IEEE Workshop on Nonlinear Circuit Networks
December 7-8, 2018

Influence of Coupling Strength
on Synchronization in Two Ringsof Coupled van der Pol Oscillators
Daiki Nariai, Yoko Uwate and Yoshifumi Nishio
Dept. Electrical and Electronic Eng.,
Tokushima University,
2-1 Minami-Josanjima,
E-mail: {nariai, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract—In this study, we investigate the synchronization
phenomena in two rings of van der Pol oscillators coupled byresistors. We propose a novel coupled oscillatory system comprising two rings of van der Pol oscillators with different couplingschema. We focus on the coupling strengths of the coupled vander Pol oscillators. By computer simulation, we investigate how
the synchronization phenomena change by changing the couplingstrengths. In these results, we observe various synchronization
phenomena.

synchronization phenomena due to difference in circuit structure, we consider two kinds of circuit structures. We consider
2k +1 - 2k +3 coupled van der Pol oscillators(k = 1, 2, 3). By
computer simulation, we investigate the synchronization phenomena observed in the proposed circuit system by changing
the coupling strengths between two rings.

II. S YSTEM MODEL
I. I NTRODUCTION
The synchronization phenomena of coupled oscillators arefamiliar. Synchronization phenomena have been studied invarious fields for many years, such as in electrical systems,
mechanical systems and biological systems. Among them,the synchronization phenomena of van der Pol oscillators are
similar to natural phenomena when the frequency is changed.
A coupled system of van der Pol oscillators is simple and easy
to handle. Many researchers have proposed various coupled oscillatory networks of van der Pol oscillators [1]-[3]. However,there are many portions not yet been clarified with respect tothe phenomenon that the mechanism and attendant. Therefore,
we investigate synchronization phenomena between two ringsof van der Pol oscillators where different synchronization
phenomena occur.
The van der Pol oscillator is a simple circuit. It consists ofa resistor, inductor, capacitor and nonlinear resistor. It wasinvented by the electrical engineer Balthasar van der Pol.
The equation of a van der Pol oscillator is a second-orderdifferential equation.
In this study, we propose coupled oscillatory system including two rings of van der Pol oscillators. The ring consists ofvan der Pol oscillators connected by inductors and resistors.
The phase difference of the ring depends on number ofoscillators inside ring. When the number of oscillators is an
odd number, the oscillators of the ring exhibit some phasesynchronization. For example, five oscillators of the ringexhibit five-phase synchronization. We coupled two rings ofdifferent size by the resistor. Because we consider changes inFigure 1 shows the circuit used in this study. We call thiscircuit NC. Figure 2 shows model of the system. We use tworings of van der Pol oscillators. The NC of the first and thesecond ring are connected by inductors and resistors. The firstring consist of 2k+1 NC and the second ring consists of 2k+3
NC (k = 1, 2, 3). We coupled two rings of different size bythe resistors R. We observe the synchronization phenomena ofadjacent oscillators. We investigate how the synchronization
phenomena change upon changing the value of R.

ignvCnFig. 1.

NCThe circuit used in this study.

The circuit equations of the first ring are given as follows:- 87 - dv1nC dt

1an2L didt

di1bn

 2L dt= −ig1n − i1an
−i1bn − R1 (v1n − v2n )= v1n − (i1an + i1b(n+1) )R′(1)= v1n − (i1bn + i1a(n−1) )R′ .

The circuit equations of the second ring are given as follows:ẋ1nẏanẏbn


NCNC
NCi1bi1aRR

Ri2bi2aNC
NC2LNCNCNCNCR'
C dvdt2n
 dv2nC dt
di2an
 2L dt
di2bn
 2L dt= x1n − β(y1an + y1b(n+1) )System model.

= −ig2n − i2an − i2bn
− R1 (v2n − v1n )= x1n − β(y1bn + y1a(n−1) )(n = 1 ∼ 2n + 1)
= −ig2n − i2an − i2bn(2)(n = 2n + 2, 2n + 3)

where the parameters ε is non-linear strength. The parameters
α and γ denote the coupling strengths of the resistor R′ andresistor R, respectively.

= v2n − (i2an + i2b(n+1) )R′

III. S IMULATION RESULTS= v2n − (i2bn + i2a(n−1) )R′First, we show the simuration results of k = 1. Figure 3
shows system model of k = 1.

where 1n and 2n denote the number of the circuit of the firstring and the second ring (1n = 1 2k + 1, 2n = 1 2k + 3.)

ign= −g1 vn + g3 vn3

NCNCNCA nonlinear resistor defined as follows:

RRR(3)NCNC

By changing the variables and parameters as follows.
√√
g1
g1 CLCτ, vn = 3gx,
i=nan
3g3 L yan ,3√√
g1 CL
ibn = 3gy,
ε=gbn
1C,3L
√√
L, α = R′ Cγ = R1 C
L
t=(5)and the normalized equations of the second ring are givenas:ẋ2n = ε(x2n − x32n ) − y2an





−y2bn + γ(x2n − x1n )




(n = 1 ∼ 2n + 1)ẋ2n = ε(x2n − x32n ) − y2an − y2bn(6)(n = 2n + 2, 2n + 3)





ẏ2an = x2n − β(y2an + y2b(n+1) )






 ẏ2bn = x2n − β(y2bn + y2a(n−1) )2LFig. 2.

