IEEE Workshop on Nonlinear Circuit Networks
December 9-10, 2016

Synchronization Phenomena in Frustration
Networks of van der Pol Oscillators
Kazuki Ueta, Yoko Uwate and Yoshifumi nishio
Dept. Electrical and Electronic Eng.,
Tokushima University
2-1 Minami-Josanjima,
Tokushima 770-8506, Japan
Email: kazuki, uwate, nishio@ee.tokushima-u.ac.jp

Abstract—Synchronization phenomena of frustration networkby coupled oscillators has been studied in a wide range of fields,such as medicine and engineering. It is investigated towards
various systems up to now. However, analysis of regarding morecomplex systems are little. In my study, We developed the systemmodel so that a basic minimum unit even in more complex
systems. The system is coupled by resistor and inductor withvan del Pol oscillator. In addition, we observed synchronization
phenomena about the system.

II. S YSTEM MODEL
Figure 2 shows a system model constituted van del pol
oscilators (VDP-A and VDP-B). We couple each VDP-B via
inductor L and ground by coupling resistor R. In addition, wecouple VDP-A via resistor r. VDP-A is the only one central
circuit which is connected to all VDP-B in this system byresistor r.

I. I NTRODUCTION
There are a lot of synchronization phenomena in this world.
This is one of the nonlinear phenomena that we can often
observe by natural animate beings which do collective actions.
For example, firefly luminescence, cry of birds and frogs,
applause of many people and so on. Synchronazation phenomena have a feature that the set of small power can produce
very big power by synchronizing at a time. Therefore studyof synchronization phenomena have been widely reported not
only engineering but also the physical and biological fields[1]–[6]. Investigation of coupled oscillators attention from many
researchers because coupled oscillatory network produces interesting phase synchronization such as the phase propagation
wave, clustering and complex patterns. In addition, it has theadvantage of being able to manufacture for circuit on theboard.
In this study, we focus on the synchronization phenomena
coupled by van del pol oscillators containing ring and starstructures. Then, we obserb the synchronization phenomena
with computer simulation. van der pol oscillator is shown inFig. 1.

CiA

iAg

ikgvAL

VDP B

VDP Ar

C

VDP A

vkCr
r

2L

2Lr

R
2L
VDP B

2L
R
VDP BR2L

2L
R

VDP B

Fig. 2. System model.

In the computer simulations, we assume that the voltage
and current characteristics of the nonlinear resistor in eachoscillator are given by the follows:ig = −g1 v + g3 v 3 ,L

(1)

(g1 , g3 > 0).

Fig. 1. van der pol oscillator.

The charactaristic of ring coupling has in-phase, anti-phaseand N-phase. The charactaristic of star coupling has in-phaseand anti-phase.

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First, the circuit equations of VDP-A are given as follows:
dvA1


= −iA − iAg + (N vA − v1 − v2 − ... − vN ),
 C
dtr
(2)
diA

= vA ,
 Ldt
where N denotes the number of VDP-B.
On the other hand, VDP-B is connected to the adjacent VDPB and VDP-A. The circuit equations of VDP-B are given asfollows:dvk1
= −ika − ikb − ikg − (vk − vA ),
C
dtr

dika2L= vk − R(ika + ik+1,b ),(3)dt 2L dikb = v − R(i + ikkbk−1,a ),dtyy2xy4x

15x2x3x45Fig. 3. Attractor between adjacent oscillators for N = 4 (horizontal axis:xk ,vertical axis:yk ) (k = 1, 2, 3, 4, 5).

R2L2L
VDP B

VDP B

rr

2L
R

(k = 1, 2, ..., N ).
By using the following parameters and variables:
√√
√
g1 C
g1 C
g1 C
iA =
yA , ika =yka , ikb =ykb ,3g L
3g3 L
3g3 L
√ 3
√g1
g1
vA =x A , vk =xk ,3g33g3√√dL, α = g1,t = LCτ, “ · ” =dτC√√
C1 L
β=, γ=R,r CL

y32LR

VDP Arr2L2L 2L2L

VDP B

VDP B

R(4)
Fig. 4. System model of N = 4.

where α is the nonlinearity, β is the coupling strength between
VDP-A and VDP-B. γ indicates the coupling strength between
VDP-B. The normalized circuit equations of VDP-A are given
as follows:(1 ) x˙A = αxA 1 − x2A −yA3(5)+β(N xA − x1 − x2 − ... − xN ),y˙A = xA .

