2004 RISP International Workshop on Nonlinear
Circuit and Signal Processing (NCSP'04)
Hawaii, USA, Mar. 5-7, 2004.

Penetration of Phase-Inversion Waves on Coupled Oscillators
by Inductors as a Cross
Miki Akazawa† , Masayuki Yamauchi†† and Yoshifumi Nishio††Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470, Fax:+81-88-656-7471
Email: akazawa@ee.tokushima-u.ac.jp††Sophia University

1. Introduction
Large number of coupled limit-cycle oscillators are useful as models for a wide variety of systems in natural ﬁelds,
for example, diverse physiological organs including gastrointestinal tracts and axial ﬁber 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 in coupled oscillators in order toclarify mechanisms of generations or in order to control thegenerating-conditions of various phenomena in such naturalsystems. In the ﬁeld of the electrical engineering, a lot ofstudies on synchronization phenomena of coupled van der
Pol oscillators have been carried out up to now.
Recently, we discovered continuously existing wave ofchanging phase states between two adjacent oscillators fromin-phase to anti-phase or from anti-phase to in-phase in coupled van der Pol oscillators by inductors as a ladder [3]-[5].
This phenomenon is observed in steady state. We call thisphenomenon as “phase-inversion wave”. And, the mechanisms of “propagation,” “disappearance,” “reﬂection in themiddle of the array” and “reﬂection at an edge of the array,”which were the basic characters of the phase-inversion waves
were clariﬁed [3].
In this study, four ladders of van der Pol oscillators are coupled as a cross. Each ladder is composed by van der Pol oscillators which are coupled by inductors. The phase-inversion
waves are observed in this system. We investigate various
phenomena of the phase-inversion waves by changing initialvalues and parameters. Especially, we pay our attention topenetration of the phase-inversion waves at the crosspoint.
Firstly, propagation mechanism of a pair of phase-inversion
waves on a ladder is made clear by using the relationship ofphase diﬀerence between adjacent oscillators and instantaneous frequency of each oscillator. And, penetration mechanism of two pairs of phase-inversion waves are made clear.
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
L2 . We carried out computer calculations for the cases of

N = 8. In the computer calculations, we assume the v − icharacteristics of the nonlinear negative resistors in each circuit as the following function.
3ir (vj,k ) = −g1 vj,k + g3 vj,k(g1 , g3 > 0)(1)

The circuit equations governing the circuit in Fig.1 arewritten as:Ladder 3

L

L 22

22

L 22OSCLadder 1

L 22

Ladder 4

Ladder 2(a)L 21
Ladder j

L 21O
SCO
S
C

oscillator j,1

oscillator j,2

L 21

L 21

O
S
CO
S
C

oscillator j,3oscillator j,N(b)O
S
Ci j,kL1

C v j,k(c)Figure 1: Circuit Model. (a) Coupled ladders system. (b) Coupled oscillators as a ladder. (c) van der
Pol oscillator.
[Center Oscillator]ẋc = yc

- 471 -(2)ẏc = −xc + α1 (Mxi,N − M xc ) + ε(yc −i=11 3y )3 c[Edge Oscillators] (j=1 ∼ 4)ẋj,1 = yj,1(3)ẏj,1 = −xj,1 + α1 (xj,2 − xj,1 ) + ε(yj,1 −1 3y )3 j,1All oscillators are in-phase synchronization and phaseinversion waves are generated at the edge of ladder1 andladder2.
Throughout this paper, center oscillator is shown OSCc
and instantaneous frequency of OSCj,k is shown fj,k andphase diﬀerence between OSCj,k and OSCj,k+1 is shownΦj(k,k+1) .
3. A Pair of Phase-Inversion Waves on a Ladder[Middle Oscillators] (j=1 ∼ 4, k=2 ∼ N-1)
ẋj,k = yj,k(4)ẏj,k = −xj,k + α1 (xj,k+1 − 2xj,k + xj,k−1 )1 3+ ε(yj,k − yj,k)3[Adjacent Oscillators of Center Oscillator] (j=1 ∼ 4)ẋj,N = yj,N(5)ẏj,N = −xj,N + α1 (xj,N−1 − xj,N ) + α2 (xc − xN )1 3+ ε(yj,N − yj,N)3
wheret=√
L1 Cτ, ij,k =Cg1xj,k , vj,k =3L1 g3

L1L1, α2 =, ε = g1
α1 =L21
L22

g1yj,k ,3g3

L1 d,= “ · ”.
C dτIt 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 N = 8, α1 = 0.050,
ε = 0.250 and ∆τ = 0.01 and calculate (2)-(5) by using thefourth-order Runge-Kutta method.
Next, we explain the mechanism of the generation of thephase-inversion wave by using the change of the instantaneous oscillation frequencies according to the synchronization
states and phase diﬀerence between adjacent oscillators. It
has been already known that oscillation frequency of in-phasesynchronization 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 of anti-phase synchronization. Further, the diﬀerence betweens fin and fanti increases
as coupling inductance increases [1].
Throughout the paper, we deﬁne the phase diﬀerence between two adjacent oscillators and the instantaneous frequency of OSCj,k as follows:12(τk (n) − τk (n − 1))

