International Symposium on Nonlinear Theory and its Applications
Xi’an, PRC, October 7–11, 2002

Generation Mechanisms of Two Pairs of Waves in a Ladder of Oscillators
Masayuki YAMAUCHI, Yoshifumi NISHIO and Akio USHIDA
Dept. of Electrical and Electronic Engineering, Tokushima University, Japane-mail {masa,nishio,ushida}@ee.tokushima-u.ac.jp

1. Introduction
Large number of coupled limit-cycle oscillators is useful
as models for a wide variety of systems in natural ﬁelds, forexample, diverse physiological organs including gastrointestinal tracts and axial ﬁber of nervous systems, convecting ﬂuids, arrays of Josephson junctions and so on. Hence, it isvery important to analyze synchronization and the related
phenomena observed in coupled oscillators in order to clarify mechanisms of the generations or in order to control thegenerating-conditions of various phenomena in such naturalsystems [1]. In the ﬁeld of the electrical engineering, a lotof studies on synchronization phenomena of coupled van der
Pol oscillators have been carried out up to now [2],[3].
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 byinductors [4]-[6]. In the study, we named the continuously
existing wave of changing phase states between two adjacentoscillators from in-phase to anti-phase or from anti-phase toin-phase as “phase-inversion-wave” and explained the generation mechanism by using the relation between the instantaneous oscillation frequencies and the phase diﬀerences of two
adjacent oscillators.
In this study, a propagating wave of phase diﬀerence between two adjacent oscillators is observed. It is called “phasewave.” The phase-waves are observed in transient states andchange to the phase-inversion-waves or disappear. In thecomputer calculations, we produce phase-waves as follows.
Almost same initial conditions are given for all oscillators toproduce complete in-phase synchronization which is one ofstable steady states in the system. After the system settles
in the complete in-phase synchronization, the voltage and thecurrent with arbitrary phase diﬀerence are input to the ﬁrst
oscillator. The observed phase-waves in coupled 20 oscillators
are classiﬁed to four patterns and the diﬀerence is analyzedin detail. We observe that the behavior of the phase-waves
generated by giving phase diﬀerence of plus value is diﬀerent from those generated by giving phase diﬀerence of minus
value. We can also observe the generation of two pairs of thephase-waves. The mechanisms of the complicated phenomena on the phase-waves are also explained.

L0O
SCi L 01

OSCL0L0i L 02

i L 0 N-2Oscillator 1 Oscillator 2L0OSCiL 0 N-1

Oscillator N -1

OSC

Oscillator N

Figure 1: Coupled van der Pol oscillators as a ladder.
L0 . We carried out computer calculations for the cases of
N = 20. In the computer calculations, we assume the v −i characteristics of the nonlinear negative resistors in eachcircuit as 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)ẏ1 = −x1 + α(x2 − x1 ) + ε y1 −1 3y13

[Middle Oscillators]ẋk = yk(3)

ẏk = −xk + α(xk+1 − 2xk + xk−1 ) + ε yk −

1 3yk
3



(k = 2 ∼ N−1)
[Last Oscillator]ẋN = yN(4)

2. Circuit Model
The circuit model used in this study is shown in Fig. 1.
N van der Pol oscillators are coupled by coupling inductors-579-ẏN = −xN + α(xN−1 − xN ) + ε yN −1 3y3 Nwheret=√
L1 Cτ , iL1 k =L1, ε = g1
α=L0Cg1xk , vk =3L1 g3dL1,=“·” .
Cdτg1yk ,3g3(5)

It should be noted that α corresponds to the coupling of
the oscillators and ε corresponds to the nonlinearity of theoscillators. Throughout the paper, we ﬁx α = 0.050 and ε =0.30 and calculate (2)-(4) by using the fourth-order RungeKutta method.

(a)3. Phase Wave
3.1. 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 phasewaves disappear at the edge of the array (see Fig. 2(a)).

(b)(b) One time reflection+65◦ ≤ Φin ≤ +70◦ : The phase-waves propagate andreﬂect 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 edgeof the array (see Fig. 2(b)).
(c)(c) Two times reflection
Φin ∼= +71◦ : The phase-waves propagate and reﬂect
two times at the edges of the array. The phase diﬀerence
decreases while the phase-waves are propagating. Thephase-waves disappear at the edge of the array (see Fig.
2(c)).
(d) Change to phase-inversion-waves+72◦ ≤ Φin ≤ +180◦ and −68◦ ≤ Φin ≤ −180◦ :The phase diﬀerence increases while the phase-waves
are propagating. The phase-waves change to the phaseinversion-waves when the phase-waves reﬂect at the edgeof the array or in the middle of the array (see Fig. 2(d)).

