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PAPER

Phase-Waves in a Ladder of Oscillators
Masayuki YAMAUCHI†a) , Yoshifumi NISHIO† , Regular Members, and Akio USHIDA† , Fellow

SUMMARY In this study, wave propagation phenomena ofphase diﬀerences observed in van der Pol oscillators coupled byinductors as a ladder are investigated. The phenomena are called“phase waves.” We classify the observed phenomena and analyze
the diﬀerence in detail. We observe that the behavior of the phasewaves generated by giving a phase diﬀerence of positive value is
diﬀerent from the behavior of those generated by giving a phase
diﬀerence of negative value. We can also observe the generationof two pairs of phase waves. We clarify the mechanisms of thesecomplicated phenomena. Finally, for the case of nine oscillators,
we carry out both computer calculations and circuit experiments.
Circuit experimental results agree well with computer calculated
results qualitatively.
key words: coupled oscillators, phase diﬀerence, phase waves,phase-inversion waves

1.

Introduction

A large number of coupled limit-cycle oscillators areuseful as models for a wide variety of systems in naturalﬁelds, for example, diverse physiological organs including gastrointestinal tracts and axial ﬁbers of nervous
systems, convecting ﬂuids, arrays of Josephson junctions and so on. Hence, it is very important to analyze
synchronization and the related phenomena observed incoupled oscillators in order to clarify the mechanismsof the generations or in order to control the generating
conditions of various phenomena in such natural systems [1]–[3]. In the ﬁeld of electrical engineering, numerous studies on synchronization phenomena of coupled van der Pol oscillators have been carried out up to
now [4]–[8].
Recently, we have discovered very interesting wave
propagation phenomena of phase states between two
adjacent oscillators in an array of van der Pol oscillators coupled by inductors [9]–[11]. As far as weknow, to date, such phenomena have not been reported in simple real physical systems. In the study,
we named the continuously existing wave of changing
phase states between two adjacent oscillators from inphase to anti-phase or from anti-phase to in-phase as“phase-inversion wave” and explained the generation
Manuscript received June 18, 2002.
Manuscript revised October 3, 2002.
Final manuscript received December 9, 2002.
†
The authors are with the Department of Electrical andElectronic Engineering, Tokushima University, Tokushimashi, 770-8506 Japan.
a) E-mail: masa@ee.tokushima-u.ac.jp

mechanism by using the relation between the instantaneous oscillation frequencies and the phase diﬀerencesof two adjacent oscillators. In the past study, we researched four basic properties of the phase-inversion
waves (propagation, reﬂection at an edge of the array,reﬂection by collision of two waves and extinction bycollision of two waves) [9]. However, phase propagations observed in transient states have not been analyzed and they have a rich variety of unknown phenomena.
In this study, a propagating wave of phase diﬀerence between two adjacent oscillators is observed. It
is called “phase wave.” The phase waves are observedin transient states and change to the phase-inversion
waves or disappear. In the computer calculations, we
produce phase waves as follows. Almost the same initialconditions are given for all oscillators to produce complete in-phase synchronization which is one of the stablesteady states in the system. After the system settles in
the complete in-phase synchronization, the voltage andthe current with arbitrary phase diﬀerence are input
to the ﬁrst oscillator. The observed phase waves areclassiﬁed into four patterns and the diﬀerence is analyzed in detail. Furthermore, the mechanisms of thecomplicated phenomena for the phase waves are also
explained. Finally, for the case of nine oscillators, we
carry out both computer calculations and circuit experiments. Circuit experimental results agree well withcomputer calculated results qualitatively.
2.

Circuit Model

The circuit model used in this study is shown in Fig. 1.

Fig. 1

Coupled van der Pol oscillators as a ladder.

