2018 International Symposium on Nonlinear Theory and Its Applications,NOLTA2018, Tarragona, Spain, September 2-6, 2018

Influence of Coupling Strengths on Synchronization Phenomena
in Two Rings of Coupled van der Pol Oscillators
Daiki Nariai, Yoko Uwate and Yoshifumi Nishio
Department. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
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 by resistors. We propose a novel coupled oscillatory system comprising two rings of van der Pol oscillatorswith diﬀerent coupling schema. We focus on the couplingstrengths of the coupled van der Pol oscillators. By computer simulation, we investigate how the synchronization
phenomena change by changing the coupling strengths. In
these results, we observe various synchronization phenomena.
1. Introduction
The synchronization phenomena of coupled oscillators
are familiar. Synchronization phenomena have been studied in various 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 thefrequency is changed. A coupled system of van der Poloscillators is simple and easy to handle. Many researchers
have proposed various coupled oscillatory networks of vander Pol oscillators [1]-[3]. However, there are many portions not yet been clarified with respect to the phenomenon
that the mechanism and attendant. Therefore, we investigate synchronization phenomena between two rings ofvan der Pol oscillators where diﬀerent synchronization phenomena occur.
The van der Pol oscillator is a simple circuit. It consistsof a resistor, inductor, capacitor and nonlinear resistor. It
was invented by the electrical engineer Balthasar van derPol. The equation of a van der Pol oscillator is a secondorder diﬀerential equation.
In this study, we propose coupled oscillatory system including two rings of van der Pol oscillators. The ring consists of van der Pol oscillators connected by inductors andresistors. The phase diﬀerence of the ring depends on number of oscillators inside ring. When the number of oscillators is an odd number, the oscillators of the ring exhibit
some phase synchronization. For example, five oscillatorsof the ring exhibit five-phase synchronization. We coupled
two rings of diﬀerent size by the resistor. Because we consider changes in synchronization phenomena due to diﬀerence in circuit structure, we consider two kinds of circuitstructures. We consider (2k + 1) - (2k + 3) coupled vander Pol oscillators (k = 1, 2). By computer simulation, weinvestigate the synchronization phenomena observed in theproposed circuit system by changing the coupling strengthsbetween two rings.

2. System Model
Figure 1 shows the circuit used in this study. We call thiscircuit NC. Figure 2 shows model of the system. We use
two rings of van der Pol oscillators. The NC of the first
and the second ring are connected by inductors and resistors. The first ring consist of 2k + 1 NC and the second ringconsists of 2k + 3 NC (k = 1, 2). We coupled two rings ofdiﬀerent size by the resistors R. We observe the synchronization phenomena of adjacent oscillators. We investigate
how the synchronization phenomena change upon changing the value of R.

ignvCnNC

Figure 1: The circuit used in this study.
The circuit equations of the first ring are given as follows: dv1n

C dt








2L didt1an






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:- 483 -ẋ1nẏan

ẏbn


NC

NC
NC

i1bi1aRR

Ri2bi2aNC
NC2LNC

NC

NC

NCNC



ẋ2n
















 ẋ2n














ẏ2an






 ẏ2bn

Figure 2: System model.

=−ig2n − i2an − i2bn− R1 (v2n − v1n )(n = 1 ∼ 2n + 1)=

−y1bn + γ(x1n − x2n )=x1n − β(y1an + y1b(n+1) )=x1n − β(y1bn + y1a(n−1) )−ig2n − i2an − i2bn(2)(n = 2n + 2, 2n + 3)=

v2n − (i2an + i2b(n+1) )R′=

v2n − (i2bn + i2a(n−1) )R′=3
ε(x2n − x2n) − y2an−y2bn + γ(x2n − x1n )(n = 1 ∼ 2n + 1)=3
ε(x2n − x2n) − y2an − y2bn

(n = 2n + 2, 2n + 3)=

x2n − β(y2an + y2b(n+1) )=x2n − β(y2bn + y2a(n−1) )where the parameters ε is non-linear strength. The parameters α and γ denote the coupling strengths of the resistor
R′ and resistor R, respectively.

