IEICE TRANS. FUNDAMENTALS, VOL.E82–A, NO.11 NOVEMBER 1999

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PAPER

Wave Propagation Phenomena of Phase States in
Oscillators Coupled by Inductors as a Ladder
Masayuki YAMAUCHI† , Masahiro WADA† , Student Members, Yoshifumi NISHIO† ,and Akio USHIDA† , Members

SUMMARY
In this study, wave propagation phenomena ofphase states are observed at van der Pol oscillators coupled byinductors as a ladder. For the case of 17 oscillators, interesting
wave propagation phenomena of phase states are found. By using
the relationship between phase states and oscillation frequencies,
the mechanisms of the propagation and the reﬂection of wave are
explained. Circuit experimental results agree well with computer
calculated results qualitatively.
key words: coupled oscillators, wave propagation phenomena,phase-diﬀerence, oscillation frequency, wave propagation speed1.

ir (vk ) = −g1 vk + g3 vk3

Circuit Model

Manuscript received January 18, 1999.
Manuscript revised April 24, 1999.
†
The authors are with the Faculty of Engineering,
Tokushima University, Tokushima-shi, 770–8506 Japan.

(1)[First Oscillator]ẋ1 = y11
ẏ1 = −x1 + α(x2 − x1 ) + ε y1 − y133(2)[Middle Oscillators]ẋk = yk1(3)
ẏk = −xk +α(xk+1 −2xk +xk−1 )+ε yk − yk33(k = 2, 3, 4, · · ·, N − 1)
[Last Oscillator]ẋN = yN



1 3
(4)
ẏN = −xN + α(xN −1 − xN ) + ε yN − yN
3

where

t = L1 Cτ, iL1 k =L1α=, ε = g1
L0




Cg1
xk , vk =3L1 g3

L1 d
,
=“·”
C dτ



g1
yk ,
3g3
(5)

It should be noted that α corresponds to the coupling
and that ε corresponds to the nonlinearity. Equations (2)–(4) are calculated by using the fourth-order
Runge-Kutta method.
3.

Circuit model is shown in Fig. 1. N van der Pol oscillators are coupled by coupling inductors L0 . We carriedout computer calculations for the cases of N = 5 · · · 17
and circuit experiments for the case of N = 5. In the(g1 , g3 > 0)

The circuit equations governing the circuit in Fig. 1
are expressed as

Introduction

A lot of studies on synchronization phenomena of coupled oscillators have been carried out up to now. Endoet al. have reported a details of theoretical analysis
and circuit experiments about some coupled oscillators
as a ladder, ring and two-dimensional array [1]–[3]. Recently, wave propagation phenomenon observed fromcoupled chaotic circuits is also reported [4], [5]. However, such studies treat only transient states for a given
set of initial conditions and there seems to be very few
studies on continuously existing wave propagation phenomenon observed simple coupled oscillators circuits.
In this study, we investigate wave propagation phenomena observed from van der Pol oscillators coupled
by inductors as a ladder. By computer calculations forthe case of 17 oscillators, we can ﬁnd various interesting wave propagation phenomena of phase states. By
using the relationship between phase states and oscillation frequencies, we can explain why does the wave
propagation and why does the wave reﬂection. Further,
for the case of 5 oscillators, we carry out both computer
calculations and circuits experiments. Circuit experimental results agree well with computer calculated results qualitatively.
2.

computer calculations, we assume the v − i characteristics of nonlinear negative resistors in the circuit by thefollowing functions.