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N van der Pol oscillators are coupled by coupling inductors L0 . We carried out computer calculations forthe cases of N = 9 and 20 and circuit experiments forthe case of N = 9. In the computer calculations, we
assume the v − i characteristics of the nonlinear negative resistors in each circuit as given 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
ẏ1 = −x1 + α(x2 − x1 ) + ε y1 − y133[Middle Oscillators]ẋk = yk(3)

ẏk = −xk + α(xk+1 − 2xk + xk−1 )1+ ε yk − yk33(k = 2 ∼ N−1)ẋN = yN

(4)



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

t = L1 Cτ, iL1 k =L1α=, ε = g1
L0




Cg1
xk , vk =3L1 g3

L1 d,= “ · ”.
C dτ

Φk,k+1 (n) =τk (n) − τk+1 (n)×πτk (n) − τk (n − 1)

(6)

<Instantaneous Frequency>
fk (n) =1,
2 × (τk (n) − τk (n − 1))(7)

where τk (n) is the time at which the voltage of OSCk
crosses 0 [V] at n-th time.
Figure 2 shows, typical examples of observed phasewaves, which originate from the propagation of thephase diﬀerence between adjacent oscillators, and thephase-inversion waves which originate from the propagation of the phase states (in-phase synchronizationor anti-phase synchronization) between adjacent oscillators. These results are obtained for the same parameter values by changing the initial conditions as follows:
1. Set the initial conditions of all oscillators as thesame.
2. Input an arbitrary phase diﬀerence Φin to the ﬁrst
oscillatorof OSC2 and Φ1,2 is the phase of OSC2 based on thephase of OSC1 . The current (x) and the voltage (y) are
obtained from the following conditional equations:yxsin Φin ∼,cos Φin ∼
. (8)
=−
=
2
2
2
x +y
x +y 2[Last Oscillator]

where<Phase Diﬀerence>



g1
yk ,
3g3

In the diagrams of Fig. 2, the vertical axis is thesum of the voltages of adjacent oscillators, and the horizontal axis is time. Hence, the diagrams show how thephase diﬀerences between adjacent oscillators changewith time. Figure 2(a) shows a pair of phase waves
changing from in-phase synchronization to a state withphase diﬀerence and changing from the state with phase
diﬀerence to in-phase synchronization. Figure 2(b)(5)

It should be noted that α corresponds to the coupling of
the oscillators and ε corresponds to the nonlinearity ofthe oscillators. Throughout the paper, we ﬁx α = 0.050
and ε = 0.30 and calculate (2)–(4) by using the fourthorder Runge-Kutta method.
3.

Phase Waves

In this section, wave propagation phenomena observed
from the circuit with 20 oscillators are investigated.
3.1 Phase Diﬀerence
We deﬁne phase diﬀerence between two adjacent oscillators and instantaneous frequency as

Fig. 2 Typical examples of waves. α = 0.050 and ε = 0.30. (a)Phase waves Φin = 45◦ . (b) Phase-inversion waves Φin = 180◦ .

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Fig. 3 Phase waves. α = 0.050 and ε = 0.30. (a) Φin = +60◦ .
(b) Φin = +68◦ . (c) Φin = +71◦ . (d) Φin = +80◦ .

shows a pair of phase-inversion waves changing fromin-phase synchronization to anti-phase synchronizationand changing from anti-phase synchronization to inphase synchronization. The propagation speed of thephase waves becomes higher as Φin increases. And thephase-inversion waves propagate at the highest speed.
3.2 Classiﬁcation
Observed phenomena are classiﬁed into four patterns
by the input phase diﬀerence as follows:(a) No reﬂection−67◦  Φin  +64◦ : The phase diﬀerence decreases while the phase waves are propagating.
The phase waves disappear at the edge of the array(see Fig. 3(a) and Fig. 4(a)).

Fig. 4 Phase waves (Phase Diﬀerence – Time). α = 0.050 and
ε = 0.30. (a) Φin = +60◦ . (b) Φin = +68◦ . (c) Φin = +71◦ .
(d) Φin = +80◦ .

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Fig. 5 Domain of observed phenomena generated by the phase
diﬀerence of positive or negative value. α = 0.050 and ε = 0.30.

(b) One-time reﬂection+65◦  Φin  +70◦ : The phase waves propagate
and reﬂect at the edge of the array. The phase difference decreases while the phase waves are propagating after the reﬂection. The phase waves disappear at the edge of the array (see Fig. 3(b) and
Fig. 4(b)).
(c) Two-times reﬂection
Φin ∼= +71◦ : The phase waves propagate andreﬂect two times at the edges of the array. The
phase diﬀerence decreases while the phase waves
are propagating. The phase waves disappear at
the edge of the array (see Fig. 3(c) and Fig. 4(c)).
(d) Change to phase-inversion waves+72◦  Φin  +180◦ and −68◦  Φin  −180◦ :The phase diﬀerence increases while the phasewaves are propagating. The phase waves change
to the phase-inversion waves when the phase waves
are reﬂected at the edge of the array or in the middle of the array (see Fig. 3(d) and Fig. 4(d)).
Figure 5 shows the domain of the observed phenomena.
We may be able to observe a phenomenon that thephase waves reﬂect three times or more and disappear
by changing N . When N is very large, we can observea phenomenon that the phase waves disappear in the
middle of the array.