(d)

Figure 2: Phase-waves. α = 0.050 and ε = 0.30. (a)Φin = +60◦ . (b) Φin = +68◦ . (c) Φin = +71◦ . (d)◦Figure 3 shows domain of the observed phenomena. We may Φin = +80 .
be able to observe a phenomenon that the phase-waves reﬂect
three times or more and disappear by changing parameter.
When N is very large, we can observe a phenomenon thatthe phase-waves disappear in the middle of the array.

(b) (c)(a)064

3.2. Two Pairs of Phase-Waves
Two pairs of waves (two pairs of phase-waves or a pair ofphase-waves and a pair of phase-inversion-waves) are generated in the domain marked with diagonal line of Fig. 3 (seeFig. 4). For Φin = −68◦ , two pairs of phase-waves are generated as shown in Fig. 4(a). For Φin = −70◦ , a pair of phasewaves and a pair of phase-inversion-waves are generated asFig. 4(b). For Φin = −69◦ , two pairs of phase-inversionwaves exist as Fig. 4(c). But, when waves are generated, a(d)
70

72

(a)
0

Φin

180(d)
6783

Φin180Figure 3: Domain of observed phenomena generated by
the phase diﬀerence of plus or minus value. α = 0.050
and ε = 0.30.

-580-Instantaneous Frequency [ 1/τ ]

(a)

f1
f2
f3
f4
f50.176
0.174
fanti
0.172
0.170
0.168
0.166
0.164
f1

f2

f3

100200f4

f50.162
0.160
fin
0.158
0300

Time [ τ ]400500600(a) Frequency.
Phase Difference [ degree ]

80(b)

Φ1,2
Φ2,3
Φ3,4
Φ4,56040
Φ1,220

Φ2,3

Φ3,4

Φ4,5

Second Waves(Phase-Waves)

0-20-40First Waves(Phase-Waves)

-60

(c)

0100200300

Time [ τ ]400500600(b) Phase-Diﬀerence.
Figure 4: Two pairs of phase-waves. α = 0.050 and ε =◦◦◦0.30. (a) Φin = −68 . (b) Φin = −70 . (c) Φin = −69 . Figure 5: Mechanism of simultaneously generated twopairs of the phase-waves. Example of generated phasewaves by phase diﬀerence of minus value (Φin = −68◦ ).

pair of phase-inversion-waves are phase-waves and theychange to phase-inversion-waves. Therefore, Fig. 4(c) has
the same mechanism as Fig. 4(b) when waves are generated.
Further, two pairs of waves can be generated when waves
reﬂect at the ﬁrst time. This phenomenon is observed whenphase diﬀerence at the reﬂection exists in the domain marked
with diagonal line in Fig. 3. Therefore, the mechanism of thisphenomenon is the same as the mechanism of generating two
pairs of waves.

(3) Φ2,3 changes from 0, because f2 is higher than fin .
Therefore, f3 starts to change toward fanti .
<ﬁrst phase-waves>

3.3. Generation of Two Pairs of Phase-Waves(4) Propagating of the ﬁrst phase-waves begin.
<secound phase-waves>

We can observe two types of two pairs of phase-waves (seesection 3.3);
1. Two pairs of phase-waves (Fig. 4(a))
2. A pair of phase-waves and a pair of phase-inversion-waves(Figs. 4(b) and (c)).
3.3.1. Mechanism of Figure 5 (a)When all oscillators are in in-phase synchronization, phase
diﬀerence of minus is given to the ﬁrst oscillator.
(1) Because Φ1,2 is not equal to 0, f1 and f2 is larger than
fin . But, f3 is equal to fin , because Φ2,3 and Φ3,4 areequal to 0.

(2) Φ1,2 is attracted to 0 in a moment, because Φ1,2 is notlarge enough to be attracted to anti-phase synchronization. Therefore, f1 and f2 change toward fin .

(4) Φ2,3 is decreasing during f2 > f3 .
(5) f2 starts to change from decreasing to fin to increasing
to fanti again.
(6) f1 changes toward fin and becomes nearly equal to f2 ,because transition of f2 is very slow by the above reason.
Therefore, Φ1,2 keeps almost same value during that theﬁrst pair of phase-waves propagate.
(7) f2 starts to change to increase to fin again. f1 andf2 decrease to fin together, because Φ1,2 is not large
enough to be attracted to anti-phase synchronization.
Therefore, f1 , f2 and Φ1,2 decrease slowly. The second
pair of phase-waves generates at this time.
(8) Φ2,3 changes toward 0, because f2 < f3 .