Wave Propagation Phenomenon

In this section, wave propagation phenomenon observed
from the circuit with 17 oscillators is investigated. Although we introduce the results only for 17 oscillators,
the similar phenomena are observed from both of evenand odd. Further, we could observe those from 100
oscillators.
Figure 2 shows typical examples of observed wave

YAMAUCHI et al: WAVE PROPAGATION PHENOMENA OF PHASE STATES

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Fig. 1

Coupled van der Pol oscillators as a ladder.

propagation phenomena. These results are obtained
for the same parameter values by changing initial conditions as follows: 1. Set initial conditions of all oscillators same. 2. Invert the sign of the voltages of one or
two oscillators.
In upper diagrams, vertical axes are sum of voltages of adjacent oscillators and horizontal axes are time.
Hence, the diagrams show how phase diﬀerences between adjacent oscillators change as time goes. White
regions in the diagram correspond to the states that
two adjacent oscillators are anti-phase synchronization.
While, black regions correspond to the in-phase synchronization. In lower ﬁgures, snapshots of attractor ofeach oscillator and phase states between adjacent oscillators are shown.
In Fig. 2 (a) wave vanishes and phase states settle
down to regular in-phase synchronization mode. On
the other hand in other ﬁgures we can see that wave
propagation phenomena continue to exist. As far as
we know, such a continuously existing wave propagation phenomena of phase states in simple real circuits
have never been reported. Further, we can see that
there are two diﬀerent scenario for the collision of two
waves. Namely, in Fig. 2 (a) two waves extinct when
they collides. While, in Fig. 2 (b) two waves reﬂect andthe wave phenomenon continues to exist.
Next, we explain the mechanism of the generationof the wave by using the change of the oscillation frequencies according to the synchronization states. It has
been already known that oscillation frequency of inphase synchronization of oscillators coupled by inductors is diﬀerent from that of anti-phase synchronization.
Namely, fin , oscillation frequency of in-phase synchronization, is smaller than fanti , oscillation frequency ofanti-phase synchronization. Further, the diﬀerence betweens fin and fanti increases as coupling inductanceincreases [6].
Throughout the paper, we deﬁne the phase diﬀerence between two adjacent oscillators and the frequency
of OSCk as follows:τk (n) − τk+1 (n)×πτk (n) − τk (n − 1)1
fk (n) =2(τk (n) − τk (n − 1))
Φk,k+1 (n) =(6)where τk (n) is time when the voltage of OSCk crosses
0[V] at n-th time.
3.1 Mechanism of Wave Propagation
1. Let us assume that OSC1 – OSC6 are in-phasesynchronization and that the wave changing fromin-phase into anti-phase is going to reach OSC6
from the direction of OSC17. (τ = τ1 in Fig. 3.)
2. As Φ6,7 approaches π, oscillation frequency of
OSC6 f6 changes from fin to fanti . (τ = τ2 in
Fig. 3.)
3. The change of oscillation frequency of OSC6 causes
increase of phase diﬀerence between OSC5 andOSC6. Namely Φ5,6 increases. Speed of the increase is decided by the diﬀerence between f5 andf6 . This means that propagation speed of the wave
is decided by the diﬀerence between fin and fanti .
4. When Φ5,6 reaches almost π, f6 is equal to fanti .
(τ = τ3 in Fig. 3.)
Although the above explanation is for the case ofchanging from in-phase to anti-phase, changing fromanti-phase to in-phase can be explained in a similar

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Fig. 2

Typical examples of wave propagation phenomena. α = 0.05, ε = 0.3 and ∆τ = 0.001.

manner.
Figure 4 shows computer calculated results ofphase diﬀerences and oscillation frequencies in the middle of the array. In the ﬁgure we can see two waves;
one is the changing from in-phase to anti-phase andthe other is the changing from anti-phase to in-phase.

change. However, it cannot reach fanti , because
OSC1 is at an edge and there are no eﬀect fromthe other side. Hence, Φ1,2 continues to increase
until reaching 2π.
5. By the eﬀect of the decrease of (Φ1,2 mod 2π), f2
begins to decreases again from fanti toward fin .

3.2 Mechanism of Wave Reﬂection at the Edges of Array

Figure 5 shows computer calculated results ofphase diﬀerences and oscillation frequencies at the edgeof the array. In the ﬁgure we can see that a wave reﬂects
as the manner explained above.