Figure 2: Example of a pair of phase-inversion waves.
3.1. Mechanism of propagation
Propagation mechanism of a pair of phase-inversion waves
are explained according to the phase diﬀerences and the instantaneous frequencies. Figures 3 shows phase diﬀerences
and instantaneous frequencies, where Φk,k+1 is phase diﬀerence between OSCk and OSCk+1 and fk is instantaneous
frequency of OSCk .
1. Let us assume that phase-inversion waves are going toreach OSC4 from OSC1 .
2. First phase-inversion wave which changes phase diﬀerence from in-phase synchronization to anti-phase synchronization Φ3,4 changes from 0 to π.
3. As Φ3,4 approaches π, f4 changes from fin to fanti .
4. The change of f4 causes increase of Φ4,5 .
5. Second phase-inversion wave which changes phase difference from anti-phase synchronization to in-phase synchronization reaches OSC3 before Φ3,4 reaches π. Therefore, Φ3,4 starts to change to 0 again.

τk (n) − τk+1 (n)×πΦj(k,k+1) (n) =τk (n) − τk (n − 1)
fj,k (n) =Figure 2 shows an example of phase-inversion waves on aladder which is composed by eight oscillators. Vertical axes
are sum of two voltages of adjacent oscillators and horizontal
axes are time. White regions in the diagram correspond to
the states that sum of voltages of the two oscillators are close
to zero, namely adjacent two oscillators synchronized at antiphase. While, black regions correspond to the states thatsum of voltages of the two oscillators with large amplitude,
namely adjacent two oscillators synchronized at in-phase. In
this ﬁgure, we can see a pair of phase-inversion waves reﬂects
at the both edges of the array and continuously exists.

(6)

where τj,k (n) is time when the voltage of OSCj,k crosses 0[V]at n-th time.

6. f4 starts to change to fin again before f4 reaches fanti .
7. f4 reaches fin after Φ3,4 reaches 0 and OSC1 ∼ OSC4
become stable as in-phase synchronization.

- 472 -Instantaneous Frequency [ 1/τ ]0.174

fantif3
f4f5

0.172
0.170
0.168
0.166

(4)

(3)
0.164
0.162

(7)

0.160

fin
0.158

050100150200250300350Time [ τ ](a) Instantaneous Frequency
20

Phase Difference [ degree ]0Figure 4: Alternating propagation of phase-inversion
waves

-20
-40

(4) (5)-60

fc starts to change quickly from fin to fanti (aroundτ =188.3).

(2)

-80
-100

-140
-160

3. Φ3(8,c) and Φ4(8,c) begin to increase (around τ =220).

Φ(3,4)
Φ(4,5)

-120

050

100

150

200

250

300

4. f1,8 and f2,8 starts to change to fin again by second
phase-inversion waves(around τ =248.6).
350

5. Φ1(8,c) and Φ2(8,c) are not large values by the inﬂuencewhich fc increased quickly (360-308=52[degree] whenaround τ =242.7).

Time [ τ ](b) Phase Diﬀerence

Figure 3: Mechanisum of propagation at the middle ofarray.

6. Because, f1,8 and f2,8 slowly decrease and Φ1(6,7) andΦ2(6,7) are values close to 0◦ , f1,7 and f2,7 slowly decrease. Threfore, Φ1(7,8) and Φ2(7,8) also change toward0◦ slowly(around τ =250).

4. Two Pairs of Phase-Inversion Waves on a Cross

7. Because, Φ1(7,8) and Φ2(7,8) are smaller than −100◦ andfc is larger than f1,8 and f2,8 , Φ1(8,c) and Φ2(8,c) keepincreaseing(around τ =268).

