REFLECTION AND TRANSMISSION OF PHASE-INVERSION-WAVESIN OSCILLATORS COUPLED BY TWO KINDS INDUCTORS
Masayuki YAMAUCHIy, Yoshifumi NISHIO y,zand Akio USHIDA yyDept. of Electrical and Electronic Engineering, Tokushima University, JapanzDept. of Communication Systems, Swiss Federal Institute of Technology Lausanne, Switzerland
fmasa,nishio,ushidag@ee.tokushima-u.ac.jp
L 00

ABSTRACTIn this study, nonlinear wave phenomena related to reections and transmissions of the phase-inversion-waves aroundthe discontinuity of the system consisting of two parts ofcoupled van der Pol oscillators are investigated. Behavior of
the phase-inversion-waves around a discontinuity in a coupled system consisting of two kinds of arrays of van der Poloscillators classi ed into 8 types by computer simulations.
Further, the mechanisms of the re ection of the phaseinversion-waves at the discontinuity are explained.

OSC

L 00

L 00i L 00 1 i L 00 2i L 00 k-1

OSC

Oscillator 1
Oscillator 2Large number of coupled limit-cycle oscillators are useful as models 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, itis 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 natural systems. In the eld of the electrical engineering, a lotof studies on synchronization phenomena of coupled vander Pol oscillators have been carried out up to now ([1]-[5]and references therein). Recently, we have discovered very
interesting wave propagation phenomena of phase states between two adjacent van der Pol oscillators coupled by inductors as a ladder [6]. 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." Further, we
reported the detail on \phase-wave" in [7], which is propagation of the phase di erence less than 180 degrees between
two adjacent oscillators. The phase-waves exist only in thetransient states unlike the phase-inversion-waves.
In this study, we observe behavior of the phase-inversionwaves around a discontinuity in a coupled system consistingof two kinds of arrays of van der Pol oscillators, namely van
der Pol oscillators coupled by inductors L00 and those coupled by L01 . Further, we explain the mechanisms of there ection of the phase-inversion-waves at the discontinuity.
2. CIRCUIT MODEL

Circuit model is shown in Fig. 1. From the 1st to theN1 th oscillators they are coupled by inductors L00 , whilefrom the N1 th to the N2 th oscillators they are coupled by
L01 . Namely, in this model there is a discontinuity at theN1 th oscillator. At rst, the v 0i characteristics of nonlinear

i L 01 ki L 01 n-2

L 01
OSCiL 01 n-1

OSCOscillator N 2 -1 Oscillator N 2

iL1kL1

vkC

Figure 1: Circuit Model.
i (v ) = 0g1 v + g3 v3 (g1 ; g3 > 0)(1)
The circuit equations governing the circuit in Fig. 1 areexpressed as[1st oscillator]
x_ 1 = y1(2)
y_1 = 0x1 + 0 (x2 0 x1 ) + f (y1 )[2nd  (N1 0 1)th oscillators]
x_ = y(3)y_ = 0x + 0 (x +1 0 2x + x 01 ) + f (y )[N1 th oscillator]
x_ 1 = y 1
y_ 1 = 0x 1 + 0 (x 1 +1 0 x 1 )(4)+ 1 (x 1 01 0 x 1 ) + f (y 1 )[(N1 + 1)th  (N2 0 1)th oscillators]
x_ = y(5)y_ = 0x + 1 (x +1 0 2x + x 01 ) + f (y )[N2 th oscillator]
x_ 2 = y 2(6)y_ 2 = 0x 2 + 1 (x 2 01 0 x 2 ) + f (y 2 )where
1 f (y ) = " y 0 y 3(7)3andrkkkkkkkNNNkkNkk

NNNNkNkkk

NNNkkNkNkkk

NNkr

r
Cg
g1
1t = L1 C ; i 1 =3L1 g3 x ; v = 3g3 y ;
r
L
L
L1 d1
10 = L ; 1 = L ; " = g1 C ; d = \ 1 ":(8)
00
01

p

negative resistors in the circuit are assumed by the following
functions.

III-767
0-7803-6685-9/01/$10.00©2001 IEEE

L 01

Oscillator N1OSC

1. INTRODUCTIONOSCL 01L kkkk

It should be noted that corresponds to the coupling
and that " corresponds to the nonlinearity. Equations (2)(8) are calculated for the cases of (N1 ; N2 ) = (9; 17) and(21; 30) by using the fourth-order Runge-Kutta method.
(a) 1 = 0:0001.

