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PAPER

Analysis of Phase-Inversion Waves in Coupled Oscillators
Synchronizing at In-and-Anti-Phase
Masayuki YAMAUCHI†a) , Student Member, Masahiro OKUDA†† , Nonmember,
Yoshifumi NISHIO† , Regular Member, and Akio USHIDA† , Fellow

SUMMARY Recently, we have discovered wave propagation
phenomena which are continuously existing waves of changing
phase states between two adjacent oscillators from in-phase toanti-phase or from anti-phase to in-phase in van der Pol oscillators coupled by inductors as a ladder. We named the phenomena
as “phase-inversion waves.” In this study, phase-inversion waves
which exist in the state of in-and-anti-phase synchronization havebeen found. We observe the phenomena by circuit experiments
and computer calculations, and investigate them.
key words:
coupled oscillators, phase diﬀerence, phaseinversion waves, in-and-anti-phase synchronization

1.

Introduction

Many studies on synchronization phenomena of coupledoscillators have been carried out up to now in various
ﬁelds, physics [1]–[3], biology [4], electrical engineering[5]–[14], and so on. Endo et al. have reported detailsof theoretical analysis and circuit experiments aboutsome coupled oscillators as a ladder, a ring and a twodimensional array [6]–[8].
Recently, the authors have discovered a wave propagation phenomena that phase states between adjacentoscillators change from in-phase to anti-phase or fromanti-phase to in-phase in oscillators coupled by inductors as a ladder [11]. We named the phenomena as“phase-inversion waves.” It is very important to analyze the phase-inversion waves and to clarify the mechanism of the generation, because it is similar to propagation phenomena of electrical information in an axial
ﬁber of nervous system. In [11], we explained the mechanism of the generation of the phase-inversion waves byusing the relationship between phase states and instantaneous oscillation frequencies. Further, we conﬁrmedthat the phenomena could be explained by a simpliﬁed
mathematical model [12].
In this study, we investigate the phase-inversion
waves observed from the same van der Pol oscillaManuscript received October 3, 2002.
Manuscript revised January 26, 2003.
Final manuscript received April 3, 2003.
†
The authors are with the Department of Electricaland Electronic Engineering, The Univsersity of Tokushima,
Tokushima-shi, 770-8506 Japan.
††
The author is with Engineering Department, Shionogi
Engineering Service Co., LTD., Amagasaki-shi, 660-0813
Japan.
a) E-mail: masa@ee.tokushima-u.ac.jp

tors ladder but in diﬀerent synchronization mode; inand-anti phase synchronization, which is similar to thephenomenon reported in [10]. In this synchronizationmode, in-phase and anti-phase synchronizations exist
alternately. An interesting point of this synchronizationis that the edge oscillators and their adjacent oscillators must be synchronized at anti-phase. This causes
various limitations of the generated phenomena with respect to the phase-inversion waves. For example, in thepast studies, only even numbers of the phase-inversion
waves can exist and the number of the oscillators do notaﬀect the observed phenomena. On the other hand, in
this study, even numbers of the phase-inversion waves
exist in even numbers of the oscillators and odd numbers of the phase-inversion waves exist in odd numbersof the oscillators. The reason of this diﬀerence is explained in Sect. 4. Further, the mechanisms of propagation, reﬂection at an edge of the array and reﬂection bycollision of two phase-inversion waves are diﬀerent fromthose reported in [11]. They are explained by using a
relationship between phase states and instantaneous oscillation frequencies in Sect. 5. The computer simulated
results are conﬁrmed by circuit experiments.
2.

Circuit Model

The circuit model used in this study is shown in Fig. 1.
N van der Pol oscillators are coupled by inductors L0 .
We carried out computer calculations for the cases of
N = 9 and 20. Further, we carried out circuit experiments for the case of N = 9. In the computer calcu-

Fig. 1

Coupled van der Pol oscillators as a ladder.

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lations, we assume the v − i characteristics of the nonlinear negative resistors in the circuit by the following
function.
ir (vk ) = −g1 vk + g3 vk3(g1 , g3 > 0).

(1)

The circuit equations governing the circuit in Fig. 1 are
written as[First Oscillator]ẋ1 = y1

(2)



1 3
ẏ1 = −x1 + α(x2 − x1 ) + ε y1 − y13(a) Double waves (N = 29).