−y1bn + γ(x1n − x2n )NCNC

NC= ε(x1n − x31n ) − y1an

circuit 5√

the normalized equations of the first ring are given as:

Fig. 3.

(4)

NCNC

NC

circuit 1circuit 2circuit 3circuit 4

The circuit used in this study (k = 1).

The simulation results of the system model are shown inFigs.4-6. The parameters are set to ε = 0.01, α = 0.01.
We change the value of γ from 0.0000 to 0.0012 at intervalsof 0.0002. We fix the value of initial value. Figure 4 shows
the phase difference of the second ring. We focus on circuits
connected to the first ring. When the coupling strength γ isincreased, it can be observed that the phase difference of
circuit 1-circuit 2 and circuit 2-circuit 3 decrease. Figure 5- 88 -shows the time waveform of the first ring and the second ring,in the case of γ = 0.0000. The oscillators of fist ring exhibit
three-phase synchronization and the oscillators of second ringexhibit five-phase synchronization. Figure 6 shows the time
waveform of the first ring and the second ring, in the caseof γ = 0.0012. The oscillators of fist ring exhibit threephase synchronization. Circuits 1, circuit 2 and cicuit 3 ofsecond ring exhibit the synchronization similar to three-phasesynchronization when the coupling strength γ increases.

Fig. 6.

Time waveform (γ = 0.0012).
NCNC
NC

NC

NC

RR
R

R

R

NC

NC

circuit 7

Fig. 4.

NCNC

circuit 1

Phase difference of the seccond ring (k = 1).

Fig. 7.

NC

NC

NC

circuit 2

circuit 3

circuit 4

circuit 6

circuit 5

The circuit used in this study (k = 2).

The first ring

The second ring

Fig. 5.

Time waveform (γ = 0.0000).
Fig. 8.

Next, we show simuration result of k = 2. Figure 7 shows
system model of k = 2.
Figure 8 shows the phase difference of the second ring.
Circuits the second ring connected to the first ring exhibit
the synchronization similar to five-phase synchronization when
the coupling strength R increases. It is easier to observe thisphenomena result of k = 1 than result of k = 2.
Finaly, we show simuration result of k = 3. Figure 9 shows
system model of k = 3.

Phase difference of the seccond ring (k = 2).

Figure 10 shows the phase difference of the second ring.
When γ is 0, circuits of second ring exhibit nine phasesynchronization. Circuits the second ring connected to thefirst ring exhibit the synchronization similar to seven-phasesynchronization when the coupling strength γ increases. Evenif γ increase from 0.004 the phase difference do not change.
It is easier to observe this phenomena result of k = 1 andk = 2 than result of k = 3.

- 89 -R EFERENCESNCNC[1] M. A. Barrón, M. Sen, “Synchronization of four coupled van der Poloscillators” ,Nonlinear Dynamics, pp. 357-367, June 2009.
[2] Y. Uwate, Y. Nishio and R. Stoop: Synchronization in three coupled
van der Pol oscillators with different coupling strength, Proceedings ofRISP International Workshop on Nonlinear Circuits, Communications andSignal Processing (NCSP’10), pp. 109-112, Mar. 2010.
[3] K. Matsumura, T. Nagai, Y. Uwate and Y. Nishio: Analysis of synchronization phenomenon in coupled oscillator chains, Proceedings of IEEE
International Symposium on Circuits and Systems (ISCAS’12), pp. 620623, May 2012.
[4] H. Kume, Y. Uwate and Y. Nishio: Synchronization phenomena indifferent sizes of rings of coupled oscillators, Proceedings of International
Symposium on Nonlinear Theory and its Applications (NOLTA’14), pp.
369-372, Sep. 2014.

NC

NC
NC

NC

NC

R

R
R

R
R

R

R

NC

NCNCNC

circuit 9

circuit 1

Fig. 9.

Fig. 10.

circuit 2

circuit 8

NCNCNCNCNC

circuit 3

circuit 3

circuit 5

circuit 6

circuit 7

The circuit used in this study (k = 3).

Phase difference of the seccond ring (k = 3).

Circuits of the second ring connected to the first ringare affected by the first ring. When the coupling strength
R increases, in the circuit of the second ring connected to
the first ring, the synchronization phenomena similar to thesynchronization phenomenon of the first ring exhibit. Whenk was large, we were able to observe the synchronization
phenomena similar to the synchronization phenomenon of thefirst ring.

IV. C ONCLUSIONS
We have proposed a system model using two rings of coupled van der Pol oscillators. We observe time waveform andphase difference with using computer simulation. We observevarious synchronization phenomena by changing the couplingstrengths. Circuits of the second ring connected to the firstring are affected by the first ring. When the coupling strength
R increases, in the circuit of the second ring connected tothe first ring, the synchronization phenomena similar to thesynchronization phenomenon of the first ring exhibit. Whenk was large, we were able to observe this phenomena. In thefuture, we will investigate synchronization phenomena using
other parameters and analyze the proposed circuit model.

- 90 -