3. REFLECTION AND TRANSMISSION

In this section we observe re ection and transmission ofthe phase-inversion-waves for the case of (N1 ; N2 ) = (9; 17).
We x the parameters " = 0:20 and 0 = 0:050 and 1 isvaried as a control parameter. In the computer calculations, we produced phase-inversion-waves as follows. Almost same initial conditions are given for all oscillators toproduce complete in-phase synchronization, which is one
stable steady states in the system. Next, invert the voltage
and the current of the rst oscillator.
Fig. 2 shows the observed result. In diagrams, verticalaxes are sum of voltages of adjacent oscillators and horizontal axes are time. The observed phenomena are classi ed
into eight types as follows.
(a) around 0.001: The phase-inversion-waves are reected at the discontinuity almost completely.
(b) around 0.0020.018: The phase-inversion-waves arere ected and transmitted. But, the phase-inversionwaves change to the phase-waves after the transmission.
(c) around 0.0190.025: The phase-inversion-waves arere ected and transmitted and do not change to thephase-waves after the transmission.
(d) around 0.0260.043: The phase-inversion-waves arere ected and transmitted. But, the phase-inversionwaves change to the phase-waves after the re ection.
(e) around 0.0440.056: The phase-inversion-waves aretransmitted almost completely. The phase-inversionwaves transmitted completely when 1 is equal to0.050.
(f) around 0.0570.130: The phase-inversion-waves aretransmitted. The phase-waves are generated after thephase-inversion-waves are transmitted.
(g) around 0.1310.173: The phase-inversion-waves arere ected and transmitted. But, the phase-inversionwaves change to the phase-waves after the re ection
and transmission.
(h) around 0.174 : The phase-inversion-waves are reected and transmitted. But, the phase-inversionwaves change to the phase-waves after the transmission.
It should be noted that we could observe di erent transmission phenomena for the cases of L01 >> L00 and L01 <<L00 . Namely, for L01 >> L00 ( 1 << 0:05) phase-inversionwave is not transmitted beyond the discontinuity. While,for L 01 << L00 ( 1 >> 0:05) phase-inversion-wave is transmitted as phase-wave beyond the discontinuity. This interesting di erence will be explained in the Sec. 5.
Figure 3 summarizes the above bifurcation.

(b) 1 = 0:018.

(c) 1 = 0:021.

(d) 1 = 0:027.

(e) 1 = 0:046.

(f) 1 = 0:100.

(g) 1 = 0:160.

(h) 1 = 0:240.
Figure 2: Observation of the re ection and the transmission of the phase-inversion-waves. N1 = 9, N2 = 17,0 = 0:050 and " = 0:20.
III-768

0.05

ab cdef

0.000 0.018 0.043
0.001 0.025 0.056

g0.130h0.173α1Figure 3: Bifurcation diagram. N1 = 9, N2 = 17, 0 =0:050 and " = 0:20.
Phase-di erence vs. time.

4. COMPARISON WITH WAVES INTRANSMISSION LINEThe re ection and transmission observed in our systemseem to be similar to wave phenomena observed in two
transmission lines with di erent characteristic impedances
joined in tandem. However, we found two major di erent
phenomena in our system as follows:
1. After the re ection or the transmission, the phaseinversion-waves often changes their characteristics,
namely we can observe that the phase-inversion-waves
change to phase-waves with phase di erence less than
180 degrees for wide range of control parameter value.
2. The re ection or the transmission may occurs twotimes when the phase-inversion-waves arrives at thediscontinuity. The reason of this phenomena is could
be explained as follows. The oscillator at the discontinuity needs longer time to returns to in-phasesynchronization frequency than time the other oscillators need, after the phase-inversion-waves arrived
at the oscillator. This produces new phase di erence
between the oscillator at the discontinuity and its adjacent oscillators and it results the generation of thesecond phase-wave.
5. MECHANISM OF REFLECTION

2. Oscillation frequency of OSC20 f20 begins to change
from f toward f , because in-phase synchronization between OSC19 and OSC20 breaks by the phaseinversion-wave (see (a) in Fig.4).
3. As oscillation frequency of OSC20 f20 changes fromf to f , phase di erence between OSC20 andOSC21 820 21 approaches 0 (see (b) in Fig.4).
4. As 820 21 approaches 0, oscillation frequency of OSC21
f21 begins to change from f to f(see (c) inFig.4).
5. Change of f21 does not propagate to OSC22 immediately, because L01 is much larger than L00 (see (d)in Fig.4).
6. Hence, f21 cannot reach f and begins returning
to f (see (e) in Fig.4).
7. As f21 returns to f , 820 21 continues to increase
until reaching 02 (see (f) in Fig.4).
8. By the e ect of the decrease of 820 21 , f20 begins
to decreases again from f toward f (see (g) inFig.4).
9. 821 22 becomes , because in spite of constant value
which is f22 , f21 changes from f and returns to fagain (see (h) in Fig.4).
ininIn this section, the re ection mechanism of the phaseinversion-waves at the discontinuity is explained according
to the computer calculated results for the case of (N1 ; N2 ) =(21; 30).
It has been already known that oscillation frequency
of in-phase synchronization of oscillators coupled by inductors is di erent from that of anti-phase synchronization.
Namely, f , oscillation frequency of in-phase synchronization, is smaller than f , oscillation frequency of antiphase synchronization. Further, the di erence between f
and f increases as coupling inductance increases. We
use this characteristic to explain the mechanism of the reection.
5.1 When L01 >> L00 (i.e.
1 << 0 ).
1. Let us assume that OSC20OSC30 are in-phase synchronization and that a phase-inversion-wave fromin-phase to anti-phase reaches at OSC20 from thedirection of OSC1.
in