1. Let us assume that only OSC1 and OSC2 are inphase synchronization and that a wave from inphase to anti-phase reaches at the edge.
2. Oscillation frequency of OSC2 f2 begins to change
from fin toward fanti , because in-phase synchronization between OSC2 and OSC3 breaks.
3. The change of f2 causes slipping of Φ1,2 .
4. Oscillation frequency of OSC1 f1 also begins to

3.3 Mechanism of Wave Reﬂection at the Middle ofArray
1. Let us assume that OSC7 – OSC10 are in-phasesynchronization and that the waves changing fromin-phase into anti-phase are going to reach OSC7
and OSC10 at the almost same time from the di-YAMAUCHI et al: WAVE PROPAGATION PHENOMENA OF PHASE STATES

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Fig. 3Fig. 4

Mechanism of wave propagation (outline charts).

Wave propagation (computer calculated results).

rections of OSC1 and OSC17, respectively.
2. Waves reach OSC7 and OSC10 almost equal timing. Φ7,8 and Φ9,10 approach π and −π by almost
equal timing.
3. Because Φ7,8 and Φ9,10 approach π and −π respec-

Fig. 5 Wave reﬂection at the edge of the array (computer
calculated results).

tively, oscillation frequencies of OSC8 f8 and OSC9f9 change from fin toward fanti .
4. However because f8 and f9 change almost simultaneously, in-phase synchronization between OSC8

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and OSC9 does not break. Namely, Φ8,9 remains
almost 0. Hence, f8 and f9 do not reach fanti . Accordingly Φ7,8 and Φ9,10 continue to change until
reaching 2π and −2π, respectively.
5. By the eﬀect of the decreases of (Φ7,8 mod 2π) and(Φ9,10 mod 2π), f8 and f9 begin to decrease toward
fin again.
Waves changing from anti-phase to in-phase canbe explained in a similar manner.
Figure 6 shows computer calculated results ofphase diﬀerences and oscillation frequencies in the middle of the array. In the ﬁgure we can see that a wave
reﬂects as the manner explained above.
3.4 Mechanism of Wave Extinction
1. Let us assume that OSC5 – OSC7 are anti-phasesynchronization and that the waves changing fromanti-phase into in-phase are going to reach OSC5
and OSC7 at the almost same time from the directions of OSC1 and OSC17 respectively.
2. Reaching the waves causes the changes of oscillation frequencies of OSC5 f5 and OSC7 f7 changes
from fanti to fin .
3. Accordingly Φ5,6 and Φ6,7 change from π to 0 and
2π respectively.
4. Changes of Φ5,6 and Φ6,7 cause that the change ofoscillation frequency of OSC6 f6 from fanti to fin .
5. When Φ5,6 and Φ6,7 reach 2π and 0 respectively,
f6 become to be equal to fin . The whole array
results in stable in-phase synchronization.
Waves changing from in-phase to anti-phase canbe explained in a similar manner.
Figure 7 shows computer calculated results ofphase diﬀerences and oscillation frequencies in the middle of the array. In the ﬁgure we can see that a wave
extinction as the manner explained above.
According to the above-explained mechanisms of
the wave reﬂection and the wave extinction in the middle of the array, we can conclude that the two waves
colliding in the middle of the array will;
1. Reﬂect when the waves reach OSCk and OSCk + 1
at the almost equal timing.
2. Extinct when the waves reach OSCk − 1 andOSCk + 1 at the almost equal timing.

Fig. 6 Wave reﬂection at the middle array (computer
calculated results).

Fig. 7

Wave extinction (computer calculated results).

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(a)
(a)

(b)

(b)

(c)
Fig. 9 Circuit experimental results (N = 5). (a) L0 =1200 mH, (b) L0 = 950 mH, (c) L0 = 650 mH. (i) v1 vs. v2 ,(ii) v2 vs. v3 , (iii) v3 vs. v4 , (iv) v4 vs. v5 .

(c)Fig. 8 Computer calculated results (N = 5). (a) α = 0.10,(b) α = 0.15, (c) α = 0.2.

4.

Circuit Experiments

In this section circuit experimental results are shown.
Because it is diﬃcult to carry out circuit experiments for the array with the size of N = 17, we carriedout both computer calculations and circuit experiments
for the array with the size of N = 5.

