent from equivalent ina frequency. In the future, we will investigate the synchronization phenomena using diﬀerent frequency of the threeTVRs.

The normalized equations of VDP circuit are given asfollows:x˙1x˙2 x˙3y˙n=ε(x1 − x31 ) − y1 + γ1 (x2 + x3 − 2x1 )===ε(x2 − x32 ) − y2 + γ1 (x1 − x2 ) + γ2 (x3 − x2 )ε(x3 − x33 ) − y3 + γ1 (x1 − x3 ) + γ2 (x2 − x3 )
xn .

(1)

In these equations, n is the number of circuit and n = 1,2,3.
ε denotes the nonlinearity of the oscillators, γ1 , and γ2 denote the coupling strengths of the TVR1 and TVR2.

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