Next, the time waveforms of the voltage of each capacitor
C after sufficient time has elapsed are shown in Fig. 5. And
the phase differences between the adjacent oscillator of thiscase is equal to the result as shown in Fig. 6. It seems that twoelectric currents of VDP-B are piled up as we underestand it
from the figures like 3 phase.

xntimeThe normalized circuit equations of VDP-B are given asfollows:1x˙k = αxk (1 − x2k ) − yka − ykb − β(xA − xk ),
3
1(6)yka˙ = {xk − γ(yka + yk+1,b )},
2
1

 ykb
˙ = {xk − γ(yka + yk−1,b )}.
2
III. S IMULATION R ESULTS
We calculate Eqs. (5) and (6) using the Runge-Kutta methodwith the step size h = 0.02. We show the simulation resultof the synchronization phenomena when N = 4 in Fig. 3.
In this figure, we show the attractor of each oscillator andthe horizontal axis is the voltage of each oscillator, and thevertical axis is the electric current of each oscillator. We set
the parameters α = 0.1, β = 0.0075 and γ = 0.02. In addition,
we show the system model of N = 4 in Fig. 4.

Fig. 5. The Time waveforms of the each oscillator for N = 4.

3x

2x

4

32xxxx

5x

1xxx

4xx

121x

545xx

1x

2x

3x

3Fig. 6. Lissajous figures of N = 4.

Second, the simulation results of the system model containing six circuits are shown in Fig. 7. The value of the- 12 -parameters are fixed with β=0.001, 0.0085 and 0.05. In thecase of β=0.001, 5 phase synchronization appeared because
the coupling strength of VDP-A is weak. The current of theVDP-A and one of the current of the VDP-B are in phaseat that time. Therefore, we assume 5 phase synchronization.
When the value of β sets 0.0085, some time waveforms ofVDP-B come close on in-phase synchronization. When we
increase coupling strength, most of two electric currents ofVDP-B are piled up as we understand it from the Lissajousfigures.
In the case of β=0.05, 5 phase synchronization become inphase synchronization. And then VDP-A becomes anti-phasesynchronization with VDP-B by increasing the value of β. We
understand that VDP-B synchronizes even if we read eitherfigure.

x4x3x2x1x1x5x1

Finaly, we summarize the simulation results in Fig. 8. In
the figure, we show the results when we increase the cuircuit
numbers N = 3, 4, ..., 7. The phase difference is based on one
voltage waveform of VDP-B. The broken line in the figure
represent asynchronous. The solid line in the figure representsynchronous. From this result, It turns out that an even numbercircuits become in-phase as increasing the coupling strength.
Similarly, It turns out that an even number circuits become inphase as increasing the coupling strength. However, we could’t
confirm the synchronization state between β=0 and β=0.016
in an odd number circuits.

x6x1x1Fig. 8. Relationship between coupling strength and phase difference.

(a) β=0.001.

xxxxx

1x

432x

16x

1IV. C ONCLUSIONSx5x11(b) β=0.0085.

x4x3x2x1x1x5x1

In this study, we have proposed a system model using fiveand six circuits that are combined the ring and star structures.
We have observed the synchronization phenomena by increasing the coupling strength of ring. When the coupling strengthis sufficiently small, system model becomes like function ofring coupling therefore, 5 phase synchronization can be observed. By increasing the coupling strength, time wave forms
of VDP-B have come close in-phase synchronization. Whenthe coupling strength is sufficiently large, time wave forms of
VDP-A and VDP-B become anti-phase synchronization. In thefuture, we investigate synchronization phenomena using othercircuits.

x6x1

R EFERENCESx1(c) β=0.05.
Fig. 7. Simulation Results for N=5 (α = 0.1 and γ = 0.02). time-waveform. Red and other colors denote xc and xN respectively. (N= 1,2,...,6)[1] L. L. Bonilla, C. J. Paerez Vicente and R. Spigler, “Time-Periodic
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[7] K. Kaneko, “Clustering, Coding, Switching, Hierarchical Ordering, andControl in a Network of Chaotic Elements”, Physica D, vol. 41, pp. 137172, 1990.
[8] M. Shinriki, M. Yamamoto and S. Mori, “Multimode Oscillations ina Modified van der Pol Oscillator Containing a Positive Nonlinear
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