2019 RISP International Workshop on Nonlinear Circuits, Communications and Signal Processing (NCSP2019)
Honolulu, Hawaii, USA, March 4-7, 2019

Synchronization Phenomena in Coupled van der Pol Oscillators Chains with Hourglass Structure
Takumi Nara, Daiki Nariai, Yoko Uwate and Yoshifumi Nishio
Dept. Electrical and Electronic Engineering,
Tokushima University
2-1 Minami-Josanjima, Tokushima 7708506, Japan
Email:{nara, nariai, uwate, nishio}@ee.tokushima-u.ac.jp

abstract

2. Previous study

Figure 2 shows the previous study circuit model [1]. Bottom, middle and top are coupled with oscillators by resistors
r. Bottom several numbers of oscillator are coupled by resistors Ri . Further, top several numbers of oscillator are coupled to ground via resistors Ra through the inductors. Middle
oscillators are not coupled with oscillators located in the horizontal direction but coupled vertically. In this network, wecoupled the oscillators of bottom several numbers of oscillator in the chains to tend to produce in-phase synchronization.
1. Introduction
Further, we coupled the oscillators of the top several numbers of oscillator in the chains to tend to produce anti-phase
Synchronization phenomena are the most familiar phe- synchronization. Middle oscillators are coupled with bottom
nomena that exist in nature and it has been studied in various and top several numbers of oscillator. However, they are notfields, such as in electrical, in mechanics, in biological sys- coupled with the other chains.
tems, basically everywhere. Synchronization phenomena are2L
2L
2L
2L
2L
2L
observed everywhere in our life. For example, we can coni
iii
R i
R i
firm flashing firefly lights, metronome, gate patterns of fourr
rr
leg animals, beating rhythm of the heart and so on. Amongiiithem, van der Pol oscillators can observe phenomena similarvvv
CCCto natural phenomena. Therefore we have been interested incoupled oscillators which synchronization phenomena fromrrrcircuit network.
ii
In this study, we investigate synchronization phenomenaiiiivv
LL
Lv
CC
C
from a circuit network, which is composed of several numbers of van der Pol oscillator chains. The circuit model ofvan der Pol oscillator is shown in Fig. 1. This circuit consistsof an inductor, a capacitor and a nonlinear resistance. FromRR
the computer simulations, we observe relatively interestingiiiiiisynchronization phenomena. We focus on synchronization
vv
v
LLLCCCphenomena in the coupled nonlinear circuits.
In this study, we investigate synchronization phenomena fromcircuit network, which is composed of several numbers of vander Pol oscillator chains. We use van der Pol oscillators which
are coupled by resistors. We investigate how synchronization
phenomena changes by changing the number of van der Poloscillators. From the computer simulations, we observe relatively interesting synchronization phenomena.

31b31a32aa 32bg31a 33bg32

g33

C31

g21

C32

L21

g22

C21

C33

L22

L11

C11

g23

L23

C23

C22ii

g11

33a

g12

L12

C12g13

L13

C13

Figure 2: Previous study system model.
CLFirst, we assume that the v − i characteristics of the nonlinear resistor in each oscillator are given by the follows:
ign = −g1 vn + g3 vn3Figure 1: Circuit of van der Pol.
where g1 , g3 > 0.

490

(1)

2019 RISP International Workshop on Nonlinear Circuits, Communications and Signal Processing (NCSP2019)
Honolulu, Hawaii, USA, March 4-7, 2019

3. System model(3) Top oscillators :Next in this study, we investigate to change the number of
middle oscillators. Figure 3 shows the circuit model. Thismodel is named “Hourglass Structure”. The difference fromthe previous study is that it has one middle oscillator.
We define the bottom several numbers of oscillator as
Osc11 , Osc12 , and Osc1m from the left. Further the middle
several numbers of oscillator as Osc21 , Osc22 , and Osc2mfrom the left and the top several numbers of oscillator as
Osc31 , Osc32 , and Osc3m from the left.

dvc3k+ ig3k + ir + i3ka + i3kb = 0,C

dt



diL31a+ Ra (i31a + i32b ) = vc31 ,2L

dt



diL31b
= vc31 ,2L


dt



diL3la2L+ Ra (i3la + i3(l+1)b ) = vc3l ,dt
diL3lb 2L dt + Ra (i3(l−1)a + i3lb ) = vc3l ,
diL3ma
= vc3m ,2L
dt
diL3mb 2L dt + Ra (i3(m−1)a + i3mb ) = vc3m .