First, we show the simulation results in k = 1. Figure 3
shows system model of k = 1. In this model, the first ringconsists of 3 circuits and the second ring consists of 5 circuits.

NCNC
NC

A nonlinear resistor defined as follows:
R=−g1 vn + g3 v3n .

RNCNCcircuit 5√√
g1
g1 Cx,
in=LCτ, vn = 3gn
a3g3 L ya n,3√√ib n = 3gg31 L yb n, ε = g1 CL ,√√γ = R1 CL , α = R′ CLR(3)

By changing the variables and parameters as follows.

t=(6)2.1. Simulation Results

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

ign

(5)

and the normalized equations of the second ring are given
as:R' dv2n
C dt
 C dvdt2n

2L didt2an



di2bn

 2L dt3
ε(x1n − x1n) − y1an

NC2L

NC=√NCNC

NC

circuit 1

circuit 2

circuit 3

circuit 4

Figure 3: System model (k = 1).
(4)

The normalized equations of the first ring are given as:

The simulation results of the system model are shown in
Figs.4-6. The parameters are set to ε = 0.01, α = 0.01. Wechange the value of γ from 0.0000 to 0.0012 at intervals of0.0002. We fix the value of initial value. Figure 4 shows thephase diﬀerence of the second ring. We focus on circuits
connected to the first ring. When the coupling strength γ
is increased, it can be observed that the phase diﬀerence of- 484 -circuit 1-circuit 2 and circuit 2-circuit 3 decrease. Figure 5
shows the time waveform of the first ring and the second
ring, in the case of γ = 0.0000. The oscillators of fist ringexhibit three-phase synchronization and the oscillators ofsecond ring exhibit five-phase synchronization. Figure 6
shows the time waveform of the first ring and the secondring, in the case of γ = 0.005. The oscillators of fist ring exhibit three-phase synchronization. Circuit 1, circuit 2 andcicuit 3 of second ring exhibit the synchronization similar
to three-phase synchronization when the coupling strength
R increases. When k was large, we were able to observe thesynchronization phenomena similar to the synchronization
phenomenon of the first ring.

The first ring

The sec o nd ring

Figure 6: Time waveform (γ = 0.0012).
NC

NC
NC

NC

NC

R

R
R

R

R

NC

NC

circuit 7NCNC

circuit 1

NC

NC

NC

circuit 2

circuit 3

circuit 4

circuit 6

circuit 5

Figure 7: System model (k = 2).
Figure 4: Phase diﬀerence of the seccond ring (k = 1).

The first ring

The sec o nd ring

Figure 5: Time waveform (γ = 0.0000).
Next, we show simulation result in k = 2. Figure 7 shows
system model of k = 2. In this model, the first ring consistsof 5 circuits and the second ring consists of 7 circuits.
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 this phenomena of k = 1 than result of k = 2.
Circuits of the second ring connected to the first ring areaﬀected 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 toFigure 8: Phase diﬀerence of the seccond ring (k = 2).
the synchronization phenomenon of the first ring exhibit.
When k was large, we were able to observe the synchronization phenomena similar to the synchronization phenomenon of the first ring.
3. Conclusion
We have proposed a system model using two rings ofcoupled van der Pol oscillators. We observe time waveform and phase diﬀerence with using computer simulation.
We observe various synchronization phenomena by changing the coupling strengths. Circuits of the second ring connected to the first ring are aﬀected by the first ring. Whenthe coupling strength R increases, in the circuit of the second ring connected to the first ring, the synchronization- 485 -

phenomena similar to the synchronization phenomenon of
the first ring exhibit. When k was large, we were able toobserve this phenomena. In the future, we will investigate
synchronization phenomena using other parameters and analyze the proposed circuit model.
Acknowledgment
This work was partly supported by JSPS Grant-in-Aidfor Scientific Research 16K06357.
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Symposium on Circuits and Systems (ISCAS’12), pp.
620-623, May 2012.
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