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

Phase-Inversion-Waves on Coupled Oscillators by Inductors as a Cross
Miki AKAZAWA, Masayuki YAMAUCHI, Yoshifumi NISHIO and Akio USHIDA
Dept. of Electrical and Electronic Engineering, Tokushima University, Japane-mail {akazawa,masa,nishio,ushida}@ee.tokushima-u.ac.jp

1. Introduction
Many kinds of digital circuits of for various applications
have been designed and developed. These circuits use signals
generated by oscillators. Therefore, oscillators always keep
an important position, and various oscillators still continued
to be designed and put to practical use. Various researches
about systems of coupled oscillators have been carried out
up to now. And, a lot of interesting phenomena have beenreported [1]-[2]. But, discoveries of many new phenomena
still continue. Therefore, coupled oscillators catch interest ofresearchers.
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 arecoupled by an oscillator and four inductors as a cross. Eachladder is composed by van der Pol oscillators which are coupled by inductors. The phase-inversion-waves are observedin this system. We investigate various phenomena of thephase-inversion-waves such as “reﬂection,” “disappearance”
and “penetration” at a coupling point of four ladders, bychanging initial values and parameters. The observed phenomena are classiﬁed.

Ladder 4

L 22

L 22

L 22

Ladder 1

OSCcenter

Ladder 3

L 22
Ladder 2(a)L 21
Ladder j

L 21

OSCO
S
C

oscillator j,1

oscillator j,2

L 21

L 21

OSCOSC

oscillator j,3oscillator j,N(b)i j,kOSCL1C v j,k(c)Figure 1: Circuit Model. (a) Coupled ladders system. (b) Coupled oscillators as a ladder. (c) Van der
Pol oscillator.

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 L21and L22 . We carried out computer calculations for the casesof N = 8. In the computer calculations, we assume the v − icharacteristics of the nonlinear negative resistors in each circuit as the following function.

[Center Oscillator]ẋc = yc(2)Mẏc = −xc + α2 (xi,N − M xc ) + ε(yc −i=1

1 3y )
3 c[Edge Oscillators]
3ir (vj,k ) = −g1 vj,k + g3 vj,k(g1 , g3 > 0)(1)

The circuit equations governing the circuit in Fig. 1 are
written as:ẋj,1 = yj,1ẏj,1 = −xj,1 + α1 (xj,2 − xj,1 ) + ε(yj,1 −(j = 1 ∼ 4)-575-(3)1 3yj,1 )3[Middle Oscillators]

Table 1: Phase-inversion-waves generated at OSC1,1ẋj,k = yj,k(4)domain number
1−(a)ẏj,k = −xj,k + α1 (xj,k+1 − 2xj,k + xj,k−1 )1 3+ε(yj,k − yj,k)31−(b)
1−(c)(j = 1 ∼ 4, k = 2 ∼ N − 1)

α2∼0.0080.084∼0.104
0.111∼0.136
0.168∼

sample ﬁgure
2(a)
2(b)
2(c)

[Adjacent Oscillators of Center Oscillator]ẋj,N = yj,N(5)ẏj,N = −xj,N + α1 (xj,N−1 − xj,N ) + α2 (xc − xj,N )1 3+ε(yj,N − yj,N)3(j = 1 ∼ 4)
wheret=√
L1 Cτ, ij,k =Cg1xj,k , vj,k =3L1 g3

L1L1, α2 =, ε = g1
α1 =L21L22

g1yj,k ,3g3

L1 d,= “ · ”.
C dτ(a)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 N = 8, α1 = 0.050,
ε = 0.250 and ∆τ = 0.01 and calculate (3)-(2) by using thefourth-order Runge-Kutta method.
3. Phase-Inversion-Waves
Value of α2 and oscillator which are generated phaseinversion-waves are changed.
And we observe phaseinversion-waves in this circuit. Further, we ﬁx that a pairof phase-inversion-waves is generated at edge oscillators.
Complex phenomena are observed in domain of α2 which
is not written in tables. These phenomena can not beenclassiﬁed.

(b)3.1. When phase-inversion-waves are generated atOSC1,1
Observed phenomena are classiﬁed into 3 patterns bychanging α2 (see Table 1).
1−(a) The phase-inversion-waves propagate to the oscillatorof center and reﬂect. The phase-inversion-waves do notpenetrate to other ladders and continuously exist.
1−(b) The phase-inversion-waves propagate to the oscillatorof center. And the phase-inversion-waves reﬂect at thecenter and penetrate to other three ladders. The phaseinversion-waves are continuously existing on all ladders.
1−(c) The phase-inversion-waves propagate to the oscillator ofcenter and disappear.

