2005 RISP International Workshop on Nonlinear
Circuit and Signal Processing (NCSP'05)
Hawaii, USA, Mar. 4-6, 2005.

Asymmetric Alternating Propagation in Coupled Oscillators
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, nishio}@ee.tokushima-u.ac.jp††Sophia University
Email: yamauchi@ieee.org

1. Introduction
Many kinds of digital circuits for various applications havebeen designed and developed. These circuits use signals generated by oscillators. Therefore, oscillators always keep animportant position, and various oscillators are still continued to be designed and are put to practical use. Various
researches about systems of coupled oscillators have been carried out up to now. And, a lot of interesting phenomena havebeen reported. However, discoveries of many new phenomena
still continue. Therefore, coupled oscillators catch interestsof researchers [1]-[4].
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. Thisphenomenon is observed in steady state. We call this phenomenon as “phase-inversion wave.” And, the mechanismsof “propagation,” “disapearance,” “reﬂection in the middleof the array” and “reﬂection at an edge of the array,” which
are the basic characters of the phase-inversion waves, havebeen clariﬁed [5]. Further, when even-number of oscillator
ladders are coupled as a star and the phase-inversion waves
are generated from the edges of the half numbers of ladders,the phase-inversion waves propagate to the center oscillator,
do not reﬂect and penetrate to other ladders. We call thisphenomenon as “alternating propagation”[6]-[7].
In this study, many ladders which are composed by van
der Pol oscillators are coupled at the center oscillator. The
van der Pol oscillators in each ladder are coupled by inductors. The phase-inversion waves are observed by computer
calculations in this system. We investigate asymmetric alternating propagation of the phase-inversion waves by changing
parameters, the number of generating phase-inversion wavesand the number of the ladders. Further, we investigate thisphenomenon by changing the value of some inductors used in
the coupling between the center oscillator and the ladders.

inductors L21 as Fig. 1(b). For computer calculations, we
assume the v − i characteristics of the nonlinear negative
resistors in all the oscillators as the following function.
ir (v) = −g1 v + g3 v 3(g1 , g3 > 0)(1)

The circuit equations governing the circuit in Fig. 1 are
written as:

L 22

L 22

Ladder 2

Ladder M

L 22
Ladder 1i c L1

C vc(a)L 21
Ladder j

L 21

L 21

L 21

OSCO
SCOSCOSCoscillator j,1oscillator j,2oscillator j,3oscillator j,N(b)OSCi j,kL1C v j,k(c)2. Circuit Model
The circuit model used in this study is shown in Fig. 1. In
each ladder, N van der Pol oscillators are coupled by coupling

Figure 1: Circuit Model. (a) Oscillator ladders coupledat the center oscillator. (b) Coupled oscillators as a ladder. (c) van der Pol oscillator.

- 101 -[Center Oscillator]ẋc = ycẏc = −xc + α1

M


(2)

xi,N − M xc



+ ε yc −1 3yc3





i=1



[Edge Oscillators] (j = 1 ∼ M )ẋj,1 = yj,1(3)ẏj,1 = −xj,1 + α1 (xj,2 − xj,1 ) + ε yj,1 −1 3yj,13

the other two ladders; the 3rd and the 4th ladders. After reﬂecting at the edges of the 3rd and the 4th ladders, the waves
return to the 1st and the 2nd ladders via the penetration at
the center oscillator. When even-number of ladders are coupled and the phase-inversion waves are generated from theedges of the half number of ladders, this phenomenon can beobserved. In other words, when M g, the number of the ladders in which the waves are generated, is equal to M/2, thealternating propagation can be observed. The range of α2 forwhich this phenomenon can be observed is 0.071∼0.216.

[Middle Oscillators] (j = 1 ∼ M , 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,k3[Adjacent Oscillators of Center Oscillator] (j = 1 ∼ M )ẋj,N = yj,N(5)ẏj,N = −xj,N + α1 (xj,N−1 − xj,N ) + α2 (xc − xN )
1 3+ ε yj,N − yj,N3
wheret=√
L1 Cτ, ij,k =vj,k =

Cg1xj,k , ic =3L1 g3

g1yj,k , vc =3g3

L1, ε = g1
α2 =L22Figure 2: Alternating propagation for M = 4. Waves are
generated from OSC1,1 and OSC2,1 .

