ISCAS 2000 - IEEE International Symposium on Circuits and Systems, May 28-31, 2000, Geneva, Switzerland

COLLISIONS BETWEEN TWO PHASE-INVERSION-WAVESIN AN ARRAY OF OSCILLATORS
Masayuki YAMAUCHI, Masahiro WADA, Y oshifumi NISHIOand Akio USHIDA

Dept. of Electrical and Electronic Engineering, Tokushima Universit y, Japanfmasa,wada,nishio,ushidag@ee.tokushima-u.ac.jp

ABSTRACTIn this study, the phenomena related to collisions
betw een t w o phase-in
version-waves in an array of vander P ol oscillators are in vestigated. Behavior of thetw o phase-inversion-waves after they collide with eachother is classi ed b y computer simulations. F urther,
the mechanisms of the complete extinction and re ection of the two waves b y using the relationship between
phase states and oscillation frequencies are explained.
1. Introduction

A lot of studies on synchronization phenomena of
coupled oscillators ha vebeen carried out up to no w.
Endo et al. have reported a details of theoretical analysis and circuit experiments about some coupled oscillators as a ladder, a ring and a tw o-dimensional array [1][3]. Recently, wave propagation phenomenon observed
from coupled chaotic circuits is also reported [4][5].
How ev er,such studies treat only transient states fora giv en set of initial conditions and there seems to bevery few studies on continuously existing wave propagation phenomenon observed simple coupled oscillators
circuits.
On the other hand, w eha vereported in our past
study [6] wave propagation phenomena of phase statesin van der Pol oscillators coupled by inductors as a ladder. In that study, w efound \phase-inversion-wave"
which is continuously existing wave of changing phasestates between tw oadjacent oscillators from in-phaseto an ti-phase or an ti-phase to in-phase. F urther, w e
explained the mechanisms of the propagation and there ection at the edge of the ladder by using the relationship betw een phase states and oscillation frequency.
In this study, we pay our attentions on the phenomena related to collisions between tw o phase-inversionw aves in the same circuit. We can con rm b y computer simulations that the behavior of the tw o phaseinversion-waves after they collide with eac h other isclassi ed into three types; completely extinction, completely re ection and intermediate complex phenomenon.
F urther, w ecan explain the mechanisms of the completely extinction and the completely re ection of thetwo waves by using the relationship betw een phase statesand oscillation frequency.
2. Circuit Model

Circuit model is shown in Fig. 1. N van der P ol
oscillators are coupled b y coupling inductors L0 . We
carry out computer calculations for the case of N =17. In the computer calculations, we assume the v 0
i characteristics of nonlinear negative resistors in thecircuit by the following functions.
i (v ) = 0g0 v + g3 v3 (g1 ; g3 > 0):
(1)
rkkk

The circuit equations governing the circuit in Fig. 1
are expressed asx_ = y(2)1
3y_ = 0x + (x +1 0 2x + x 01 ) + " y 0 y3kkkkkkkkk(k = 1; 2; 1 1 1; N; x0 = x1 ; x +1 = x )NNwheresrg
pCg1
1t = L1 C ; i 1 =x ;v =3L1 g33g3 y ;r
L1(3)= L ; " = g1 LC1 ; dd = \ 1 ":L kkkk0

It should be noted that corresponds to the coupling
and that " corresponds to the nonlinearity. Equations (2)are calculated by using the fourth-order Runge-Kuttamethod.
L0OSCL0

L0

iL 1 OiL 2 OSCS
C00

L0

iL 3

iL N-1 O

00

SCsubcircuit 1 subcircuit 2 subcircuit 3subcircuit Ni L1 kOSCL1Cvk

Figure 1: Array of van der Pol oscillators.

Figure 2 shows a typical example of observed phaseinversion-waves. In upper diagrams, vertical axes are
sum of voltages of adjacent oscillators and horizontal
axes are time. Hence, the diagrams show ho wphase
di erences betw een adjacent oscillators change as timegoes. White regions in the diagram correspond to thestates that tw o adjacent oscillators are anti-phase synchronization. While, black regions correspond to thein-phase synchronization. In lower gures, snapshotsof attractor of each oscillator and phase states betw een
adjacent oscillators are shown.
In our previous study [6], we have investigated thepropagation and the re ection at the edge of the ladderof the wave.