Figure 4 shows two pair of phase-inversion waves that continuously exist on the cross. These phase-inversion waves
continuously exist alterneting ladders.
Two pairs of phase-inversion waves are generated at
OSC1,1 and OSC2,1 and propagate to the oscillator of center.
And two pairs of phase-inversion waves do not reﬂect andpenetrate to other two ladders. The penetration mechanismof two pairs of phase-inversion waves are explained according to the phase diﬀerences and the instantaneous frequencies
too.
4.1. Alternating Propagation Mecanisums
Mechanism of penetration at a cross point are explained
according to the phase diﬀerences and the instantaneous frequencies. Figures 5 shows phase diﬀerences and instantaneous frequencies.
Initial values of Φ1(8,c) , Φ2(8,c) , Φ3(8,c) and Φ4(8,c) are 360◦
instead of 0◦ in Figure 5(c), because the graphs are better tobe understood.
1. Let us assume that ﬁrst phase-inversion waves on ladder1 and ladder2 are going to reach OSC1,8 and OSC2,8 .
2. Φ1(8,c) and Φ2(8,c) begin to change simultaneously. And

8. f1,8 and f2,8 start to increase again(around τ =268).
9. Φ1(8,c) and Φ2(8,c) become mostly in-phase. Φ3(8,c) andΦ4(8,c) are large values. Because, Φ3(8,c) and Φ4(8,c)
are attracted into anti-phase once, fc quickly increasesagain(around τ =268).
10. Φ1(8,c) and Φ2(8,c) start to change toward anti-phase synchronization(around τ =280).
11. Because phase states between the OSCc and adjacent
OSCc do not stabilize in anti-phase with these parameters, f3,8 and f4,8 start to change toward fin again.
Therefore, Φ3(8,c) and Φ4(8,c) keep increasing. Φ3(8,c)
and Φ4(8,c) exceed 540◦ and start to change toward 720◦ ,fc begins to decrease quickly toward fin again(aroundτ =280).
12. Because, between OSC1,8 and OSCc and between
OSC2,8 and OSCc are attracted to anti-phase synchronization, f1,8 and f2,8 change to fanti once and start tochange toward fin again(around τ =290).
13. Because, Φ1(8,c) and Φ2(8,c) reache to 540◦ , decreasing
speed of fc slow. fc starts to decrease toward fin again
because Φ1(8,c) and Φ2(8,c) keep changing toward 720◦ .

- 473 -

15. Φ3(8,c) and Φ4(8,c) are almost 600◦ . The phase diﬀerences are far from 740◦ . Therefore, decreasing speed offc slows once. And, f3,8 and f4,8 begin to change toward
fanti again (around τ =300)
16. f3,8 and f4,8 begin to increase. But, fc is larger thanf3,8 and f4,8 till around 335τ . Therefore, Φ3(8,c) andΦ4(8,c) keep to increase till 720◦ (around τ =310).

0.21

Instantaneous Frequency [ 1/τ ]14. Phase states between OSCc and OSC1,8 and between
OSCc and OSC2,8 become in-phase synchronization andstable because Φ1(8,c) and Φ2(8,c) reach to 720◦ and f1,8and f2,8 reach to fin (around τ =330).

17. Phase states between OSCc and OSC3,8 and OSCc andOSC4,8 become to in-phase syncronizarion. Phase states
between OSCc and OSC1,8 and OSCc and OSC2,8 also
become to in-phase synchronization. fc changes to finquickly.

5. Conclusions
In this study, four ladders of van der Pol oscillators werecoupled by an oscillator and four inductors as a cross. Whentwo pairs of phase-inversion waves were generated at eachedges of two ladders, the phase-inversion waves which alternated in two pairs of ladders were observed. By using the relationship between phase states and instantaneous oscillation
frequencies, we explained the mechanisms of the propagationand the alternating propagation of wave.

0.19

0.18

fanti
0.17

0.16

fin
0.15

0

[4] 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.
[5] 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.

200250300

350400450350400450350400

450f3,7 , f4,7f3,8 , f4,8fc0.200.190.18

fanti
0.170.16

fin0.15050

100

150200250300

Time [ τ ](b) Frequency(ladder3,center)

Phase Difference [ degree ][3] 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.

1500.21900[2] T. Endo and S. Mori, “Mode Analysis of a Multimode
Ladder Oscillator,” IEEE Trans. Circuits and Syst., vol.
23, pp. 100-113, Feb. 1976.

100(a) Frequency(ladder1,center)

References[1] T. Suezaki and S. Mori, “Mutual Synchronization of
Two Oscillators,” Trans of IECE, vol. 48, no. 9, pp.
1551–1557, Sep. 1965.

50

Time [ τ ]Instantaneous Frequency [ 1/τ ]18. f3,8 and f4,8 are also attracted to fin .
As observed above, the phase-inversion wave is penetrated from ladder1,2 to ladder3,4.

f1,7 , f2,7
f1,8 , f2,8fc

0.20

Φ1(6,7) , Φ2(6,7)Φ1(7,8) , Φ2(7,8)Φ1(8,c) , Φ2(8,c)Φ3(6,7) , Φ4(6,7)Φ3(7,8) , Φ4(7,8)Φ3(8,c) , Φ4(8,c)

800
700
600
500
400
300
200
1000-100-200

050100150200250300

Time [ τ ](c) Phase Diﬀerence

Figure 5: Machanisum of alternating propagation ofphase-inversion waves.

- 474 -