[Middle Oscillators]ẋk = yk

(3)


1 3
ẏk = −xk + α(xk+1 − 2xk + xk−1 ) + ε yk − yk3(k = 2 ∼ N−1)

[Last Oscillator](b) Double waves (N = 30).

ẋN = yN

1 3
ẏN = −xN + α(xN −1 − xN ) + ε yN − yN
3
where

t = L1 Cτ, iL1 k =L1, ε = g1
α=
L0




Cg1
xk , vk =3L1 g3L1 d,= “ · ”.
C dτ

(4)


,

g1
yk ,
3g3
(5)

(c) Extinction by collision of waves (N = 30).

It should be noted that α corresponds to the couplingof the oscillators and ε corresponds to the nonlinearity of the oscillators. Throughout the paper, we ﬁx
α = 0.10 and ε = 0.30 and calculate (2)–(4) by using
the fourth-order Runge-Kutta method with the stepsize∆τ = 0.01.
3.

Example of Phase-Inversion Waves

Figure 2 shows some typical examples of phaseinversion waves reported previously. They are computer calculated results from the circuit with the sizesof N = 29 and 30. In the ﬁgures, the vertical axes are
the sum of the voltages of adjacent oscillators and thehorizontal axis is time. White regions in the diagram
correspond to the states that the sum of the voltages
is close to zero, namely the adjacent two oscillators aresynchronized at anti-phase. While, black regions correspond to the states that the sum of the voltages haslarge amplitude. We can see that the adjacent two oscillators are synchronized at in-phase in the black regions from Fig. 3, which shows the same phenomenon
as Fig. 2(b), but the vertical axes are the diﬀerence between the voltages of adjacent oscillators. Namely, we(d) Reﬂection by collision of two waves (N = 30).

(e) Decuple waves (N = 30).
Fig. 2 Examples of phase-inversion waves. yk + yk+1 vs. time(k = 1 ∼ N − 1). α = 0.10, ε = 0.30 and ∆τ = 0.01.

YAMAUCHI et al.: ANALYSIS OF PHASE-INVERSION WAVES IN COUPLED OSCILLATORS

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Double waves (N = 30).
Fig. 3 Examples of phase-inversion waves. yk − yk+1 vs. time(k = 1 ∼ N − 1). α = 0.10, ε = 0.30 and ∆τ = 0.01.

(a)

(b)

Fig. 4 Example of in-and-anti-phase synchronization. (a) Circuit experimental result for L0 =1170 mH, L1 =200 mH, C=68 nF
and r=1 kΩ. (b) Computer calculated result for α = 0.10,
ε = 0.30 and ∆τ = 0.01.

can say that the phase-inversion waves in Fig. 2 propagate in the state of in-phase synchronization.
Remark A:
Strictly speaking, when the phase-inversion waves propagate, the whole circuits are not synchronized. However, because the concept of synchronization is useful
to explain the phenomena, the phase-inversion waves
are described such as they propagate in a synchronization state.
In Figs. 2 and 3, we can see the wave propagation,
reﬂection and extinction. Even numbers of the phaseinversion waves exist in both odd and even numbers ofcoupled oscillators.
It has been known that two van der Pol oscillators
coupled by an inductor have two stable phase states,
namely in-phase synchronization and anti-phase synchronization. Further, the oscillation frequency for thein-phase synchronization is smaller than the oscillation
frequency for the anti-phase synchronization. By using
these features, we could explain the mechanism of thephenomena in Fig. 2 by using the relationship between
phase states and instantaneous oscillation frequencies(See [11] for the details).
4.

(a) No wave (N = 6).

(b) Single wave (N = 9).

Phase-Inversion Waves in the State of Inand-Anti-Phase Synchronization

4.1 In-and-Anti-Phase Synchronization
In the circuit model, we can observe another type ofsynchronization as shown in Fig. 4. The vertical axes
are the amplitudes of the voltages and the horizontal
axis is time. We name this synchronization state as“in-and-anti-phase synchronization” because in-phaseand anti-phase exist alternately. Also, in this synchronization state, the edge oscillators and their adjacentoscillators can not be synchronized at in-phase. Hence,
this synchronization state can be observed only when
N is an even number. The oscillation frequency forthe in-and-anti-phase synchronization fmid is almost
the average of the oscillation frequency for the in-phasesynchronization flow and the oscillation frequency for(c) Double waves (N = 8).
Fig. 5 Experimental results. Circuit experiments: L0 =1170 mH, L1 = 200 mH, C = 68 nF and r=1 kΩ. Computer calculations: α = 0.10, ε = 0.30 and ∆τ = 0.01.

the anti-phase synchronization fhigh .
4.2 Phase-Inversion Waves
Figures 5 and 6 show examples of phase-inversion waves
in the state of in-and-anti-phase synchronization. The
vertical axes of Figs. 5 and 6 are the sum of the voltagesof adjacent oscillators and the horizontal axis is time.