(c)

Figure 2: Phase-inversion-waves generated at OSC1,1 .
(a) α2 = 0.001. (b) α2 = 0.084. (c) α2 = 0.168.

-576-

3.2. When phase-inversion-waves are generated atOSC1,1 and OSC2,1
Observed phenomena are classiﬁed into 4 patterns bychanging α2 (see Table 2).
2−(a) The phase-inversion-waves propagate to the oscillatorof center and reﬂect. The phase-inversion-waves do not
penetrate to other ladders and continuously exist.
2−(b) The phase-inversion-waves propagate to the oscillator ofcenter. And the phase-inversion-waves do not reﬂect andpenetrate to other two ladders. The phase-inversionwaves continuously exist on two ladders.

(a)2−(c) The phase-inversion-waves propagate to the oscillatorof center. And the phase-inversion-waves reﬂect at theoscillator of center and penetrate to other two ladders.
2−(d) The phase-inversion-waves become extinction while thephase-inversion-waves are propagating.

Table 2: Phase-inversion-waves generated at OSC1,1 andOSC2,1domain number
2−(a)
2−(b)
2−(c)
2−(d)

α2
∼0.0080.071∼0.216
0.237∼0.354
0.381∼0.426
0.531∼

sample ﬁgure
3(a)
3(b)
3(c)
3(d)

(b)

3.3. When phase-inversion-waves are generated at
OSC1,1 , OSC2,1 and OSC3,1
Observed phenomena are classiﬁed into 2 patterns bychanging α2 (see Table 3).
3−(a) The phase-inversion-waves propagate to the oscillator ofcenter and reﬂect. And the phase-inversion-waves continuously exist.
3−(b) The phase-inversion-waves propagate to the oscillator ofcenter. And the phase-inversion-waves reﬂect and penetrate to other one ladder.
(c)

Table 3: Phase-inversion-waves generated at OSC1,1 ,OSC2,1 and OSC3,1
domain number
3−(a)
3−(b)

α2
∼0.0080.059∼

sample ﬁgure4(a)4(b)

3.4. When phase-inversion-waves are generated at
all oscillators of edges
Observed phenomena are classiﬁed into 2 patterns bychanging α2 (see Table 4).
4−(a) The phase-inversion-waves propagate to the oscillator ofcenter and reﬂect. And the phase-inversion-waves continuously exist.
4−(b) The phase-inversion-waves become extinction while thephase-inversion-waves are propagating.

(d)Figure 3: Phase-inversion-waves generated at OSC1,1
and OSC2,1 . (a) α2 = 0.001. (b) α2 = 0.080. (c)α2 = 0.237. (d) α2 = 0.531.

-577-(a)

(a)

(b)

(b)

Figure 4: Phase-inversion-waves generated at OSC1,1 , Figure 5: Phase-inversion-waves generated at all edge
OSC2,1 and OSC3,1 . (a) α2 = 0.001. (b) α2 = 0.100.
oscillators. (a) α2 = 0.001. (b) α2 = 0.042.
Table 4: Phase-inversion-waves generated at all edge oscillators
domain number
4−(a)
4−(b)
4−(a)

α2
∼0.0110.014∼0.016
0.042∼0.045
0.046∼

sample ﬁgure5(a)References[1] T. Suezaki and S. Mori, “Mutual Synchronization ofTwo Oscillators,” Trans of IECE, vol. 48, no. 9, pp.
1551–1557, Sep. 1965.

5(b)[2] T. Endo and S. Mori, “Mode Analysis of a MultimodeLadder Oscillator,” IEEE Trans. Circuits and Syst., vol.
23, pp. 100-113, Feb. 1976.

5(a)4. Conclusions
In this study, four ladders of van der Pol oscillators werecoupled by an oscillator and four inductors as a cross. We
investigated various phenomena of the phase-inversion-waves
such as “reﬂection,” “disappearance” and “penetration” at a
coupling point of four ladders, by changing initial values andparameters. When the phase-inversion-waves were generated
at an oscillator of an edge, the observed phenomena wereclassiﬁed into 3 pattern. When the phase-inversion-waves
were generated at each oscillators of two edge, the observed
phenomena were classiﬁed into 4 pattern. When the phaseinversion-waves were generated at each oscillators of threeedge, the observed phenomena were classiﬁed into 2 pattern.
When the phase-inversion-waves were generated at each oscillators of all edge, the observed phenomena were classiﬁedinto 2 pattern.

-578-[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.
[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.