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

Synchronization Phenomena and Oscillations
Stopping in Coupled van der Pol with Diodes
Yuya Ikezoe

Toshiya Matsunashi

Yoko Uwate

Yoshifumi Nishio

Dept. of Electrical and Electronic Eng., Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email:{ikezoe, matsunashi, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract—We investigate synchronization phenomena in vander Pol oscillators. We compare the coupled systems. Systems arefour van der Pol oscillations coupled with diodes and resistors.
In this study, the diode is assumed that it shows cubic function.
We investigate the difference between the coupled systems. And
we investigate that stop oscillation partially.

= NCI. I NTRODUCTION

Fig. 1.

Synchronization phenomena observe around us. For example, beat of the heart, firefly luminescence, cry of birds,and so on. Synchronization phenomena are used for thesemathematical models. Therefore, they have been studied invarious fields (Biology, Network systems ...)We use van der Pol oscillators. Synchronization can beproduced it by circuits. Because it is easy to observe thesynchronization phenomena of van der Pol oscillators bychanging frequency. This circuit is superior that it can fit
products immediately.
The van der Pol oscillator is a simple circuit. It is consistedof resistor, inductor, capacitor and nonlinear resistor. Equation
assumed that second‐order differential equation.
We use the coupled van der Pol systems. Four van der Pol
oscillations coupled with diodes and resistors. And resistor anddiode have coupling strength each. The coupled model in theonly resistors have been studied [1],[2].
In this study, we investigate synchronization phenomena ofvan der Pol oscillators coupled models. And we compare thesystems. This study has two purposes. The first, we find out thedifferences of synchronization phenomena between resistors
and diodes. The second, we verify oscillation stop partiallyby changing the coupling strength.
II. S YSTEN MODELS
Figure 1 shows basic van der Pol oscillator. Figures 2 and3 show system models constituted van der Pol oscillators(VDP). We use a ring of van der Pol osciilators, four VDP areconnected by resistors or diodes. Figure 2 is all VDP coupledwith resistors, but Fig. 3 is coupled with diodes. Figures 4and 5 show system models coupled with resistors and diodes.
We investigate synchronization phenomena, how changing
synchronization phenomena by the connection method.

van der Pol oscillator.

Fig. 2.

System model 1.

Fig. 3.

System model 2.

The normalized equations of VDP circuit connected withresistors are given as follows:- 80 -
x˙j


yja
˙
y˙
jb

yjc
˙= ε(xj − x3j ) − yja − yjb − yjcx

= − β3 (yja + ylb ) + 3jx= − β3 (yjb + yla ) + 3jxjβ= − 3 (yjc ) + 3 .

(1)


 x˙k
yka˙
ykb˙ykc
˙{Fig. 4.

System model 3.

Fig. 5.

System model 4.

System model 5.

Fig. 7.

System model 6.

= ε(xk − x3k ) − yka − ykb − ykc= − β3 (yka + ynb ) + x3k= − β3 (ykb + yna ) + x3k= − β3 (ykc + ymc ) + x3k .
√C
β=RL= ε(xj − x3j ) − yja − yjb − yjc√x= − γ3 3 yja + ylb + 3jxγ√= − 3 3 yjb + yla + 3jxγ√
j= − 3 3 yjc + 3 .
= ε(xk − x3k ) − yka − ykb − ykc√= − γ3 3 yka + ynb + x3kγ√= − 3 3 ykb + yna + x3k√= − γ3 3 ykc + ymc + x3k .
√Lg3
γ= 3
Cgg1

III. S IMULATION RESULTS

(2)

(3)

The normalized equations of VDP circuit connected withdiodes are given as follows:x˙j
yja˙˙ yjbyjc˙x˙ kyka˙y˙kbykc
˙{Fig. 6.

(4)

The simulation results of the system model are shown fromFig. 8 to 11. The value of the parameter is set to ε = 0.05,
β = 0.005, γ = 0.005.
Circuit1-3 shows in-phase, and circuit1-2 shows anti-phasein Fig. 8. Figures 8 and 9 show almost the same synchronization phenomena. It shows there are few differences betweenmodels. The coupling schema is changed from resistors todiodes partially in Fig. 10. Circuit2-3, and circuit1-4 show inphase synchronization phenomena in Fig. 10. The others aresynchronized at anti-phase.
Circuit1-3, circuit3-4 show in-phase, and circuit2-3 shows
anti-phase in Fig. 11. It shows s partial change causes thedifferences of synchronization phenomena.

(5)

x1x1

ign

= −g1 vn + g3 vn3

(7)

x1

- 81 -x1x3x3x3x4

Phase differences of system 1.

x4x1x2x2

x1x3x2x4x1x2x2x4x4

Phase differences of system 2.

x2x4x3Fig. 9.

x3

In these equations, j and k are the numbers of circuit.
Parameters l, n, m are related to j and k. j=1 l=2,4, j=3 l=4,2,k=2 n=3,1 m=4, k=4 n=1,3 m=2. ε denotes the nonlinearity ofthe oscillators. Figures 6 and 7 show systems which observe
oscillations stop by the change of the parameter γ.

x1

Fig. 8.

x3

Nonlinear resistorare defined as follows:x2x2x3(6)

x2

x1

x4

x1
x3

x3x4x4

Fig. 10.

Phase differences of system 3.

x4

x1x2

Fig. 11.

x2x2x1x3x3x4

Phase differences of system 4.

x4

Next simulation results of the system model 5 ,6 are shownby Figs. 12 to 13. Horizon axis indicates time τ in Fig. 12 and13. The value of the parameters is set to ε = 0.05, β = 0.005.
Parameter γ is variable. The value of parameter of γ is set toγ=0.08 in Fig. 12, γ=0.135 in Fig. 13.
Figure 12 shows wave3 stop oscillation. The others continue
to oscillate. Figure 13 shows wave1 and wave4 stop oscillation.
So these circuits make stopping oscillation partially.

Fig. 12.

Stopping oscillation of systems 5.

Fig. 13.

Stoppng oscillation of system 6.

IV. C ONCLUSION
We have proposed a system model using a ring of four
van der Pol oscillators coupled by resistors and diodes. We
can observe various synchronization phenomena by changing
resistors into diodes partially. When all resistors change into
diodes, these synchronization phenomena show almost thesame. However, when we change resistors into diodes partially,
we observe the difference synchronization phenomena bychanging only one part. We will investigate the synchronization phenomena using difference system model with diodes.
We observe oscillation stop partially. Future work, wemeasure data of the amplitude, and sort the date. We would
like to presume a parameter and oscillatory relations.
R EFERENCES[1] Yoko UWATE, Yoshifumi NISHIO and Ruedi STOOP ”Synchronizationin Two Polygonal Oscillatory Networks Sharing a Branch” Proceedingsof International Workshop on Nonlinear Dynamics of Electronic Systems(NDES’10), pp. 62-65, May 2010.
[2] Yoko UWATE and Yoshifumi NISHIO ”Synchronizing Coupled Oscillators in Polygonal Networks with Frustration” Proceedings of IEEE
International Symposium on Circuits and Systems (ISCAS’11), pp. 745748, May 2011.

- 82 -