Synchronization Phenomena in a Ring of van der Pol Oscillators
Coupled by Time-Varying Resistor
Toshiya Matsunashi, Daiki Nariai, Yoko Uwate and Yoshifumi Nishio
Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
E-mail: {matsunahi, nariai, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract

2. System model

Figure 1 shows van der Pol oscillator and Fig. 2 shows asystem model constituted van der Pol oscillators (VDP). We
use a ring of van der Pol oscillators, three VDPs are connected by Time-Varying Resistor (TVR1, TVR2 and TVR3).
We realize the TVR by switching a positive and a negative
resistor periodically as shown in Fig. 3. We investigate thesynchronization phenomena, how changing by the frequencyof TVR1, TVR2 and TVR3 and the duty ratio. When threeTVRs are connected by resistors, it stables in-phase synchronization phenomena. On the other hand, when three TVRs
are connected by negative resistors, it stables three phase syn1. Introduction
chronization phenomena. When the duty ratio p is larger than0, it stables three phase synchronization phenomena. WhenThe synchronization phenomena are the fundamental phe- the duty ratio p is smaller than 0, it stables in-phase synchronomena in nature and the synchronization phenomena are ob- nization phenomena.
served over the various fields; for example, biological fields,
Firstly, the igk -vk characteristics of the nonlinear resistor
physical fields and electrical fields. The natural synchroniza- are defined as follows,tion phenomena can be imitated with electric circuit. The
reason why we use electric circuit is guaranteed high reigk = −g1 vk + g3 vk3 .
(1)
producibility of the experiment and short experimental time.
Hence, many researchers are investigating the relationship By changing variable and the parameters,
among the synchronization phenomena coupled by oscilla√ √√√g1
g1 Ctors and the natural synchronization phenomena.
x k , ik =yk , t = LCτ(2)
vk =This time, we use van der Pol oscillators. van der Pol osg3
g3 L
cillator has very simple structure, this oscillator consists of√√nonlinear resistor, conductor, and inductor. van der Pol oscilL1
Llators are used to model natural phenomena.
ε = g1, γ=,(3)CrCIn our previous research, we have investigated the synchronization phenomena in three van der Pol oscillators coupled by Time-Varying Resistor (TVR) [1]. TVR is a device
which positive resistor and negative resister are periodicallyswitched. We change the frequency of TVR, when the fregCLquency is high, positive resistor and negative resistor switch
quickly.
In this study, we propose coupled oscillatory systems suchas a ring of van der Pol oscillators coupled by TVR. We use
three TVRs, and each TVR has different frequencies. We inFigure 1: van der Pol oscillator.
vestigate the synchronization phenomena observed in the proposed circuit system by changing the frequency of TVR[2].
In this study, we investigate the synchronization phenomena
in a ring of van der Pol oscillators coupled by Time-Varying
Resistor (TVR). We realize the TVR by switching a positive
and a negative resistor periodically. When the frequency ishigh, positive resistor and negative resistor switch quickly.
We use three TVRs, and each TVR has different frequencies.
And, we observe the synchronization phenomena in the proposed circuit system by changing the frequency of TVRs andthe duty ratio.

VDP- 168 -circuit1= f3 = 0.1, we observe only three phase synchronization phenomena, it is observed regardless of initial value.

R3TVTV
R1

VDP
VDP

TVR2

(a)in-phase.

VDP(b)3-phase.

Figure 4: Phase difference (f1 = f2 = f3 = 0.09).

circuit3

circuit2

Figure 2: Circuit model.
γ γ γ3(a) in-phase.
(b) 3-phase.
Figure 5: Phase difference (f1 = f2 = f3 = 0.1).

20.01

03.2 Different frequencyωτNext, we investigate the influence of the duty ratio. Theduty ratio sets with p. If p is larger than 0, synchronization stables three phase synchronization phenomena. If p issmaller than 0, synchronization stables in-phase synchronization phenomena.
Figure 6 shows the characteristics of the TVR in the caseof the duty ratio p included.

−0.01

Figure 3: Characteristics of the TVR.
The normalized equations of VDP circuit are given as follows:x˙1 = ε(x1 − x31 ) − y1 + γ1 (x2 − x1 ) + γ3 (x3 − x1 )x˙2 = ε(x2 − x32 ) − y2 + γ1 (x1 − x2 ) + γ2 (x3 − x2 )3
 x˙3 = ε(x3 − x3 ) − y3 + γ2 (x1 − x3 ) + γ3 (x2 − x3 )y˙n = xn .