2L2L2L2L2L

2L

i31b

i31a

Ra i32bi32a

Ra i33b
r

i33ar

r

ig31

ig32
vC31

ig33
vC32

C

ig21

vC33

C

iL21
vC21

C

Lr

Ri

iL
vC11

ig12

11CL

By using the following variables and parameters:√√g1
LCτ,v=xkjt=kj
g3√√
g1 CCrr

ig11C(4)Riig13iL12
vC12CL

iL13
vC13C

ikj =ykj , ε = g1g3 LL√√√11CLL

α=R,α=,β=.
ai
 aL
RiCrCL(5)

Figure 3: Hourglass Structure.
The normalized equation of this circuit is given as follows:
The circuit equations are given as follows:(1) Bottom oscillators :(1) Bottom oscillators :dvc1j+ iL1j + ig1j + iRi + ir = 0,Cdt dvc1l
C+ iL1l + ig1j + iRi + iRi + ir = 0,dt diL1k L dt = vc1k .

(2)

2

 x˙1j = εx1j (1 − x1j ) − y1j−αi (x1j − x1l ) − β(x1j − x21 ),x˙1l = εx1l (1 − x1l 2 ) − y1l+αi (x1(l−1) − 2x1l + x1(l+1) ) − β(x1l − x21 ),



 y1k
˙ = x1k .
(6)(2) Middle oscillators :(2) Middle oscillators :dvc21
 C dt + iL21 + ig21 + 2mir = 0,diL21
= vc21 .
 Ldt(3)x˙ = εx21 (1 − x21 2 ) − y21 21+β(x11 + ... + x1m − 2mx21 + x31 + ... + x3m ),˙ = x21 .
 y21(7)491

2019 RISP International Workshop on Nonlinear Circuits, Communications and Signal Processing (NCSP2019)
Honolulu, Hawaii, USA, March 4-7, 2019

x 11(3) Top oscillators :x 12x 13

x3k
˙ = εx3k (1 − x3k 2 )−(y3ka + y3kb ) − β(x3k − x21 ),



1



y31a
˙ = {x31 − αa (y31a + y32b )},

2

1


y31b
˙ = x31 ,

2




1y3la
˙ = {x3l − αa (y3la + y3(l+1)b )},
2




1


y3lb
˙ = {x3l − αa (y3(l−1)a + y3lb )},


2





1


y3ma
˙ = x3m ,


2




1



y ˙ = {x3m − αa (y3(m−1)a + y3mb )}.
 3mb
2x 21x 22x 23x 31x 32
x 33
→τ(8)

Figure 5: Computer simulation results (time waveforms)
for previous study.

(B) Case of hourglass structure (m = 3)
Figures 6 and 7 show the computer simulation results forthe case of hourglass structure. The circuit parameters arechosen at the same.
In comparison with (A), the phase shift with Osc11 is approaching in-phase synchronization or anti-phase synchronization.

( j = 1, m ・ k = 1, 2, ..., m ・ l = 2, 3, ..., m − 1 ) .

x31

x32
x11x33x11x11x21

4. Simulation results

x11(A) Case of previous study

x11Figures 4 and 5 show the computer simulation results forthe case of the previous study model. The circuit parameters
are chosen as ε = 0.10, αa = αi = 0.50 and β = 0.02.

x12x11x13x11x11

The bottom three oscillators are become in-phase synchro- Figure 6: Computer simulation results (phase shift)for hourglass structure (m = 3).
nization and the top three oscillators are become anti-phasesynchronization.
x11x32x31

x33

x12x13x11x21x11x22x11x21x23x31
x32

x11x11x11x12x11x11x33→τx13x11x11

Figure 7: Computer simulation results (time waveforms)
for hourglass structure (m= 3).