0-7803-5482-6/99/$10.00 ©2000 IEEE

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must be compared with f and f , which is oscillation frequency observed from the two coupled oscillators, we use the term frequency.
inanti

3.1 Mechanism of complete extinction

1. Let us assume that OSC5  OSC7 are anti-phasesynchronization and that the waves changing froman ti-phase into in-phase are going to reach OSC5and OSC7 at the almost same time from the directions of OSC1 and OSC17 respectively.
2. Reaching the w aves causes the changes of oscillation frequencies of OSC5 f5 and OSC7 f7changes from f to f .
3. Accordingly 85 6 and 86 7 change from  to 0and 2 respectively.
4. Changes of 85 6 and 86 7 cause that the changeof oscillation frequency of OSC6 f6 from f tof .
5. When 85 6 and 86 7 reach 2 and 0 respectively,
f6 become to be equal to f . The wholearra yresults in stable in-phase synchronization.
Waves changing from in-phase to anti-phase can beexplained in a similar manner.
Figure 3 sho ws computer calculated results corresponding to the completely extinction. We can see aw ave extinction occurs as the manner explained abov e.
antiin;

;;

;

anti

Figure 2: T ypical example of phase-inv ersionw a evs.

Remark: Wave propagation phenomenon seems to beobserved for an y number of coupled oscillators. We
could observe similar phenomenon from both of evenand odd numbers of oscillators and from both of small(5) and large (100) numbers.

We can see in Fig. 2 that the tw o phase-inversionw aves completely re ect when they collide with eac hother. Howev er, b y giving di erent initial conditions
w e can also observe that they completely extinct after
the collision (see Fig. 3).
In order to make clear the condition of the generation of the tw ocompletely di erent phenomena, w einvestigate the mechanisms of these phenomena by using the relationship betw een phasestates and oscillation frequency. It has been already known that oscillation frequency of in-phase synchronization of oscillators
coupled by inductors is di erent from that of anti-phasesynchronization. Namely, f , oscillation frequency ofin-phase synchronization, is smaller than f , oscillation frequency of anti-phase synchronization. F urther,
the di erence betw eensf and f increases as coupling inductance increases [7].
Throughout the paper, we de ne the phase di erence betw een t w o adjacent oscillators and the frequencyof OSCk as follo ws:(n) 2 8 +1 (n) =  ((nn))00 (+1n 0 1)
1f (n) =(4)2( (n) 0  (n 0 1))
where  (n) is time when the voltage of OSCk crosses
0[V] at n-th time.
Remark: The f (n) in Eq. (4) should be called zerocrossing-time instead of frequency. However, since f (n)inantikanti

kk;kkkkk;;

in

3.2 Mechanism of complete re ection

3. Collision between Two PhaseInv ersion-Wa v esininkkkk

1. Let us assume that OSC7  OSC10 are in-phasesynchronization and that the waves changing fromin-phase into an ti-phase are going to reach OSC7and OSC10 at the almost same time from the directions of OSC1 and OSC17, respectively.
2. Waves reach OSC7 and OSC10 almost equal timing. 87 8 and 89 10 approach  and 0 by almost
equal timing.
3. Because 87 8 and 89 10 approach  and 0 respectively, oscillation frequencies of OSC8 f8 andOSC9 f9 change from f tow ardf .
4. How ev er,because f8 and f9 change almost simultaneously, in-phase synchronization betw een
OSC8 and OSC9 does not break. Namely, 88 9
remains almost 0. Hence, f8 and f9 do not reachf . Accordingly 87 8 and 89 10 continue tochange until reac hing 2 and 02, respectively.
5. By the e ect of the decreases of (87 8 mod 2)and (89 10 mod 2), f8 and f9 begin to decrease
toward f again.
Waves changing from anti-phase to in-phase can beexplained in a similar manner.
Figure 4 sho ws computer calculated results corresponding to the completely re ection. We can see wave
re ections occur as the manner explained above.
According to the abo ve-explainedmechanisms ofthe completely extinction and the completely re ection, w e can conclude that the tw o phase-inversionw aves colliding each other will;
;

;

;

;

inanti;anti;;;;inI-680

Figure 3: Completely extinction.

Figure 4: Completely re ection.