IEICE TRANS. FUNDAMENTALS, VOL.E86–A, NO.7 JULY 2003

1802(a) Single wave (N = 29).

Figure 5 shows circuit experimental results and thecorresponding computer calculated results obtained forthe cases of N ≤ 9. In Fig. 5(a), phase-inversion waves
do not exist. A phase-inversion-wave exists continuously in Fig. 5(b). We can see that the change of thephase states from in-phase to anti-phase or from antiphase to in-phase propagates. In Fig. 5(c), a pair ofphase-inversion-waves exist.
Figure 6 shows computer calculated results obtained for the cases of N = 29 and 30. We can observe
phase-inversion waves similar to Fig. 5.
In in-and-anti-phase synchronization, we could notobserve wave extinction by collision of two waves likeFig. 2(c).
4.3 Single Phase-Inversion Wave(b) Triple waves (N = 29).

(c) Double waves (N = 30).

(d) Double waves (N = 30).

In this subsection, We clarify the relationship between
the number of oscillators and the number of phaseinversion waves.
<in In-Phase Synchronization>
In the state of the in-phase synchronization, theedge oscillators and their adjacent oscillators must be
synchronized at in-phase. Therefore, in spite of thenumber of the oscillators in the array, only even numbers of phase-inversion waves can exist [11],<in In-and-Anti-Phase Synchronization>
In the state of the in-and-anti-phase synchronizing, the edge oscillators and their adjacent oscillators
must be synchronized at anti-phase. Therefore, whenan even number of oscillators are coupled, even numbers of phase-inversion waves can exist, while when an
odd number of oscillators are coupled, odd numbers ofphase-inversion waves can exist.
Remark B:
As explained in Sect. 4.1, the in-and-anti-phase synchronization is not stable in the array of odd numbersof oscillators. However, when odd numbers of phaseinversion waves exist in the array, the in-and-anti-phasesynchronization can be observed. Furthermore, for theﬁrst time, odd numbers of phase-inversion waves including the single wave are observed.
5.

Mechanisms of Phase-Inversion Waves

Figures 7, 8 and 9 show phase diﬀerences and instantaneous frequencies, where Φk,k+1 is phase diﬀerence
between OSCk and OSCk+1 and fk is instantaneous
frequency of OSCk . We deﬁne them as follows:(e) Quadruple waves (N = 30).
Fig. 6 Examples of phase-inversion waves. α = 0.10, ε = 0.30
and ∆τ = 0.01.

τk (n) − τk+1 (n)×πτk (n) − τk (n − 1)1
fk (n) =2(τk (n) − τk (n − 1))
Φk,k+1 =(6)(7)

where τk (n) is time when the voltage of OSCk crosses

YAMAUCHI et al.: ANALYSIS OF PHASE-INVERSION WAVES IN COUPLED OSCILLATORS

1803(a) Phase diﬀerence.
(a) Phase diﬀerence.

(b) Instantaneous frequency.
(b) Instantaneous frequency.
Fig. 7 Mechanism of propagation (computer calculated
results).

Fig. 9 Mechanism of reﬂection by collision of two waves (computer calculated results).

0 [V] at n-th time. It has been known that f changes asΦ changes. We explain the mechanisms of the phaseinversion waves by using the relationship between f andΦ.
5.1 Mechanism of Propagation(a) Phase diﬀerence.

Figure 7 shows propagating single phase-inversion
wave. The mechanism of the wave propagation can be
explained in Table 1.
After the wave reﬂects at an edge of the array,
when the propagating wave reaches OSC4 from the direction of OSCN , phase states and instantaneous fre8 in Fig. 7(a)quencies change in a similar manner (see 
and (b)).
5.2 Mechanism of Reﬂection at an Edge
Figure 8 shows reﬂecting single phase-inversion wave at
an edge of the array. The mechanism of the reﬂection
at an edge of the array can be explained in Table 2.
5.3 Mechanism of Reﬂection by Collision(b) Instantaneous frequency.
Fig. 8 Mechanism of reﬂection at an edge of the array(computer calculated results).