Fig. 6 Two pairs of phase waves. α = 0.050 and ε = 0.30. (a)Φin = −68◦ . (b) Φin = −70◦ . (c) Φin = −69◦ .

consider that the phase waves are generated at OSC20 ,Φin is Φ19,20 . In other words, a phase diﬀerence ofnegative value is input as Φin . Thus, this phenomenon
is the same as those generated by a negative value of
Φin . Therefore, the mechanism of this phenomenon is
the same as the mechanism of generating two pairs ofwaves.
4.

Mechanisms of Phase Waves

4.1 Mechanism of Propagation
3.3 Two Pairs of Phase Waves
Two pairs of waves (two pairs of phase waves or a pairof phase waves and a pair of phase-inversion waves) are
generated in the domain of the shaded area of Fig. 5(see Fig. 6). For Φin = −68◦ , two pairs of phase waves
are generated as shown in Fig. 6(a). For Φin = −70◦ , apair of phase waves and a pair of phase-inversion waves
are generated as shown in Fig. 6(b). For Φin = −69◦ ,two pairs of phase-inversion waves exist as shown in
Fig. 6(c). However, when waves are generated, a pairof phase-inversion waves is ﬁrst generated as phasewaves and they waves change to phase-inversion waves.
Therefore, Fig. 6(c) exhibits the same mechanism as
Fig. 6(b) when waves are generated.
Furthermore, two pairs of waves can be generated
when waves reﬂect at the ﬁrst time. When Φin has
a positive value, Φk,k+1 has a negative value. If we

Figure 7 shows how the instantaneous frequencies andthe phase diﬀerences change. We deﬁne fin as the oscillation frequency of the two coupled oscillators being
in-phase. Also, we deﬁne fanti as the oscillation frequency of the two coupled oscillators being anti-phase.
For the parameter values used in this study, the two
frequencies are numerically obtained as fin = 0.1584
and fanti = 0.1736.
Phase waves are propagated by this mechanism(see Table 1). When the propagating phase diﬀerence is
suﬃciently large to be attracted to anti-phase synchronization, the phase waves change to the phase-inversion
waves.

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Table 2time [τ ]Mechanism of Fig. 6(a).

sequence of generated phenomena

When all oscillators are in in-phase synchronization, a negative
phase diﬀerence is given to the ﬁrst oscillator.

around 0

around 20

Because Φ1,2 is not equal to 0, f1 and f2 are larger
than fin . However, f3 is equal to fin , because Φ2,3
and Φ3,4 are equal to 0.
Φ1,2 is attracted to 0 in a moment, because Φ1,2 is
not suﬃciently large to be attracted to anti-phasesynchronization. Therefore, f1 and f2 change towardfin .
Φ2,3 changes from 0, because f2 is higher than fin .
Therefore, f3 starts to change toward fanti .

around 50

around 80

Fig. 7 Mechanism of propagation of phase waves. Example
of generated phase waves by phase diﬀerence of positive value(Φin = +60◦ ).

Table 1time [τ ]around 90

around 130

fk+1 changes from fin toward fanti .

around 225

Because Φk−1,k is relatively small, the phase difference tends to decrease in in-phase synchronization. Therefore, fk changes toward fin .

Φ2,3 is decreasing during f2 > f3 .

f2 starts to change from decreasing to fin to increasing to fantiagain.

around 125

Φ2,3 changes toward 0, because
f2 < f3 by the ﬁrst waves.

around 150

f2 starts to change to decrease to
fin again. f1 and f2 decrease
to fin together, because Φ1,2 is
not suﬃciently large to be attracted to anti-phase synchronization. Therefore, f1 decreases to
fin . Because f1 and f2 becomenearly equal, the decrease of Φ1,2
becomes very slow. The second
pair of phase waves is generated at
this time.

around 170

f3 starts to change toward fin by
the ﬁrst waves.

around 200

Because the decrease of Φ1,2 isvery slow, rates of decrease of f1
and f2 are small values.

sequence of generated phenomena

The phase diﬀerence between the k-th and the(k + 1)-th oscillator (Φk,k+1 ) changes from 0 toward −π, because fk is higher than fk+1 .