Cg1xc ,
3L1 g3

4. Asymmetric Alternating
Phase-inversion Waves

L1g1
yc , α1 =,3g3
L21

L1 d,
= “ · ”.
C dτ

It should be noted that α1 and α2 correspond to the couplingof 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.
3. Alternating Propagation of the Phase-Inversion
Waves
Figure 2 shows an alternating propagation of two pairs
phase-inversion waves on four ladders. Vertical axes are sumof two voltages of adjacent oscillators and horizontal axes aretime. White regions in the diagram correspond to the statesthat sum of voltages of the two oscillators are close to zero,
namely adjacent two oscillators synchronized at anti-phase.
While, black regions correspond to the states that sum ofvoltages of the two oscillators with large amplitude, namelyadjacent two oscillators synchronized at in-phase.
In this ﬁgure, we can see that two pairs of phase-inversion
waves in the 1st and the 2nd ladders arrive at the center oscillator, and that the waves do not reﬂect and penetrate toPropagation

oftheIn this study, we investigate the alternating propagation of
the phase-inversion waves for the case of M g=M/2. We callthis type of phenomenon as asymmetric alternating propagation.
4.1. Asymmetric alternating propagation 1
In this section, we observe asymmetric alternating propagation phenomena, when many ladders are coupled. Figure
3 shows an asymmetric alternating propagation of the phaseinversion waves, which are generated from 5 ladders, on nineladders, namely M = 9 and M g = 5. When ﬁve pairs ofphase-inversion waves arrive the center oscillator, the phaseinversion waves do not reﬂect and penetrate to the other fourladders (when around 250[τ ]). The range of α2 at which thisphenomenon is observed are around 0.104∼0.186.
Table 1 shows the number of M g for which symmetricand asymmetric alternating propagations are observed. As
the number of the ladders increases, asymmetric alternating
propagation becomes easier to be observed.
4.2. Asymmetric alternating propagation 2
In this section, the values of some L22 are replaced by L23 .
Namely, the coupling structure becomes asymmetric. The- 102 -Table 1: Number of M g for which alternating propagation
is observed.

Figure 3: Asymmetric alternating propagation for M = 9and M g = 5.
parameter of α3 is newly added for L23 as follows;
α3 =L1.
L23(6)

4.2.1. M=3 and Mg=2
Figure 4 shows an asymmetric alternating propagation observed for M = 3. The coupling parameter between the thirdladder and the center oscillator is changed to α3 . Phaseinversion waves are generated from 2 ladders.
(a) Two pairs of phase-inversion waves propagate to the center oscilator and penetrate to another ladder.
(b) This graph shows the ranges of α2 and α3 generating
asymmetric alternating propagation. The upper line
indicates the maximum value of α3 for each α2 . The
lower line indicates the minimum value of α3 for each
α2 . Namely, this phenomenon can be observed when α3is about 2α2 .
ladder3 (generated from 2 ladders)
0.18

0.14

maximum value of
α3 at each α22α2=α3α3 0.10

minimum value of
α3 at each α20.06

maximum value
minimum value
probable value

0.02
0.02

(a)

0.06

0.10

α20.140.18(b)Figure 4: Asymmetric alternating propagation. (a) M =3 and M g = 2. (b) Range where the phenomenon can beobserved.

- 103 -

M
34
567
8
910
11
12
13
1415
16
1718
1920
2122232425
2627
28
2930
3132
33
34
35
36373839
40···

Symmetric×
2×
3×
4×
5×
6×
7×
8×
9×
10×
11×
12×
13×
14×15×16×
17×18×19×20···

Asymmetric××××××
4,5×5,6×6,76,87,87,98,98 , 10
9 , 109 , 11
9 , 10 , 11 , 1210 , 12
10 , 11 , 12 , 13
11 , 13
11 , 12 , 13 , 14
12 , 14
12 , 13 , 14 , 15
13 , 15
13 , 14 , 15 , 16
13 , 14 , 16 , 17
14 , 15 , 16 , 17
14 , 15 , 17 , 18
15 , 16 , 17 , 18
15 , 16 , 18 , 19
16 , 17 , 18 , 19
16 , 17 , 19 , 20
16 , 17 , 18 , 19 , 20 , 21
17 , 18 , 20 , 21
17 , 18 , 19 , 20 , 21 , 22
18 , 19 , 21 , 22···