Figure 9 shows reﬂection by collision of two phaseinversion waves in the middle of the array. When thetwo phase-inversion waves reach two adjacent oscillators at the same time, the phase-inversion waves reﬂect.
The mechanism of the wave reﬂection can be explained

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Table 1time [τ ]Mechanism of propagation.
sequence of propagation

Let us assume that the circuit synchronizes at in-and-antiphase and that single wave reaches OSC6 from the directionof OSCN .

around 24

around 27

around 32

around 36

around 62

Table 2time [τ ]f5 starts to change from fmid toward flow (see
3 in Fig. 7(b)).

sequence of reﬂection at an edge

Let us assume that the circuit synchronizes at in-and-antiphase and that single wave reaches OSC3 from the directionof OSCN . The phase state between OSC1 and OSC2 must be
anti-phase and the phase state between OSC2 and OSC3 must
be in-phase (see Sect. 4.1).

f6 changes from fmid toward fhigh , because thephase state between OSC6 and OSC7 changes
1 in Fig. 7(b)).
from in-phase to anti-phase (see 

The phase state of OSC6 advances, because f6changes from fmid toward fhigh . Hence, Φ5,6
starts to change from −π to 0 (see Eq. (6) and
2 in Fig. 7(a)).

Mechanism of reﬂection at an edge.

around 80

f2 starts to change from fmid toward fhigh , because the phase state between OSC2 and OSC3
1 inchanges from in-phase to anti-phase (see 
Figs. 8(a) and (b)).

around 84

2 inΦ1,2 starts to change from −π to 0 (see 
Fig. 8(b)).

around 89

3 in
f1 starts to degrease toward flow (see 
Fig. 8(a)).

around 122

f2 starts to decrease to fmid again before f2
reaches fhigh , because the phase states between
OSC1 and OSC2 and between OSC2 and OSC3
4 in Fig. 8(b)).
start to interchange (see 

around 148

When Φ1,2 reaches 0, f1 reaches flow and f2
reaches fmid . However, the in-phase synchronization of OSC1 and OSC2 is not stable, because f1 is not equal to f2 . Φ1,2 changes toπ, because f1 is smaller than f2 . Further, f1
5 in
and f2 start to increase toward fhigh (see 
Figs. 8(a) and (b)).

around 154

Φ2,3 starts to change to −2π, because f2 is larger
6 in Fig. 8(a)).
than f3 (see 

around 160

7 in Fig. 8(b)).
f3 changes toward flow . (see 

around 170

f2 changes to fmid again before f2 reaches fhigh ,because the phase states between OSC1 andOSC2 and between OSC2 and OSC3 start to in8 in Fig. 8(b)).
terchange (see 

around 200

When Φ1,2 reaches π, f1 and f2 reaches fmid
9 in Figs. 8(a) and (b)).
again (see 

4 in Fig. 7(a)).
Φ4,5 changes from 0 to −π (see 

f6 changes to fmid again before f6 reaches fhigh ,because the phase states between OSC5 andOSC6 and between OSC6 and OSC7 start to in5 in Fig. 7(b)).
terchange. (see 

around 79

f5 changes to fmid again before f5 reaches flow
because the phase states between OSC5 andOSC6 and between OSC6 and OSC7 start to in6 in Fig. 7(b)).
terchange (see 

around 114

f5 becomes to fmid when the phase states between OSC4 and OSC5 and between OSC5 and
7 in Fig. 7(a)OSC6 ﬁnish to interchange (see 
and (b)).

in Table 3.
5.4 Extinction by Collision
We could observe extinction of the phase-inversion
waves by collision in the past study [11] (see Fig. 2(c)).
However, such an extinction can not be observed in thisstudy. The reason would be explained as follows. In thepast study, when two phase-inversion waves reach an
oscillator at the same time, the phase-inversion waves
disappeare. Because the instantaneous frequency of theoscillator changes from flow to fhigh or fhigh to flow .
However, in this study, the instantaneous frequency ofthe oscillator can not change as the past study, because
the changing directions of the instantaneous frequency
are diﬀerent between the two sides of the oscillator, forexample, from fmid to fhigh at the right side and fromfmid to flow at the left side. Therefore, we can notobserve extinction by collision.
6.