-581-

(9) f3 starts to change toward fin .
(10) Rate of decrease of Φ2,3 changes to small value, because
f3 becomes nearly equal to f2 which decrease slowly.
The second pair of phase-waves propagate by the above
mechanism.
Two pairs of the phase-waves generate by this mechanism.
Figure 7 shows how the instantaneous frequencies and thephase diﬀerences change when two pairs of the phase-waves
generate.

Instantaneous Frequency [ 1/τ ]3.3.2. Mechanism of Figures 5 (b) and (c)

f8

0.174
fanti

f6

f7

0.172
0.170

f4
f2

f3

0.162
0.160
fin
0.158
0

100

200

300

Time [ τ ]

400

500

600

(a) Frequency.
400Phase Difference [ degree ](4) The second pair of phase-waves is same waves which
are generated when Φin = 180◦ . Therefore, a pair ofphase-waves changes to a pair of phase-inversion-waves
in a moment. This mechanism is same mechanism of
reﬂection of the phase-inversion-wave [4].

In this study, the phase-waves were investigated by numerical calculation for the case of coupled 20 oscillators as a ladder. The observed phase-waves were classiﬁed to 4 patterns
and the diﬀerence was analyzed in detail. We observed thatthe behavior of the phase-waves generated by giving phase
diﬀerence of plus value was diﬀerent from those generated
by giving phase diﬀerence of minus value. We were able toobserve the generation of two pairs of the phase-waves. Further, the mechanism of the complicated phenomena on thephase-waves were explained.

f5
0.166

f1

(3) f1 increases to middle of fin and fanti and decrease to
fin again, because edge of the array does not become
stable in anti-phase synchronization. Φ1,2 exceeds 180◦
and reaches 360◦ (in-phase synchronization).

4. Conclusions

0.168

0.164

(2) f1 changes toward fanti .

Figure 8 shows how the instantaneous frequencies and thephase diﬀerences changes when a pair of the phase-waves anda pair of phase-inversion waves generate.

f1
f2
f3
f4
f5
f6
f7
f80.176(1) Φ1,2 changes from decrease to increase, because f2 begins to increase quickly by Φ2,3 which decreases.

References

Φ1,2
Φ2,3
Φ3,4
Φ4,5
Φ5,6
Φ6,7
Φ7,8

300

200
Φ1,2

[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, issues 1, pp.
79–97, Feb. 1998.

100[2] T. Suezaki and S. Mori, “Mutual Synchronization ofTwo Oscillators,” Trans of IECE, vol. 48, no. 9, pp.
1551–1557, Sep. 1965.

0
Φ2,3-100-200

Φ3,4First Waves(Phase-Waves)

Φ4,5
Φ5,60

100

200

Φ6,7300

Time [ τ ][3] T. Endo and S. Mori, “Mode Analysis of a Multimode
Ladder Oscillator,” IEEE Trans. Circuits and Syst., vol.
23, pp. 100-113, Feb. 1976.

Second Waves

Φ7,8 (Phase-Inversion-Waves)
400500600(b) Phase-Diﬀerence.

Figure 6: Mechanism of simultaneously generated a pairof the phase-waves and a pair of the phase-inversionwaves. Example of generated phase-waves by phase difference of minus value (Φin = −70◦).
Mechanism of the ﬁrst phase-waves is the same as Fig.
4(a). But, rate of increase of f2 is larger than that when
two pairs of phase-waves generate, because minimum valueof Φ2,3 by the ﬁrst pair of phase-waves is larger than thatwhen two pairs of phase-waves generate (around τ = 100 inFigs. 5(b) and 6(b)). Mechanism of the second waves is asfollows:-582-[4] M. Yamauchi, M. Wada, Y. Nishio and A. Ushida,“Wave Propagation Phenomena of Phase States in Oscillators Coupled by Inductors as a Ladder,” IEICE
Trans. Fundamentals, vol.E82-A, no.11, pp.2592-2598,
Nov. 1999.
[5] M. Yamauchi, M. Wada, Y.Nishio and A. Ushida, “Collisions between Two Phase-Inversion-Waves in an Arrayof Oscillators,” Proc. ISCAS’00, vol.1, pp.679-682, May
2000.
[6] M. Yamauchi, Y. Nishio and A. Ushida, “Reﬂectionand Transmission of Phase-Inversion-Waves in Oscillators Coupled by Two Kinds Inductors,” Proc. ISCAS’01,vol.3, pp.767-770, May 2001.