ANALYSIS OF PHASE-WAVES IN COUPLED OSCILLATORS AS A LADDER
Masayuki YAMAUCHI, Yoshifumi NISHIO and Akio USHIDA

Dept. of Electrical and Electronic Engineering, Tokushima University, Japane-mailfmasa,nishio,ushidag@ee.tokushima-u.ac.jp

ABSTRACTIn this study, wave propagation phenomena of phase di erences observed in van der Pol oscillators coupled by inductors as a ladder are investigated. It is called \phase-wave."
We observe that the behavior of the phase-waves generated
by giving phase di erence of plus value is di erent fromthose generated by giving phase di erence of minus value.
We analyze the di erence in detail. Further,we analyze thedi erence between generated phase-waves by plus and minus value.

system settles in the complete in-phase synchronization, thevoltage and the current of arbitrary phase di erence is input in the rst oscillator. We observe that the behavior of
the phase-waves generated by giving phase di erence of plus
value is di erent from that generated by giving phase difference of minus value. We analyze the di erence in detail.
Further, we analyze the reason of the di erences between
observed phenomena by giving phase di erence of plus andminus value.
2.

1.

INTRODUCTION

Large number of coupled limit-cycle oscillators are useful asmodels for a wide variety of systems in natural elds, for example, diverse physiological organs including gastrointestinal tracts and axial ber of nervous systems, convecting
uids, 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 tothe clarify mechanisms of the generations or in order to control the generating-conditions of various phenomena in suchnatural systems. In the eld of the electrical engineering,a lot of studies on synchronization phenomena of coupled
van der Pol oscillators have been carried out up to now.
Recently, we have discovered very interesting wave propagation phenomena of phase states between two adjacentoscillators in an array of van der Pol oscillators coupled byinductors [1]-[3]. In the study, we named the continuously
existing wave of changing phase states between two adjacent oscillators from in-phase to anti-phase or from antiphase to in-phase as \phase-inversion-wave" and explained
their generation mechanisms by using the relations between
the oscillation instantaneous frequencies and the phase differences of two adjacent oscillators.
We observed \phase-wave," which is propagation of thephase di erence between two adjacent oscillators [4]. The
phase-waves exist only in the transient states unlike thephase-inversion-waves. But, these study only observed thephase-waves by inputting phase di erence between the rstoscillator and the other oscillators of plus value. In thisstudy, we investigate the phase-waves in more detail again.
In the computer calculations, we produce phase-waves asfollows. Almost same initial conditions are given for alloscillators to produce complete in-phase synchronization,
which is one stable steady states in the system. After the0-7803-7448-7/02/$17.00 ©2002 IEEE

CIRCUIT MODEL

The circuit model used in this study is shown in Fig. 1. N
van der Pol oscillators are coupled by coupling inductors
L0 . We carried out computer calculations for the cases ofN = 17. Although we introduce the results only for 17 oscillators, the similar phenomena are observed from both ofeven and odd number oscillators. In the computer calculations, we assume the v 0 i characteristics of the nonlinear
negative resistors in the each circuit as the following function.

ir (vk ) = 0g1 vk + g3 vk3 (g1 ; g3 > 0)(1)

The circuit equations governing the circuit in Fig. 1 arewritten as[First Oscillator]

x_ 1 = y1(2)

y_1 = 0x1 + (x2 0 x1 ) + " y1 0 1 y133[Middle Oscillators]x_ k = yk(3)
y_k = 0xk + (xk+1 0 2xk + xk01 ) + " yk 0 1 yk3
3(k = 2  N01)
[Last Oscillator]

V - 317x_ N = yN



y_N = 0xN + (xN 01 0 xN ) + " yN 0 1 yN33

(4)Figure 1: Coupled van der Pol oscillators as a ladder.
wherept = L1 C ; i 1 =r Cg1

r L 3Ld1g3

L kx ;v =kkr g1(i) Input phase di erence +60[deg](domain (a)).

3g3 y ;k(5)= LL1 ; " = g1 C1 ; d = \ 1 " :0
It should be noted that corresponds to the coupling of
the oscillators and " corresponds to the nonlinearity of theoscillators. In this study, we calculate (2)-(4) by using thefourth-order Runge-Kutta method.
3. PHASE-WAVES
3.1. Phase di erence

In this section, wave propagation phenomenon observed
from the circuit with 17 oscillators is investigated.
Throughout the paper, we x = 0:050 and " = 0:30
and we de ne the phase di erence between two adjacentoscillators as follows:(n) 2 
8 +1 (n) =  ((nn))00 (+1n 0 1)
where  (n) is time when the voltage of OSCk crosses 0[V]at n-th time.
Figure 2 shows typical examples of observed phase-waves
which are propagation of the phase-di erence between adjacent oscillators. These results are obtained for the same
parameter values by changing initial conditions as follows:
1. Setting the initial conditions of all oscillators as thesame.
2. Putting the arbitrary phase di erence of the voltage
and the current to one oscillator.
In the upper 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 the phase
di erences between adjacent oscillators change as time goes.
In the lower gures, the snapshots of the attractor of each
oscillator and the phase states between adjacent oscillators
are shown.
kkkk(ii) Input phase di erence +70[deg](domain (b)).

k;kk

V - 318(iii) Input phase di erence +135[deg](domain (c)).
Figure 2: Phase-waves. = 0:05; " = 0:3.

plus value.

minus value.
Figure 3: Domain of observed phenomena that are gerated
by the phase di erence of plus or minus value.