4.2.2. M=7 and Mg=4ladder7 (generated from 5 ladders)

Figure 5 shows an asymmetric alternating propagation observed for M = 7. The coupling parameters between theladder 5∼7 and the center oscillator are changed to α3 respectively. Phase-inversion waves are generated from 4 ladders.
(a) Four pairs of phase-inversion waves propagate to thecenter oscillator and penetrate to the other three ladders.
(b) This graph shows the ranges of α2 and α3 generating
asymmetric alternating propagation. The upper line
indicates the maximum value of α3 for each α2 . The
lower line indicates the minimum value of α3 for each
α2 . Namely, this phenomenon can be observed when α3is about 4/3 α2 ,0.22

ladder7 (generated from 4 ladders)maximum value of
α3 at each α2

0.16

maximum value
minimum value
probable value
0.06

(a)

0.10

0.14

α2

0.18

maximum value
minimum value
probable value

0.04
0.04

0.08

α2

0.12

0.16

0.20

(b)
(a)
Figure 6: Asymmetric alternating propagation. (a) M =7 and M g = 5. (b) Range where the phenomenon can beobserved.
As the number of the ladders increases, asymmetric alternating propagation becomes easier to be observed. Further, the asymmetric alternating propagation was able to beobserved by changing the coupling parameters between thecenter oscillator and some ladders.

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

α3 0.140.06

minimum value of
α3 at each α20.08

4α2=3α30.10

5α2=2α3

α3 0.120.18

minimum value of
α3 at each α2maximum value of
α3 at each α20.200.22(b)Figure 5: Asymmetric alternating propagation. (a) M =7 and M g = 4. (b) Range where the phenomenon can beobserved.

4.2.3. M=7 and Mg=5
Figure 6 shows an asymmetric alternating propagation observed for M = 7. The coupling parameters between theladder 6,7 and the center oscillator are changed to α3 respectively. Phase-inversion waves are generated from 5 ladders.
(a) Five pairs of phase-inversion waves propagate to the center oscillator and penetrate to the other two ladders.
(b) This graph shows the ranges of α2 and α3 generating
asymmetric alternating propagation. The upper line
indicates the maximum value of α3 for each α2 . The
lower line indicates the minimum value of α3 for each
α2 . Namely, this phenomenon can be observed when α3is about 5/2 α2 . It should be noted that the range where
this phenomenon can be observed is narrower than therange in Fig. 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.
[3] J. A. Sherratt, “Invading Wave Fronts and Their Oscillatory Wakes are Linked by a Modulated TravelingPhase Resetting Wave,” Physica D: Nonlinear Phenomena, vol. 117, issues 1-4, pp. 145-166, Jun. 1998.
[4] G. Abramson, V. M. Kenkre and A. R. Bishop, “Analytic Solutions for Nonlinear Waves in Coupled Reacting
Systems,” Physica A: Statistical Mechanics and its Applications, vol. 305, issues 3-4, pp. 427-436, Mar. 2002.
[5] 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.
[6] M. Akazawa, M. Yamauchi, Y. Nishio and A. Ushida,“Phase-Inversion-Waves on Coupled Oscillators by Inductors as a Cross,”Proc. NOLTA’02, vol. 2, pp. 575578, Oct. 2002.
[7] M. Akazawa, M. Yamauchi and Y. Nishio, “Penetrationof Phase-Inversion Waves on Coupled Oscillators by Inductors as a Cross,”Proc. NCSP’04, pp. 487-490, Mar.
2004.

5. Conclusions
In this study, many ladders which were composed by vander Pol oscillators were coupled by an oscillator and manyinductors. We investigated asymmetric alternating propagation of the phase-inversion waves by changing parameters,
number of generating phase-inversion waves and the numberof ladders.

- 104 -