Figure 4: Computer simulation results (phase shift)for previous study.

492

2019 RISP International Workshop on Nonlinear Circuits, Communications and Signal Processing (NCSP2019)
Honolulu, Hawaii, USA, March 4-7, 2019

(C) Case of hourglass structure (m = 4)
Figures 8 and 9 show the computer simulation results forthe case of hourglass structure. The circuit parameters arechosen at the same.
In comparison with (B), we increase one oscillator to thetop and bottom. As (A) and (B), the bottom oscillators are inphase synchronization and the top oscillators are anti-phasesynchronization. However, in comparison with (B), the toposcillators are not in-phase synchronization and anti-phasesynchronization with Osc11 .
x31x32x11x11x33
x11
x21

x11

x11x13x12x11x34

x11

x11

x14x11x11

the middle oscillator is not influenced and independently synchronized.

5. Conclusions
In this study, we have investigated synchronization phenomena observed by circuit networks, which are composedof several numbers of van der Pol oscillator chains. By thecomputer simulations, we could obtain of results that the bottom oscillators observe in-phase synchronization and the top
oscillators observe anti-phase synchronization. Further, wecould observe synchronization phenomena by changing withthe value of m.
In the future works, we will investigate synchronization
phenomena using other parameters. Therefore, we investigate
this case in detail. Further, in this study, we have investigatedoscillators connected symmetrically to left and right, but wewould like to investigate circuit models that are connected
asymmetrically to left and right by increasing the number ofoscillators in the middle.

Figure 8: Computer simulation results (phase shift)for hourglass structure (m = 4).

References[1] K. Matsumura, T. Nagai, Y. wate and Y. Nishio, “Analysis of Synchronization Phenomenon in Coupled Oscillator Chains,” Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS’12), pp.620-623,
May 2012.

x11x12x13x14[2] D. Nariai,M. Tran, Y. Uwate and Y. Nishio, “Synchronization in Two Rings of Coupled Three van der Pol Oscillators,” Proceedings of RISP International Workshop
on Nonlinear Circuits, Communications and Signal Processing (NCSP’17), pp.289-292, FEb.2017.

x21x31x32
x33x34→τFigure 9: Computer simulation results (time waveforms)
for hourglass structure (m= 4).

[3] V. Hien,M. Tran, Y. Uwate, Y. Nishio, “Effect of Changing Coupling Strength to Synchronization Phenomena in
Coupled van der Pol Oscillators,” Proceedings of RISPInternational Workshop on Nonlinear Circuits, Communications and Signal Processing (NCSP’18), pp. 148-151,
Mar. 2018.

From Figs. 6 and 8 there is the difference in the synchro- [4] T. Matsunashi, D. Nariai, Y. Uwate and Y. Nishio, “Synnization phenomena of the top oscillators. The top oscillators
chronization Phenomena in a Ring of van der Pol Oscilof all results are anti-phase synchronization. However, looklators Coupled by Time-Varying Resistor,” Proceedings
ing at the result of phase shift with Osc11 , when the value ofof RISP International Workshop on Nonlinear Circuits,
m is an odd number, the phase shift with Osc11 is approachCommunications and Signal Processing (NCSP’18), pp.
ing the in-phase synchronization or anti-phase synchroniza168-171, Mar. 2018.
tion, and when the value of m is an even number, the synchronization phenomena were not observed in Osc11 and toposcillators. This is because the top oscillators are counteracting the synchronization. The top oscillators produce antiphase synchronization. So, in the case of an even number,the adjacent top oscillators counteract each other. As a result,493