Propagation of
the ﬁrst phasewaves begins.

around 120

Let us assume that all oscillators are in in-phase synchronization. The phase waves are going to reach the (k − 1)-th oscillator from the ﬁrst oscillator.

around 10

second waves

f1 changes toward fin and becomes nearly equal to f2 , because
the transition of f2 is very slow
due to the above reason. Therefore, Φ1,2 maintains almost thesame value during propagation of
the ﬁrst pair of phase waves.

Mechanism of propagation.

Instantaneous frequencies of the (k − 1)-th andthe k-th oscillator (fk−1 and fk ) become higher
than the in-phase frequency (fin ), because, theanti-phase frequency (fanti ) is higher than fin .

ﬁrst waves

around 225

The rate of decrease of Φ2,3
changes to a small value, because
f3 becomes nearly equal to f2
which decrease slowly. The second
pair of phase waves is propagatedby the above mechanism.

4.2 Diﬀerence between Positive and Negative Values
We observed that the behavior of the phase waves generated by giving a phase diﬀerence of plus value is dif-

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Fig. 8 Mechanism of simultaneous generation of two pairs of
the phase waves. Example of generated phase-waves by phase
diﬀerence of negative value (Φin = −68◦ ).

ferent from that generated by giving a phase diﬀerenceof minus value as shown in Fig. 5.
If the phase waves are generated by the phase difference of +45◦ , instantaneous frequency of the ﬁrst oscillator (f1 ) changes from the in-phase frequency (fin )toward the anti-phase frequency (fanti ), and instantaneous frequency of the second oscillator (f2 ) changes
from fin toward fanti after the change of f1 . However,
f1 starts to return to fin , because a phase diﬀerence
between the ﬁrst and the second oscillators (|Φ1,2 |) is
relatively small. f2 also starts to return to fin after thechange of f1 . Therefore, a phase of the second oscillator
catches up with a phase of the ﬁrst oscillator.
If the phase waves are generated by the phase
diﬀerence of −45◦ , f1 changes from fin toward fanti .
Therefore a phase of the ﬁrst oscillator is rapidly catching up with a phase of the second oscillator.
4.3 Generation of Two Pairs of Phase Waves
We can observe two types of two pairs of phase waves(see Sect. 3.3):
1. Two pairs of phase waves (Fig. 6(a))
2. A pair of phase waves and a pair of phase-inversion
waves (Fig. 6(b) and Fig. 6(c)).

Fig. 9 Mechanism of simultaneous generation of a pair of thephase waves and a pair of the phase-inversion-waves. Example
of generated phase waves by phase diﬀerence of negative value(Φin = −70◦ ).

4.3.1

Mechanism of Fig. 6(a)Two pairs of phase waves are generated by this mechanism (see Table 2).
Figure 8 shows how the instantaneous frequencies
and the phase diﬀerences change when two pairs ofphase waves are generated.
4.3.2

Mechanism of Fig. 6(b) and Fig. 6(c)The generation mechanism of the ﬁrst phase waves is
the same as in Fig. 6(a). However, the rate of increaseof f2 is larger than that when two pairs of phase waves
are generated, because the minimum value of Φ2,3 by
the ﬁrst pair of phase waves is larger than that when
two pairs of phase waves are generated (around τ = 100
in Fig. 8(b) and Fig. 9(b)). Two pairs of phase waves
are generated by this mechanism (see Table 3).
Because the transition of Φ1,2 is slow, the ﬁrst
pair of phase waves and the second pair of phase waves
can exist separately. The second pair of phase waves
comprises the same waves which are generated whenΦin = 180◦ . Therefore, a pair of phase waves changes
to a pair of phase-inversion waves in a moment. ThisYAMAUCHI et al.: PHASE-WAVES IN A LADDER OF OSCILLATORS