Conclusions

In this study, we observed phase-inversion waves in aladder of coupled oscillators synchronizing at in-andanti-phase by both circuit experiments and computer
calculations.
In the previous study, wave propagation, reﬂection

YAMAUCHI et al.: ANALYSIS OF PHASE-INVERSION WAVES IN COUPLED OSCILLATORS

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Table 3time [τ ]Mechanism of reﬂection by collision.
sequence of reﬂection by collision

Let us assume that the circuit synchronizes at in-and-antiphase and that two phase-inversion waves are going to reach
OSC13 and OSC16 at the same time from the direction of
OSC1 and OSCN , respectively. The phase states between
OSC13 and OSC14 and between OSC15 and OSC16 must be
anti-phase and the phase state between OSC14 and OSC15
must be in-phase (see Sect. 4.1).

around 51

f14 and f15 start to change from fmid toward
flow , because Φ13,14 changes from −π to −2π.
1 in Fig. 8(b)).
Φ15,16 changes from −π to 0 (see 

around 60

Φ14,15 does not change, because f14 and f15
start to change toward flow at the same time
2 in Fig. 8(a)).
(see 

around 78

f13 and f16 start to decrease to fmid again before they reach fhigh , because the phase states
between OSC12 and OSC13 and between OSC13
and OSC14 start to interchange and the phasestates between OSC15 and OSC16 and between
OSC16 and OSC17 also start to interchange (see
3 in Fig. 8(b)).

around 103

When Φ13,14 and Φ15,16 reach −2π and 0, f14
and f15 reach flow . However, because f13 andf16 are fmid , Φ13,14 and Φ15,16 change to −3π
and π. Therefore f13 and f16 change from fmid
toward fhigh , and f14 and f15 change from flow
4 in Figs. 8(a) and (b)).
to fmid (see 

around 300

When Φ13,14 and Φ15,16 reach −3π and π, f13 ∼f16 reach fmid .

at an edge of the array, reﬂection by collision of two
waves and extinction by collision of two waves could
be observed. However, the extinction could not be observed in the in-and-anti-phase synchronization. Further, single phase-inversion wave could be observed forthe ﬁrst time.
In the state of the in-and-anti-phase synchronization, instantaneous oscillation frequencies changearound only single frequency fmid . We could explain
the mechanism of the wave propagation, the reﬂection
by collision of two waves and the reﬂection at an edgeof the array by using the relationship between phasestates and instantaneous frequencies.
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Masayuki Yamauchi
was born inOsaka, Japan, in 1974. He received the
B.E. and M.E. degrees from Tokushima
University, Tokushima, Japan, in 1998
and 2000, respectively. He is currently
working towards a Ph.D. degree at thesame university. His research interest lies
in nonlinear phenomena in coupled oscillators, mainly wave-motion in coupled oscillators as a ladder.

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Masahiro Okuda
was born in Kagawa, Japan, in 1977. He received the
B.E. and M.E. degrees from Tokushima
University, Tokushima, Japan, in 2000
and 2002, respectively. He is currently
working in the Shionogi Engineering Service Co., LTD.

Yoshifumi Nishio
received the B.E.,
M.E. and Ph.D. degrees in Electrical Engineering 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. From May 2000 hespent a year in the Laboratory of Nonlinear Systems (LANOS) at the Swiss Federal Institute of Technology Lausanne (EPFL) as a visiting professor. His research interests include analysis and application ofchaos in electrical circuits, analysis of synchronization in nonlinear circuits, development of analytical methods for nonlinear
circuits and theory and application of cellular neural networks.
Dr. Nishio is a member of the IEEE.

Akio Ushidareceived the B.E. and
M.E. degrees in Electrical Engineeringfrom Tokushima University in 1961 and
1966, respectively, and the Ph.D. degree
in Electrical Engineering from Universityof Osaka Prefecture in 1974. He was an
Associate Professor from 1973 to 1980 at
Tokushima University. Since 1980 he has
been a Professor in the Department ofElectrical and Electronic Engineering at
Tokushima University. From 1974 to 1975
he spent one year as a visiting scholar at the Department of Electrical Engineering and Computer Sciences at the University ofCalifornia, Berkeley. His current research interests include numerical methods and computer-aided analysis of nonlinear systems. Dr. Ushida is a member of IEEE.