Figure 8 shows computer calculated results for
ε = 0.30 and ∆τ = 0.01. We can see that wave propagation phenomenon appears even in this small numberof oscillators case. Further, we can conﬁrm that wave
propagation speed increases as coupling parameter increases.
Figure 9 shows circuit experimental results for
L1 = 200 mH, C = 100 nF and r = 1.0 kΩ. Circuit experimental results agree well with computer calculated
results qualitatively.
The phenomena correponding to Figs. 2 (c) and (d)are not observed in circuit experiments. We consider
that this is because the number of elements is small.
However, detailed investigation is our future study.
5.

Conclusions

In this study, we investigated wave propagation phenomena of phase states observed from van der Pol oscillators coupled by inductors as a ladder. For the caseof 17 oscillators, we found interesting wave propagation
phenomena of phase states. By using the relationship

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between phase states and oscillation frequencies, we explained the mechanisms of the propagation and the reﬂection of wave. Circuit experimental results agreed
well with computer calculated results qualitatively.

Yoshifumi Nishio
received the B.E.
and M.E. and Ph.D. degrees in ElectricalEngineering from Keio University, Yokohama, Japan, in 1988, 1990 and 1993,
respectively. In 1993, he joined the Department of Electrical and Electronic Engineering at Tokushima University, Tokushima Japan, where he is currently an Associate Professor. His research interests
are in chaos and synchronization phenomena in nonlinear circuits. Dr. Nishio is aReferences[1] T. Endo and S. Mori, “Mode analysis of a multimode ladderoscillator,” IEEE Trans. Circuits & Syst., vol.23, pp.100–113,
Feb. 1976.
[2] T. Endo and S. Mori, “Mode analysis of two-dimensional lowpass multimode oscillator,” IEEE Trans. Circuits & Syst.,
vol.23, pp.517–530, Sept. 1976.
[3] T. Endo and S. Mori, “Mode analysis of a ring of a large
number of mutually coupled van der Pol oscillators,” IEEE
Trans. Circuits & Syst., vol.25, pp.7–18, Sept. 1978.
[4] M.J. Ogorzalek, Z. Galias, A.M. Dabrowski, and W.R.
Dabrowski, “Chaotic waves and spatio-temporal patterns inlarge arrays of doubly-coupled Chua’s circuits,” IEEE Trans.
Circuits & Syst. I, vol.42, pp.706–714, Oct. 1995.
[5] L. Pivka, “Autowaves and Spatio-temporal chaos in CNNsPart I: A tutorial,” IEEE Trans. Circuits & Syst. I, vol.42,
pp.638–649, Oct. 1995.
[6] T. Suezaki and S. Mori, “Mutual synchronization of two oscillators,” IECE Trans., vol.48, no.9, pp.1551–1557, Sept.
1965.

Masayuki Yamauchi
was born inOsaka, Japan, on 1974. He received the
B.E. degrees from Tokushima University,
Tokushima, Japan, in 1998. He is currently working towards M.E. degree at thesame university. His research interest isin nonlinear phenomena in coupled oscillators, mainly wave-motion in coupled oscillators as a ladder.

Masahiro Wada
was born in Nara,
Japan, on 1972. He received the B.E. and
M.E. degrees from Tokushima University,
Tokushima, Japan, in 1995, 1997, respectively. He is currently working towards
Ph.D. degree at the same university. His
research interest is in chaos in nonlinear
circuits.

member of the IEEE.

Akio Ushidareceived the B.E.
and M.E. degrees in electrical engineering from Tokushima University in 1961
and 1966, respectively, and the Ph.D. degree in electrical engineering from University of Osaka Prefecture in 1974. He wasan associate professor from 1973 to 1980
at Tokushima University. Since 1980 hehas been a Professor in the Department
of Electrical Engineering at the university.
From 1974 to 1975 he spent one year as
a visiting scholar at the Department of Electrical Engineeringand Computer Sciences at the University of California, Berkeley. His current research interests include numerical methods
and computer-aided analysis of nonlinear systems. Dr. Ushida isa member of IEEE.