(iv) input phase di erence +175[deg](domain (d)).
Figure 2: (continued).

3.2. Classi cation
When the phase-waves are generated by inputting the phase
di erence of plus value or minus value at an edge of theladder, observed phenomena are classi ed into four patterns
by the input phase di erence as follows:(a) Between 0 and +69 and between 0 and 067 : Thephase di erence decreases while the phase-waves arepropagating. The phase-waves become extinct (1) atthe edge of the array without re ection or (2) in the
middle of the array ahter one re ection(see Fig.2(i)).

(b) Around +70 : The phase di erence increases while
the phase-waves are propagating. After the phasewaves re ect at the end of the array, they become extinct at the rst oscillator. Because the phase-waves
do not have enough phase di erences to become antiphase at the edge of the array (see Fig.2(ii)).
(c) Between +71 and +171 and around 068 : Thephase di erence increases while the phase-waves arepropagating. The phase-waves re ect at the edge ofthe array and change to the phase-inversion-waves,because the phase-waves have the phase di erences
to become anti-phase at the edge of the array (seeFig.2(iii)).

Figure 4: Input phase di erence 070[deg](domain markedwith diagonal line of Fig.3).

Figure 4 shows an example of generated two pairs ofwaves. This complicated phenomena will be discussed infuture.

3.3. Mechanism of propagation
Let us assume that all oscillators are in in-phase synchronization. The phase di erence of arbitrary plus value inputs
into the rst oscillator.

(d) Between +172 and +180 and between 069 and0180 : The phase-waves change to the phase-inversionwaves while the phase-waves are propagating (see Fig.2(iv)).
Figure 3 shows domain of the observed phenomena.
Furthermore, we found that two pairs of waves are generated in the domain marked with diagonal line of Fig.3.

V - 319(1) Therefore, instantaneous frequencies of the rst andthe second oscillators(f1 and f2 ) become high frequency from frequency of in-phase synchronization(fin ).
(The frequency of anti-phase synchronization is higher
than frequency of in-phase synchronization.)3.4. Cause of the di erences between phenomena
by phase di erence of plus value and minus value

We observed that the behavior of the phase-waves generated
by giving phase di erence of plus value is di erent fromthat generated by giving phase di erence of minus value.
We analyze the di erence in detail.
When the phase-waves are generated by minus value,
we can observe the phenomena same as those by the plusvalue in the domain between 0 and 058 . But, we can notobserve the phenomena same as those by the plus value inthe domain between 059 and 0180 .
When the phase-waves are generated by the phase difference of +70 , the phase of the second oscillator has tochange 70 to catch up with the phase of rst oscillator.
While, when the phase-waves are generated by the phase
di erence of 070 , the phase of the second oscillator has tochange 290 to catch up with the phase of rst oscillator.
Therefore, it takes long time that the rst and the second
oscillator become in-phase synchronization.

(a) Frequency.

4. CONCLUSION(b) Phase-Di erence.
Figure 5: Mechanism of propagation of phase-waves byphase di erence of plus value(+60 ).

In this study, the phase-waves were investigated by numerical calculation. We found that the behavior of the phasewaves generated by giving phase di erence of plus value isdi erent from that generated by giving phase di erence ofminus value. We analyze the di erence in detail. Whenthe phase-waves are generated by giving phase di erenceof minus value, we found the phenomena that phase-wavesof two sets generate. Further, we analyzed the di erences
between generated the phase-waves by the phase di erenceof plus and minus value.
5. REFERENCES(2) The phase di erence between the second and the thirdoscillators(82 3 ) changes from 0 toward 0, because
f2 is higher than f3 .
(3) f3 changes from f toward frequency of anti-phasesynchronization(f ).
(4) f1 becomes f , because the 1 2 is small value for becoming instantaneous anti-phase synchronization between the rst and the second oscillator.
(5) Therefore, f1 and f2 change toward f .
(6) The 82 3 changes toward 0, because f2 is smaller than
f3 .
(7) f3 becomes f .
;inanti

in;

in;

in

When value of propagating phase di erence is in theregion attracting to anti-phase synchronization, the phasewaves change toward the phase-inversion-waves. When valueof propagating phase di erence is not in the region attracting to anti-phase synchronization, the phase-waves change
toward extinction.

[1] 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.
[2] 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.
[3] M.Yamauchi, Y.Nishio and A.Ushida, \Re ection andTransmission of Phase-Inversion-Waves in Oscillators
Coupled by Two Kinds Inductors,"
Proc. ISCAS'01, vol.3, pp.767-770, May 2001.
[4] M.Yamauchi, M.Wada, Y.Nishio and A.Ushida, \Analysis of Phase Di erence Propagation in Oscillators
Coupled by Inductors as a Ladder," Proc. NOLTA'99,vol.2, pp.783-786, Nov. 1999.

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