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Table 3time [τ ]Mechanism of Figs. 6(b) and 6(c).
sequence of generated phenomena

When all oscillators are in in-phase synchronization, a negative
phase diﬀerence is given to the ﬁrst oscillator.

around 0

around 20

Because Φ1,2 is not equal to 0, f1 and f2 are larger
than fin . However, f3 is equal to fin , because Φ2,3
and Φ3,4 are equal to 0.
Φ1,2 is attracted to 0 in a moment, because Φ1,2 is
not suﬃciently large to be attracted to anti-phasesynchronization. Therefore, f1 and f2 change towardfin .
Φ2,3 changes from 0, because f2 is higher than fin .
Therefore, f3 starts to change toward fanti .

around 80

ﬁrst waves

second waves

Propagation of
the ﬁrst phasewaves begins.

Φ1,2 changes from decreasing toincreasing, because f2 begins to increase by Φ2,3 which decreases.

around 90

f1 changes toward fanti again.

around 100

The transition of Φ1,2 is very slow,
because f1 changes from decreasing to increasing and f2 changes
from decreasing to increasing.

around 215

f1 increases to the middle of fin
and fanti and decreases to fin
again, because the edge of the array does not become stable in antiphase synchronization. Φ1,2 exceeds 180◦ and reaches 360◦ (inphase synchronization).
Therefore, the second pair of phasewaves is generated.

mechanism is the same mechanism of reﬂection of thephase-inversion wave [9].
Figure 8 shows how the instantaneous frequencies
and the phase diﬀerences change when a pair of phasewaves and a pair of phase-inversion waves are generated.
5.

Fig. 10 Circuit experimental results for N = 9. L0 =1100[mH], L1 = 200[mH], C = 47[nF] and r = 1[kΩ]. (a) The
phase-inversion waves. (b) Disappearing phase waves. (c) Changing from phase waves into phase-inversion waves.

Circuit Experiments

In this section, we conﬁrm the generation of the phaseinversion waves and the phase waves by circuit experiments. The results for N = 9 are shown in Fig. 10.
The corresponding numerically calculated results areshown in Fig. 11. We observed that the circuit experimental results and the numerically calculated results
agree very well. The small diﬀerence between the two
results may come from the resistance in real inductors
or diﬃculty of setting the initial phase diﬀerence in realcircuit experiments.

6.

Conclusions

In this study, phase waves were investigated by numerical calculation for the case of 20 oscillators coupled
as a ladder. The observed phase waves were classiﬁed
into four patterns and the diﬀerences were analyzed indetail. We observed that the behavior of the phasewaves generated by giving a phase diﬀerence of positive value was diﬀerent from the behavior of those generated by giving a phase diﬀerence of negative value.
We were able to observe the generation of two pairs ofphase waves. The propagation mechanism of the phasewaves was analyzed. Furthermore, the mechanism of
the complicated phenomena of the phase waves was

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lytic solutions for nonlinear waves in coupled reacting systems” Physica A: Statistical Mechanics and Its Applications, vol.305, nos.3-4, pp.427–436, March 2002.
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vol.E82-A, no.11, pp.2592–2598, Nov. 1999.
[10] M. Yamauchi, M. Wada, Y. Nishio, and A. Ushida, “Collisions between two phase-inversion-waves in an array ofoscillators,” Proc. ISCAS’00, vol.1, pp.679–682, May 2000.
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770, May 2001.

Fig. 11 Corresponding numerically calculated results for N =9. α = 0.050 and ε = 0.30. (a) The phase-inversion waves. (b)Disappearing phase waves. (c) Changing from phase waves into
phase-inversion waves.

also explained. Finally, for the case of nine oscillators,
we carried out both computer calculations and circuit
experiments. Circuit experimental results agreed wellwith computer calculated results qualitatively.
References[1] L.L. Bonilla, C.J. Pérez Vicente, and R. Spigler, “Timeperiodic phases in populations of nonlinearly coupled oscillators with bimodal frequency distributions,” Physica D:
Nonlinear Phenomena, vol.113, no.1, pp.79–97, Feb. 1998.
[2] J.A. Sherratt, “Invading wave fronts and their oscillatory
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[3] G. Abramson, V.M. Kenkre, and A.R. Bishop, “Ana-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.

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.

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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.