γ1 γ2 γ3−p0.01(4)0

In these equations, n is the number of circuits and n = 1,2,3.
ε denotes the nonlinearity of the oscillators, γ1 , γ2 , and γ3
denote the coupling strengths of the TVR1, TVR2 and TVR3.

ωτ−0.01+p3. Simulation results

Figure 6: Characteristics of the TVR.

3.1 Same frequencyThe simulation results of the system model are shown fromFig. 4 and Fig. 5. The value of the parameters is set toε = 0.1, γ1 = ±0.01, γ2 = ±0.01 and γ3 = ±0.01. The
frequencies of TVR1, TVR2 and TVR3 sets with f1 , f2 andf3 . Where, fn = 2πωn . The figure on the left shows thephase difference when the initial condition set with in-phase.
The figure on the right shows phase difference when the initial condition set with 3-phase. Firstly, in case of f1 = f2 = f3= 0.09, and duty ratio p = 0, we can observe in-phase synchronization phenomena and 3-phase synchronization phenomena. We can observe both synchronization phenomena
due to change by initial value. Secondly, in case of f1 = f2

Figures 7 and 8 show the influence of the duty ratio p.
In case of f1 = f2 = f3 = 0.09, when p is between -0.01
and 0.005, we can observe both synchronization phenomena. In case of f1 = f2 = f3 = 0.1, when p is between -0.01
and -0.001, we can observe both synchronization phenomena.
Hence, we can handle the synchronization phenomena easily
by changing the duty ratio.
Next, we investigate the phase difference of TVR wheneach TVR has different frequencies. At first, we show thephase difference between the oscillators. The value of theparameters is set to ε = 0.1, γ1 = ±0.01, γ2 = ±0.01,
γ3 = ±0.01, and p = 0.

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When the phase difference is near at 0, this is in-phase synchronization phenomena. When the phase difference is near
at 60, this is three phase synchronization phenomena.

Figure 7: Phase difference (f1 = f2 = f3 = 0.09).

Figure 11: Phase difference (in-phase).

Figure 8: Phase difference (f1 = f2 = f3 = 0.1).
In case of f1 = 0.1, f2 = 0.2, f3 = 0.15, we can observe
only three synchronization phenomena.
From Figs. 9 to 12 show the influence of the duty ratio,(a) in-phase.
Figure 9: Phase difference (f1 = 0.1, f2 = 0.2, f3 = 0.15).

Figure 12: Phase difference (three phase).

In Fig. 11, when the duty ratio p is larger than 0, we canobserve almost three phase synchronization phenomena. And
when the duty ratio p is smaller than -0.002, we can observein-phase synchronization phenomena.
(b) three-phase.
In Fig. 12, when the duty ratio p is larger than -0.002, weFigure 10: Phase difference (f1 = 0.1, f2 = 0.2, f3 = 0.15).
can observe almost three phase synchronization phenomena.
Figures 11 and 12 show the influence of the duty ratio. In And when the duty ratio p is smaller than -0.004, we can obFig. 11, initial condition set with in-phase. In Fig. 12, initial serve in-phase synchronization phenomena.
condition set with three phase.

- 170 -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 frequency of
TVR and the duty ratio. When three frequencies of TVR (f1 ,f2 , f3 ) equal, we can observe in-phase synchronization phenomena, and 3-phase synchronization phenomena. However,when we changed the duty ratio, we can choose the kind of
the synchronization phenomena. And we realize that we canhandle the synchronization phenomena easily.
In the future, we would like to investigate the synchronization phenomena using a more complicated system by changing the frequency and the duty ratio.
References[1] Y. Uwate, Y. Nishio, “Two van der Pol Oscillators Coupled by Chaotically Varying Resistor”, Proceedings of International Workshop on Nonlinear Dynamics of ElectricSystems (NDES’06), pp.189-192, Jul. 2006.
[2] D. Nariai, M. Tran, Y. Uwate, Y. Nishio, “Synchronization in Two Rings of Coupled Three van der Pol Oscillators”, Proceedings of RISP International Workshop onNonlinear Circuits, Communications and Signal Processing (NCSP’17), pp. 325-328, Feb. 2017.

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