1. Completely re ect when the w aves reach OSCkand OSCk + 1 at the almost equal timing.
2. Completely extinct when the waves reach OSCk 0
1 and OSCk + 1 at the almost equal timing.

lev el of phase di erence remains. The phase difference grows to  and propagates as a phaseinversion-wave.
4. Figures 5(d)(f): Complex wave propagation phenomenon is observed. V arious waves of di erent
level of phase di erences are produced after thecollision and they in uence each other.
5. Figure 5(g):After the collision, t w o phase-inversionw aves completely re ect. This phenomenon areobserved for the time delay larger than 1:7T [ ].
Figure 6 shows the relationship betw een time delay
and observed phenomena. Because the time that thephase-inversion-wave propagate from one oscillator to
the next is about 5T [ ], the structure of this gurewill repeat as time delay increases. F urther, we can saythat the re ection can be generated more easily thanextinction.

3.3 Between complete extinction and complete
re ection

From the abo veconclusion, our next question iswhat kind of phenomena will be generated when thetw o phase-inversion-waves collide each other with time
di erence.
In order to investigatethe in termediate phenomena, we generate tw o phase-inversion-waves from bothedges of the arra y with time delay for one side. T
is the period of oscillation frequency of the in-phasesynchronization (T = 1=f ).
Obtained computer simulated results from various
values of the time delay are shown in Fig. 5.
1. Figure 5(a): After the collision, tw o phase-inversion4. Conclusions
w aves completely extinct.
2. Figure 5(b): After the collision, small lev el ofIn this study w einvestigated the phenomena rephase di erence remains. The phase di erence
lated to collisions betw eentw ophase-inversion-waves
propagate to one direction. It is atten uatedas
in an array of van der P ol oscillators. We could contime goes and extinct soon. This phenomenon
rm by computer simulations that the behavior of theare observed for the time delay smaller than 0.9
two phase-inversion-waves after they collide with each
T [ ] .
other was classi ed into three types; completely extinction, completely re ection and intermediate complex
3. Figure 5(c): After the collision, a relatively large
inin

in

in

in

in

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Extinction
Others(a) delay=0:00T [ ].

Reflection

in0Τ in 0.9Τin 1.7ΤinTime Delay

8.7Τin 9.6Τin[τ]

Figure 6: Domain of extinction, re ection andother phenomena.

(b) delay=1:50T [ ].
inphenomenon. F urther, w ecould explain the mechanisms of the completely extinction and the completely
re ection of the tw ow aves b y using the relationshipbetw een phase states and oscillation frequency.
1.

(c) dela y=1:00T [ ].
in(d) delay=1:25T [ ].
in(e) dela y=1:50T [ ].
in(f) delay=1:65T [ ].

REFERENCES[1] T. Endo and S. Mori, \Mode Analysis of a Multimode
Ladder Oscillator," IEEE T rans. Circuits Syst., vol.
23, pp. 100{113, Feb. 1976.
[2] T. Endo and S. Mori, \Mode Analysis of TwoDimensional Low-Pass Multimode Oscillator," IEEETr
ans. Circuits Syst., vol. 23, pp. 517{530, Sep. 1976.
[3] T. Endo and S. Mori, \Mode Analysis of a Ringof a Large Number of Mutually Coupled van der Pol
Oscillators," IEEE Trans. Circuits Syst., vol. 25, pp.
7{18, Jan. 1978.
[4] M. J. Ogorzalek, Z. Galias, A. M. Dabrowski andW. R. Dabrowski, \Chaotic Waves and SpatioT emporal Patterns in Large Arrays of Doubly-Coupled
Ch ua's Circuits,"IEEE Trans. Circuits Syst. I, vol. 42,
pp. 706{714, Oct. 1995.
[5] L. Pivka, \Autow aves and Spatio-Temporal Chaos inCNNs-Part I: A Tutorial," IEEE Trans. Circuits Syst.
I, vol. 42, pp. 638{649, Oct. 1995.
[6] M. Yamauchi, M. Wada, Y. Nishio and A. Ushida,\Wave Propagation Phenomena of Phase States in
Coupled Oscillators," Pr oc. of ECCTD'99, vol. 2,
pp.695{698, Aug. 1999.
[7] T. Suezaki and S. Mori, \Mutual Synchronization ofTwo Oscillators," T rans of IECE, vol. 48, no. 9, pp.
1551{1557, Sep. 1965.

in(g) delay=1:70T [ ].
inFigure 5: Change from complete extinction tocomplete re ection.

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