anti

in

anti

Frequency vs. time.
Figure 4: Re ection at the discontinuity for the case of
1 << 0 . N1 = 21, N2 = 30, 0 = 0:0500, 1 = 0:0001
and " = 0:20.

anti

anti;

;

in

anti

anti

inin;

;

anti

in;

in

III-769inAs L01 approaches L00, it becomes easier that changeof f21 propagates to OSC22. Hence, we could observe thatthe phase-inversion-wave is transmitted beyond the discontinuity as shown in Figs. 3(b)(c)(d). While, the re ection
caused by the fact that the change of f21 is hard to propagate becomes smaller.
5.2 When L01 << L00 (i.e.
1 >> 0 ).

Hence, when L01 is much smaller than L00 , the transmission waves always exist.
As L01 approaches L00 , it becomes more dicult thatphase states of OSC21 and OSC22 hold in-phase synchronization. Hence, the re ection of the phase-inversion-waves
is disappear as shown in Figs. 3(f)(g).
6. CONCLUSIONSIn this study, we observed behavior of the phase-inversionwaves around the discontinuity of the system consistingof two parts of coupled van der Pol oscillators. By computer calculations observed phenomena were classi ed into
8 types. Further, we could explain the mechanism of there ection of the phase-inversion-wave at the discontinuity
and made clear that di erent mechanisms re ect the waves
fro the cases of L01 > L00 and L01 < L00.
7.

Phase-di erence vs. time.

Frequency vs. time.
Figure 5: Re ection at the discontinuity for the case of1 >> 0 . N1 = 21, N2 = 30, 0 = 0:00500, 1 = 0:20000
and " = 0:20.
1. Let us assume that OSC20OSC30 are in-phase synchronization and that a wave from in-phase to antiphase reaches at OSC20 from the direction of OSC1.
2. Oscillation frequency of OSC20 f20 begins to change
from f toward f , because in-phase synchronization between OSC19 and OSC20 breaks by the phaseinversion-wave (see (a) in Fig.5).
3. As oscillation frequency of OSC20 f20 changes fromf to f820 21 approaches 0 (see (b) in Fig.5).
4. As 820 21 approaches 0 , oscillation frequency of OSC21
f21 changes from f to f(see (c) in Fig.5).
5. Change of f21 propagates to OSC22 immediately, because L01 is much smaller than L00 (see (d) in Fig.4).
6. Hence, in-phase synchronization between OSC21 andOSC22 is hard to break. (see (e) in Fig.5). And f21
and f22 return to f together (see (f) in Fig.5).
7. As f21 returns to f , 820 21 continues to increase
until reaching 02 (see (g) in Fig.5).
ininanti

anti;;

inanti[1] D. A. Linkens \Analytical Solution of Large Numbersof Mutually Coupled Nearly Sinusoidal Oscillators"
IEEE Trans. Circuits Syst., vol. 21, no. 2, pp. 294-300,
Mar. 1974.
[2] T. Endo and S. Mori \Mode Analysis of a Multimode
Ladder Oscillator" IEEE Trans. Circuits Syst., vol. 23,
no. 2, pp. 100-113, Feb. 1976.
[3] T. Endo and S. Mori \Mode Analysis of TwoDimensional Low-Pass Multimode Oscillator" IEEETrans. Circuits Syst., vol. 23, no. 9, pp. 517-530, Sep.
1976.
[4] T. Endo and S. Mori \Mode Analysis of a Ring of aLarge Number of Mutually Coupled van der Pol Oscillators" IEEE Trans. Circuits Syst., vol. 25, no. 1, pp.
7-18, Jan. 1978.
[5] S. Moro, Y. Nishio and S. Mori \Synchronization Phenomena in Oscillators Coupled by One Resistor" IEICE Trans. Fundamentals, vol. E78-A, no. 2, pp. 244253, Feb. 1995.
[6] 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. 25922598, Nov. 1999.
[7] 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.

inin

REFERENCES;III-770